PHYSICAL CHEMISTRY 1

1.The properties of gases

1.1 Introduction: observable properties of gases

Throughout much of human history, “airs” or gases were not believed to be matter at all; their apparently weightless nature and their ability to move about freely and fill all available space, while carrying with them definite physical properties such as odor and sometimes color, conferred upon them a somewhat mysterious nature. Even the scientist Robert Boyle wrote about “The Strange Subtility, Great Efficacy and Determinate Nature of Effluviums

".It's interesting, however, that around 550 BCE the Greek philospher Anaximenes maintained that all matter consists of air: "It is from air that all the things that exist , have existed, or will exist comeinto being."

The invention of the sensitive balance in the early seventeenth century showed once and for all that gases have weight and are therefore matter. Guericke's invention of the air pump (which led directly to his discovery of the vacuum) launched the “pneumatic era" of chemistry long before the existence of atoms and molecules had been accepted. Indeed, the behavior of gases was soon to prove an invaluable tool in the development of the atomic theory of matter.

The study of gases allows us to understand the behavior of matter at its simplest: individual particles, acting as individuals, almost completely uncomplicated by interactions and interferences between each other. Later on, our knowledge of gases will serve as the pathway to our understanding of the far more complicated condensed phases (liquids and solids) in which the theory of gases will no longer give us correct answers, but it will still provide us with a useful model that will at least help us to rationalize the behavior of these more complicated states of matter.

Let us start with what we can observe experimentally about gases.

•First, we know that a gas has no definite volume or shape a gas will fill whatever volume is available to it. Contrast this to the behavior of a liquid, which always has a distinct upper surface when its volume is less than that of the space it occupies.

•The other outstanding characteristic of gases is their low densities compared with those of liquids and solids. One mole of liquid water at 298 K and 1 atm pressure occupies a volume of 18.8cm3 whereas the same quantity of water vapor at the same temperature and pressure has a volume of 30200 cm3 more than 1000 times greater.

•The most remarkable property of gases, however, is that to a very good approximation,they all behave the same wayin response to changes in temperature and pressure, expanding or contractingby predictable amounts. This is very different from the behavior of liquids or  solids, in which the properties of each particular substance must be determined individually.

`           We will see later that each of these three macroscopic characteristics of gases follows directly from the microscopic view— that is, from the atomic nature of matter.

The pressure of a gas

The molecules of a gas, being in continuous motion, frequently strike the inner walls of their container. As they do so, they immediately bounce off without loss of kinetic energy, but the reversal of direction (acceleration) imparts a force to the container walls. This force, divided by the total surface area on which it acts, is the pressure of the gas.

The pressure of a gas is observed by measuring the pressure that must be applied externally in order to keep the gas from expanding or contracting. To visualize this, imagine some gas trapped in a cylinder having one end enclosed by a freely moving piston. In order to keep the gas in the container, a certain amount of weight (more precisely, a forcef) must be placed on the piston so as to exactly balance the force exerted by the gas on the bottom of the piston, and tending to push it up. The pressure of the gas is simply the quotient f/A where A is the cross-section area of the piston.

Pressure units

The unit of pressure in the SI system is the pascal (Pa), defined as a force of one newton per square metre (1 Nm–2= 1 kg m–1–2).

At the Earth's surface, the force of gravity acting on a 1 kg mass is 9.81 N. Thus if, in the aboveFigure, the weight is 1 kg and the surface area of the piston is 1 M, the pressure of the gas would be 9.81 Pa. A 1-gram weight acting on a piston of 1 cm2 cross-section would exert a pressure of 98.1 pA. (If you wonder why the pressure is higher in the second example, consider the number of cm2 contained in 1 m2)

In chemistry, it is more common to express pressures in units of atmospheres or torr:

1 atm = 101325 Pa = 760 torr.

The older unit millimetre of mercury (mm Hg) is almost the same as the torr; it is defined as

one mm of level difference in a mercury barometer at 0°C. In meteorology, the pressure unit

most commonly used is the bar:

1 bar = 106 N m–2= 0.987 atm.

In engineering work the pound per square inch is still widely used; standard atmospheric pressure is 14.7 psi.

How is pressure measured?

Atmospheric pressure and the barometer The column of air above us exerts a force on each 1-cm2of surface equivalent to a weight of about 1034 g.

This figure is obtained by solving Newton's law f = ma for m using the acceleration of gravity for a:

                                           

In the early 17th century the Italian EVANGELISTA

TORRICELLI  invented a device to measure this pressure.

The barometerconsists of a vertical glass

tube closed at the top and evacuated, and open at

the bottom, where it is immersed in a dish of a liquid.

The atmospheric pressure acting on this liquid will  force it up into the evacuated tube until the weight of the liquid column exactly balances the atmospheric  pressure. If the liquid is mercury, the height supported will be about 760 cm; this height corresponds to standard atmospheric pressure.                     

              

The density of water is only 1/13.6 that of mercury, so standard atmospheric pressure would support a water column whose height is 13.6 x 76 cm = 1034 cm, or 10.3 m. You would have to read a water barometer from a fourth-story window!

Torricelli's invention overturned the then-common belief that air (and by extension, all

gases) are weightless.

The fact that we live at the bottom of a sea of air was most spectacularly demonstrated in 1654, when two teams of eight horses were unable to pull apart two 14-inch copper hemispheres (the "Magdeburg hemispheres") which had been joined together and then evacuated with Guericke's newly-invented vacuum pump.        

The manometer

A modification of the barometer, the U-tube manometer

provides a simple device for measuring the pressure of

any gas in a container. The U-tube is partially filled with

mercury, one end is connected to container, while the

other end is left open to the atmosphere. The pressure

inside the container is found from the difference in height

between the mercury in the two sides of the U-tube.

The temperature of a gas

If two odies are at different temperatures, heat will flow from the warmer to the cooler one until their temperatures are the same. This is the principle on which thermometry is based; the temperature of an object is measured indirectly by placing a calibrated device known as a thermometer in contact with it. When thermal equilibrium is obtained, the temperature of the thermometer is the same as the temperature of the object.

Temperature scales

A thermometer makes use of some temperature-dependent quantity, such as the density of a liquid, to allow the temperature to be found indirectly through some easily measured quantity such as the length of a mercury column. The resulting scale of temperature is entirely arbitrary; it is defined by locating its zero point, and the size of the degree unit.

At one point in the 18th century, 35 different temperature scales were in use! The Celsius temperature scale locates the zero point at the freezing temperature of water; the Celsius degree (C °) 1 is defined as 1/100 of the difference between the freezing and boiling temperatures of water at 1 atm pressure.

The older Fahrenheit scale placed the zero point at the coldest temperature it was possible to obtain at the time (by mixing salt and ice.) The 100° point was set with body temperature (later found to be 98.6°F.) On this scale, water freezes at 32°F and boils at 212°F. The Fahrenheit scale is a finer one than the Celsius scale; there are 180 Fahrenheit degrees in the same temperature interval that contains 100 Celsius degrees, so 1F° = 9/5 C . Since the zero points are also different by 32F, conversion between temperatures expressed on the two scales requires the addition or subtraction of this offset, as well as multiplication by the ratio of the degree size.

You should be able to derive the formula for this conversion.

Absolute temperature

In 1787 the French mathematician and physicist JACQUES CHARLES

discovered that for each Celsius degree that the temperature of a gas is lowered, the volume of the gas will diminish by 1/273 of its volume at 0°C. The obvious implication of this is that if the temperature could be reduced to –273°C, the volume of the gas would contract to zero. Of course, all real gases condense to liquids before this happens, but at sufficiently low pressures their volumes are linear functions of the temperature (Charles' Law), and extrapolation of a plot of volume as a function of temperature predicts zero volume at -273°C. This temperature, known as absolute zero, corresponds to the total absence of thermal energy.

The temperature scale on which the zero point is –273.15°C was suggested by LORD KELVIN,

and is usually known as the Kelvin scale. Since the sizes of the Kelvin and Celsius degrees

are the same, conversion between the two scales is a simple matter of adding or subtracting

273.15; thus room temperature, 20°, is about 293 K.

1. Notice that temperature is expressed by placing the degree symbol in front of the scale

abbreviation (37°C), whereas a temperature interval is written with the degree sign following

the sumbol (2 C°).

A rather fine point to note: the degree symbol is not used with the "K", which should always be separated from the preceding number by a space. See here for an explanation.

Because the Kelvin scale is based on an absolute, rather than on an arbitrary zero of temperature, it plays a special significance in scientific calculations; most fundamental physical relations involving temperature are expressed mathematically in terms of absolute temperature. In engineering work, an absolute scale based on the Fahrenheit degree is sometimes used; this is known as the Rankine scale.

http://www.chem1.com/acad/pdf/c1gas.pdf

2. The gases laws

The "pneumatic" era of chemistry began with the discovery of the vacuum around 1650 which clearly established that gases are a form of matter. The ease with which gases could be studied soon led to the discovery of numerous emprical (experimental) laws that proved fundamental to the later development of chemistry and led indirectly to the atomic view of matter.

2.1 Basic the gases ideal laws

 Pressure-volume relations: Boyle's law

ROBERT BOYLE (1627-91) showed that the volume of air trapped by a liquid in the closed short limb of a J-shaped tube decreased in exact proportion to the pressure produced by theliquid in the long part of the tube. The trapped air acted much like a spring, exerting a force opposing its compression. Boyle called this effect “the spring of the air", and published his results in a pamphlet of that title.

The effect can be seen in a simple J-shaped tube in which air is trapped in the short limb asmercury is poured into the right side. The difference between the heights of the two mercurycolumns gives the pressure (76 cm = 1 atm), and the volume of the air is calculated from thelength of the air column and the tubing diameter. In Boyle's experiment he used a simple airpump invented by his friend Robert Hooke.n Boyle's law can be expressed as

PV = constant

P1V1 = P2V2

These relations hold true only if the number of molecules n and the temperature are constant. This is a relation of inverse proportionality; any change in the pressure is exactly compensated by an opposing change in the volume. As the pressure decreases toward zero, the volume will increase without limit. Conversely, as the pressure is increased, the volume

decreases, but can never reach zero. There will be a separate P-V plot for each temperature;

a single P-V plot is therefore called an isotherm.

Shown here are some isotherms for one mole of an ideal gas at several different temperatures. Each plot has the shape of a hyperbola— the locus of all points having the property x y = a, where a is a constant. You will see later how the value of this constant (PV=25 for the 300K isotherm shown here) is determined.

It is very important that you understand this kind of plot which governs any relationship of inverse proportionality. You should be able to sketch out such a plot when given the value of any one (x,y)pair.

 

A related type of plot with which you

 Should be familiar shows the product

 PV as a function of the pressure. You  should understand why this yields a straight line, and how this set of plots relates to the one immediately above.

How the temperature affects the volume: Charles’ law

All matter expands when heated, but gases are special in that their degree of expansion is

independent of their composition. The French scientists JACQUES CHARLES (1746-1823) and

JOSEPH GAY-LUSSAC (1778-1850) independently found that if the pressure is held constant, the volume of any gas changes by the same fractional amount (1/273 of its value) for each C° change in temperatu

A graphical expression of the law of Charles and Gay-Lussac can be seen in these plots of the volume of one mole of an ideal gas as a function of its temperature at various constant pressures.

• What do these plots show? The straight-line plots show that the ratio V/T (and thus dV/dT) is a constant at any given pressure. Thus we can express the law algebraically as

V/T = constant or V1/T1 = V2/T2. (Don’t memorize this formula!)

• What is the significance of the extrapolation to zero volume? If a gas contracts by 1/273 of its volume for each degree of cooling, it should contract to zero volume at a temperature of –273°C. This, of course, is the absolute zero of temperature, and this extrapolation of Charles' law is the first evidence of the special significance of this temperature.

• Why do the plots for different pressures have different slopes? The lower the pressure, the greater the volume (Boyle's law), so at low pressures the fraction (V/273) will have a larger value. You might say that the gas must "contract faster" to reach zero volume when its starting volume is larger.

 

Volume and the number of molecules: Avogadro’s law

Gay-Lussac noticed that when two gases react, they do so in volume ratios that can always be expressed as small whole numbers. Thus when hydrogen burns in oxygen, the volume of hydrogen consumed is always exactly twice the volume of oxygen. The Italian scientist AMADEO AVOGADRO (1776-1856) drew the crucial conclusion: these volume ratios must be related to the relative numbers of molecules that react, and thus the famous "E.V.E.N principle"

:

Avogadro's law thus predicts a directly proportional relation between the number of moles of a gas and its volume. This relationship, originally known as Avogadro's Hypothesis, was crucial in establishing the formulas of simple molecules at a time (around 1811) when the distinction between atoms and molecules was not clearly understood. In particular, the existence of diatomic molecules of elements such as H2, O2, and Cl2 was not recognized until the results of combining-volume experiments such as those depicted below could be interpreted in terms of the E.V.E.N. principle.

Once it was shown that equal volumes of hydrogen and oxygen do not combine in the manner

depicted in (1), it became clear that these elements exist as diatomic molecules and that the formula of water must be H2O rather than HO as  thought.

Molar volume of a gas: standard temperature and pressure

You will recall that the molar mass of a pure substance is the mass of 6.021023 (Avogadro's number) of particles or molecular units of that substance. Molar masses are commonly expressed in units of grams per mole (g mol–1) and are often referred to as molecular weights. As was explained in the preceding lesson, equal volumes of gases, measured at the same temperature and pressure, contain equal numbers of molecules (this is the "EVEN" principle, more formally known as Avogadro's law.)

 Dalton's law of partial pressures

The ideal gas equation of state applies to mixtures just as to pure gases. It was in fact with a gas mixture, ordinary air, that Boyle, Gay-Lussac and Charles did their early experiments. The only new concept we need in order to deal with gas mixtures is the partial pressure, a concept invented by the famous English chemist JOHN DALTON (1766-1844). Dalton reasoned that the low density and high compressibility of gases indicates that they consist mostly of empty space; from this it follows that when two or more different gases occupy the same volume, they behave entirely independently.

 

The contribution that each component of a gaseous mixture makes to the total pressure of the gas is known as the partial pressure of that gas. Dalton himself stated this law in the simple and vivid way quoted above. The usual way of stating Dalton's Law of Partial Pressures is

This is expressed algebraically as       

or, equivalently,                                  

                                                            

2.2 General equation of ideal gas

The ideal gas law is the equation of state of a hypothetical ideal gas. It is a good approximation to the behavior of many gases under many conditions, although it has several limitations. It was first stated by Émile Clapeyron in 1834 as a combination of Boyle's law and Charles's law.^[1] It can also be derived from kinetic theory, as was achieved (apparently independently) by August Krönig in 1856 and Rudolf Clausius in 1857.

The state of an amount of gas is determined by its pressure, volume, and temperature. The modern form of the equation is:

where p is the absolute pressure of the gas; V is the volume; n is the amount of substance; R is the Regnault constant, better known as universal gas constant; and T is the absolute temperature.

In SI units, p is measured in pascals; V in cubic metres; n in moles; and T in kelvin. R has the value 8.314472 J·K⁻¹·mol⁻¹ in SI units^[4]).

The temperature given in the equation of state must be an absolute temperature that begins at absolute zero. In the metric system of units, we must specify the temperature in Kelvin (degree increments as in Celsius). In the Imperial system, absolute temperature is in Rankine (degree increments as in Fahrenheit).

http://en.wikipedia.org/wiki/Ideal_gas_law

An ideal gas is defined as one in which all collisions between atoms or molecules are perfectly eleastic and in which there are no intermolecular attractive forces. One can visualize it as a collection of perfectly hard spheres which collide but which otherwise do not interact with each other. In such a gas, all the internal energy is in the form of kinetic energy and any change in internal energy is accompanied by a change in temperature.

An ideal gas can be characterized by three state variables: absolute pressure (P), volume (V), and absolute temperature (T). The relationship between them may be deduced from kinetic theory and is called the

• n = number of moles
• R = universal gas constant = 8.3145 J/mol K
• N = number of molecules
• k = Boltzmann constant = 1.38066 x 10^-23 J/K = 8.617385 x 10^-5 eV/K
• k = R/N_A
• N_A = Avogadro's number = 6.0221 x 10²³ /mol

The ideal gas law can be viewed as arising from the kinetic pressure of gas molecules colliding with the walls of a container in accordance with Newton's laws. But there is also a statistical element in the determination of the average kinetic energy of those molecules. The temperature is taken to be proportional to this average kinetic energy; this invokes the idea of kinetic temperature. One mole of an ideal gas at STP occupies 22.4 liters.

Molecular Constants

In the kinetic theory of gases, there are certain constants which constrain the ceaseless molecular activity.

A given volume V of any ideal gas will have the same number of molecules. The mass of the gas will then be proportional to the molecular mass. A convenient standard quantity is the mole, the mass of gas in grams equal to the molecular mass in amu. Avogadro's number is the number of molecules in a mole of any molecular substance. The average translational kinetic energy of any kind of molecule in an ideal gas is given by

State Variables

A state variable is a precisely measurable physical property which characterizes the state of a system, independently of how the system was brought to that state. It must be inherently single-valued to characterize a state. For example in the heat-work example, the final state is characterized by a specific temperature (a state variable) regardless of whether it was brought to that state by heating, or by having work done on it, or both.

Common examples of state variables are the pressure P, volume V, and temperature T. In the ideal gas law, the state of n moles of gas is precisely determined by these three state variables. If a property, e.g., enthalpy H, is defined as a combination of other state variables, then it too is a state variable. Enthalpy is one of the four "thermodynamic potentials", and the other three, internal energy U, Helmholtz free energy F and Gibbs free energy G are also state variables. The entropy S is also a state variable.

Some texts just use the term "thermodynamic variable" instead of the description "state variable".

The Mole

A mole (abbreviated mol) of a pure substance is a mass of the material in grams that is numerically equal to the molecular mass in atomic mass units (amu). A mole of any material will contain Avogadro's number of molecules. For example, carbon has an atomic mass of exactly 12.0 atomic mass units -- a mole of carbon is therefore 12 grams. For an isotope of a pure element, the mass number A is approximately equal to the mass in amu. The accurate masses of pure elements with their normal isotopic concentrations can be obtained from the periodic table.

One mole of an ideal gas will occupy a volume of 22.4 liters at STP (Standard Temperature and Pressure, 0°C and one atmosphere pressure).

Avogadro's number

Standard Temperature and Pressure

STP is used widely as a standard reference point for expression of the properties and processes of ideal gases. The standard temperature is the freezing point of water and the standard pressure is one standard atmosphere. These can be quantified as follows:

Standard temperature: 0°C = 273.15 K

Standard pressure = 1 atmosphere = 760 mmHg = 101.3 kPa Gauge Pressure

Does the flat tire on your automobile have zero air pressure? If it is completely flat, it still has the atmospheric pressure air in it. To be sure, it has zero useful pressure in it, and your tire gauge would read zero pounds per square inch. Most gauges read the excess of pressure over atmospheric pressure and this excess is called "gauge pressure". While a useful measurement for many practical purposes, it must be converted to absolute pressure for applications like the ideal gas law.

Since a partial vacuum will be below atmospheric pressure, the phrase "negative pressure" is often used. Certainly there is no such thing as a negative absolute pressure, but small decreases in pressure are commonly used to entrain fluids in sprayers, in carburetors for automobiles, and many other applications. In the case of respiration, we say that the lungs produce a negative pressure of about -4 mmHg to take in air, which of course means a 4 mmHg decrease from the surrounding atmospheric pressure.

When a system is at atmospheric pressure like the left image above, the gauge pressure is said to be zero. In this image, the system has been opened so that it is at equilibrium with the atmosphere. In the right image, the system has been closed and the plunger pushed down until the pressure reads about 15 lb/in². This implies that the absolute pressure has been approximately doubled by compressing the gas to half its volume (ideal gas law). Standard atmospheric pressure in these U.S. common units is 14.7 lb/in², so this must be added to the gauge pressure above to get the absolute pressure.

Standard volume of 1 mole of an ideal gas at STP: 22.4 liters

(http://hyperphysics.phy-astr.gsu.edu/hbase/kinetic/idegas.html)

3.3 Real gases

Real gases do not obey the perfect gas law exactly. Deviations from the law are particularly important at high pressures and low temperatures, especially when a gas is on the point of condensing to liquid.

Molecular interactions

Real gases show deviations from the perfect gas law because molecules interact with each other. Repulsive forces between molecules assist expansion and attractive forces assist compression.

(Atkins, peter.Page:16)

Real Gases fail to obey the Ideal Gas equation of state exactly. Why?

For exactly one mole of an ideal gas:

Plotting the experimentally determined value of (pV/RT) for exactly one mole of various real gases as a function of pressure, p, shows a deviation from ideality(The quantity (pV)/(nRT) = Z is called the COMPRESSIBILITY FACTOR and should be unity for an Ideal Gas):

The deviation from ideal behavior is large at high pressure and low temperatue At lower pressures and high temperatures, the deviation from ideal behavior is typically small, and the ideal gas law can be used to predict behavior with little error.

Deviation from ideal behavior can also be shown for a given gas (Nitrogen, in this example) as a function of temperature:

As temperature is decreased below a critical value, the deviation from ideal gas behavior becomes severe, because the gas CONDENSES to become a LIQUID.

PV = nRT

The van der Waals Equation (A fix to the Ideal Gas Equation)

One of the most useful equations to predict the behavior of real gases was developed by Johannes van der Waals (1837-1923) It takes into account Molecular Stickiness and Molecular Size

The Ideal Gas Equation:

PV= nRT

Has been presented as an Empirical relation, but it doesn't work perfectly for all gases under all conditions because it is based on imperfect assumptions.

Remember the conditions assumed for an Ideal Gas:

1.                  Molecules are perfectly elastic (no STICKINESS)

2.                  Molecules are point masses (no SIZE)

3.                  Molecules move at random

The first two of these assumptions is clearly wrong for all gases, because at low temperatures all gases CONDENSE, or form a liquid phase. This must happen because the molecules stick to one another, at least a little. We can correct the Ideal Gas equation for stickiness.

Moreover, the liquid has measurable molar volume, and this volume is simply the size of the close-packed molecules of the liquid.

At high pressures, and thus high densities, the intermolecular distances can become quite short, and attractive forces between molecules becomes significant. Neighboring molecules exert a relatively long-ranged attractive force on one another, which will reduce the momentum in which they tranfer to the container walls. The observed pressure exerted by the gas under these conditions will be less than that for an Ideal Gas

To put the observed (real) pressure into the Ideal Gas expression, we must correct for the decrease in pressure due to molecular stickiness. For Stickiness to be a factor, the two gas molecules must have a collision. The probability of a collision is the probability of two molecules being in the same place at the same time. The probability of the first molecule being at the place of the collision is proportional to the number density (n/V). The probability of the second one being in the same place is the same, (n/V). Thus the reduction in pressure due to stickiness should be proportional to (n/V)². If the proportionality constant is called a, then the ideal pressure is

Thus, correcting for molecular stickiness alone, the Ideal Gas equation would become:

As pressures and density increase, the volume of the molecules themselves becomes significant relative to the size of the container.

To correct for the effect of finite molecular volume, we must recognize that in the ideal gas equation the volume used is the "free volume" that the molecules find themselves in. The free volume is just the real (container) volume minus the volume that is taken up by the molecules of the gas itself.

where b is a constant representing the volume of a mole of gas molecules at rest. Thus, the ideal gas equation, if corrected for molecular size and stickiness, looks like:

We can rearrange this expression slightly to give the familiar form of the van der Waals equation.

 

Unlike the universal gas constant, R, The van der Waals constants a and b are different for different gases.

Substance	a (L2 atm/mol2)	b (L/mol)
He	0.0341	0.0237
H2	0.244	0.0266
O2	1.36	0.0318
H2O	5.46	0.0305
CCl4	20.4	0.1383

Numerical Example:

Use the van der Waals equation to calculate the pressure of a sample of 1.0000 mol of oxygen gas in a 22.415 L vessel at 0.0000°C (Note: For an ideal gas, this temperature and volume would lead to conditions of STP, i.e. a pressure of exactly 1.0000 atm)

V=22.4L = (0.000 + 273.15) = 273.15K

a(O₂)=1.36L²atm/mol²(FromTable)
b(O₂)=0.0318L/mol(FromTable)
(nRT)/(V-nb)=1.0014atm
-a(n/V)²=-0.0027atm
p = (nRT)/(V-nb) - a (n/V)² = 0.9987 atm

If O₂ was perfectly ideal, the pressure would be 1.0000 atm.

The first term on the right-hand-side of the van der Waals equation, (nRT)/(V-nb), represents the ideal gas pressure corrected for finite molecular volume. In other words the volume is slightly less than 22.415 L due to the mole of O₂ in it. The the molecules collide a bit more frequently with the walls of the container, because they have less room to fly around in in the middle of the container. The value of 1.0014 atm for this term represents this increase in pressure.

The second term in the equation, - a (n/V)², represents the reduction in pressure due to molecular stickiness. This correction to ideal gas behavior dominates the finite volume correction, leading to a compressibility factor, Z=pV/nRT of less than 1 (0.9987, in fact).

http://www.chem.ufl.edu/~itl/2045/lectures/lec_e.html

             3.Laws of thermodynamics

Thermodynamics is the science of energy conversion involving heat and other forms of energy, most notably mechanical work. It studies and interrelates the macroscopic variables, such as temperature, volume and pressure, which describe physical, thermodynamic systems.

Thermodynamics defines four laws which do not depend on the details of the systems under study or how they interact. Hence these laws are generally valid and can be applied to systems about which one knows nothing other than the balance of energy and matter transfer. Examples of such systems include Einstein's prediction of spontaneous emission, and ongoing research into the thermodynamics of black holes.

These four laws are:

• Zeroth law of thermodynamics: If two systems are in thermal equilibrium with a third, they are also in thermal equilibrium with each other.

This statement implies that thermal equilibrium is an equivalence relation on the set of thermodynamic systems under consideration. Systems are said to be in equilibrium if the small, random exchanges between them (eg. Brownian motion) do not lead to a net change in energy. This law is tacitly assumed in every measurement of temperature. Thus, if one seeks to decide if two bodies are at the same temperature, it is not necessary to bring them into contact and measure any changes of their observable properties in time.^[19] The law provides a fundamental definition of temperature and justification for the construction of practical thermometers.

It is interesting to note that the zeroth law was not initially recognized as a law. The need to for the zeroth law was not initially realized, so the first, second, and third laws were explicitly stated and found common acceptance in the physics community first. Once the importance of the zeroth law was realized, it was impracticable to renumber the other laws, hence the zeroth.

• First law of thermodynamics: The internal energy of an isolated system is constant.

The first law of thermodynamics is an expression of the principle of conservation of energy. It states that energy can be transformed (changed from one form to another), but cannot be created or destroyed.^[20]

The first law is usually formulated by saying that the change in the internal energy of a closed thermodynamic system is equal to the difference between the of heat supplied to the system and the amount of work done by the system on its surroundings. It is important to note that internal energy is a state of the system (see Thermodynamic state) whereas heat and work modify the state of the system. In other words, a specific internal energy of a system may be achieved by any combination of heat and work; the manner by which a system achieves a specific internal energy is path independent.

• Second law of thermodynamics: Heat cannot spontaneously flow from a colder location to a hotter location.

The second law of thermodynamics is an expression of the universal principle of decay observable in nature. The second law is an observation of the fact that over time, differences in temperature, pressure, and chemical potential tend to even out in a physical system that is isolated from the outside world. Entropy is a measure of how much this evening-out process has progressed. The entropy of an isolated system which is not in equilibrium will tend to increase over time, approaching a maximum value at equilibrium.

In classical thermodynamics, the second law is a basic postulate applicable to any system involving heat energy transfer; in statistical thermodynamics, the second law is a consequence of the assumed randomness of molecular chaos. There are many versions of the second law, but they all have the same effect, which is to explain the phenomenon of irreversibility in nature.

• Third law of thermodynamics: As a system approaches absolute zero, all processes cease and the entropy of the system approaches a minimum value.

The third law of thermodynamics is a statistical law of nature regarding entropy and the impossibility of reaching absolute zero of temperature. This law provides an absolute reference point for the determination of entropy. The entropy determined relative to this point is the absolute entropy. Alternate definitions are, "the entropy of all systems and of all states of a system is smallest at absolute zero," or equivalently "it is impossible to reach the absolute zero of temperature by any finite number of processes".

Absolute zero, at which all activity would stop if it were possible to happen, is −273.15 °C (degrees Celsius), or −459.67 °F (degrees Fahrenheit) or 0 K (kelvin).

http://en.wikipedia.org/wiki/Thermodynamics

4 The Zeroth Law

The zeroth law is a consequence of thermal equilibrium and allows us to conclude that temperature is a well-defined physical quantity. The zeroth law of thermodynamics states:

“ If a body A and a body B are both in equilibrium with each other; then a body C which is in thermal equilibrium with body B will also be in equilibrium with body Aand the temperature of body C is equal to the temperature of body A. ”

It is the zeroth law, because it preceeds the first and second laws of thermodynamics and is also a tacit assumption in both laws.

We use the zeroth law when we wish to compare the temperatures of two objects, A and B. We can do this by using a thermometer, C and placing it again object A it reaches thermal equilibrium with object A and measure the temperature of A. Placing the thermometer against object B until thermal equilibrium is reached we measure the temperature of object B. If they are the same temperature then they will be in thermal equilibrium with each other.

Figure 1. The Zeroth law of thermodynamics.

The zeroth law of thermodynamics is a generalization principle of thermal equilibrium among bodies, or thermodynamic systems, in contact.

The zeroth law states that if two systems are in thermal equilibrium with a third system, they are also in thermal equilibrium with each other.

Systems are in thermal equilibrium if they do not exchange heat. The law implies that thermal equilibrium between systems is a transitive relation, which affords the definition of an empirical physical parameter, called temperature. The temperatures are equal for all systems in thermal equilibrium. The law permits the construction of a thermometer to measure this property.

Zeroth law as equivalence relation

A system is said to be in thermal equilibrium when it experiences no net change in thermal energy. If A, B, and C are distinct thermodynamic systems, the zeroth law of thermodynamics can be expressed as:^[2]

"If A and C are each in thermodynamic equilibrium with B, A is also in equilibrium with C."

This statement asserts that thermal equilibrium is a Euclidean relation between thermodynamic systems. If we also grant that all thermodynamic systems are in thermal equilibrium with themselves, then thermal equilibrium is also a reflexive relation. Relations that are both reflexive and Euclidean are equivalence relations. One consequence of this reasoning is that thermal equilibrium has a transitive relationship between the temperature T of A, B, and C:

If T (A) = T (B)

and T (B) = T (C)

then T (A) = T (C).

Thermal equilibrium between many systems

Many systems are said to be in equilibrium if the small, random exchanges (due to Brownian motion, for example) between them do not lead to a net change in the total energy summed over all systems. A simple example illustrates why the zeroth law is necessary to complete the equilibrium description.

Consider N systems in adiabatic isolation from the rest of the universe, i.e. no heat exchange is possible outside of these N systems, all of which have a constant volume and composition, and which can only exchange heat with one another.

The combined First and Second Laws relate the fluctuations in total energy, δU, to the temperature of the ith system, and the entropy fluctuation in the ith system, as follows:

.

The adiabatic isolation of the system from the remaining universe requires that the total sum of the entropy fluctuations vanishes, or:

That is, entropy can only be exchanged between the N systems. This constraint can be used to rearrange the expression for the total energy fluctuation and obtain:

where is the temperature of any system j we may choose to single out among the N systems. Finally, equilibrium requires the total fluctuation in energy to vanish, in which case:

which can be thought of as the vanishing of the product of an antisymmetric matrix and a vector of entropy fluctuations . In order for a non-trivial solution to exist,

That is, the determinant of the matrix formed by must vanish for all choices of N. However, according to Jacobi's theorem, the determinant of a NxN antisymmetric matrix is always zero if N is odd, although for N even we find that all of the entries must vanish, , in order to obtain a vanishing determinant. Hence at equilibrium. This non-intuitive result means that an odd number of systems are always in equilibrium regardless of their temperatures and entropy fluctuations, while equality of temperatures is only required between an even number of systems to achieve equilibrium in the presence of entropy fluctuations.

The zeroth law solves this odd vs. even paradox, because it can readily be used to reduce an odd-numbered system to an even number by considering any three of the N systems and eliminating one by application of its principle, and hence reduce the problem to even N which subsequently leads to the same equilibrium condition that we expect in every case, i.e., . The same result applies to fluctuations in any extensive quantity, such as volume (yielding the equal pressure condition), or fluctuations in mass (leading to equality of chemical potentials). Hence the zeroth law has implications for a great deal more than temperature alone. In general, we see that the zeroth law breaks a certain kind of asymmetry present in the First and Second Laws.

Foundation of temperature

Max Planck and others have stated that the zeroth law implies the definition of a temperature function or more informally, that one can construct a thermometer.

In the space of thermodynamic parameters, zones of constant temperature form a surface, that provides a natural order of nearby surfaces. One may therefore construct a global temperature function that provides a continuous ordering of states. The dimensionality of a surface of constant temperature is one less than the number of thermodynamic parameters, thus, for an ideal gas described with three thermodynamic parameters P, V and n, it is a two-dimensional surface.

For example, if two systems of ideal gases are in equilibrium, then P₁V₁/N₁ = P₂V₂/N₂ where P_i is the pressure in the ith system, V_i is the volume, and N_i is the amount (in moles, or simply the number of atoms) of gas.

The surface PV/N = const defines surfaces of equal temperature, and one may label defining T so that PV/N = RT, where R is some constant. These systems can now be used as a thermometer to calibrate other systems.

The zeroth law of thermodynamics was first stated and named by Ralph H. Fowler in 1931.^[3] This law is arguably the most fundamental of the four numbered laws of thermodynamics. It was called the zeroth law because the need to state it explicitly was not understood until after the first, second, and third laws had been named and found common acceptance in the physics community. 

4.First Law of Thermodynamics

Introduction The first law of thermodynamics is the application of the conservation of energy principle to heat and thermodynamic processes:

The first law makes use of the key concepts of internal energy, heat, and system work. It is used extensively in the discussion of heat engines. The standard unit for all these quantities would be the joule, although they are sometimes expressed in calories or BTU.

It is typical for chemistry texts to write the first law as ΔU=Q+W. It is the same law, of course - the thermodynamic expression of the conservation of energy principle. It is just that W is defined as the work done on the system instead of work done by the system. In the context of physics, the common scenario is one of adding heat to a volume of gas and using the expansion of that gas to do work, as in the pushing down of a piston in an internal combustion engine. In the context of chemical reactions and process, it may be more common to deal with situations where work is done on the system rather than by it.

The System and Surroundings

Before we can start with the first law, it is a good idea to be clear on two important in thermodynamics: the system and the surroundings. The system is the region of the universe under study while the surroundings include everything else in the universe except the system

            Figure 1. The system and surroundings

The first law deals with macroscopic properties, work, energy, enthalpy, etc. One of the most fundamental laws of nature is the conservation of energy principle. It simply states that

• during an interaction, energy can change from one form to another but the total amount of energy remains constant. That is, energy cannot be created or destroyed. Or,

• during an interaction between a system and its surroundings, the amount of energy gained by the system must be exactly equal to the amount of energy lost by the surroundings. A rock

falling off a cliff, for example, picks up speed as a result of its potential energy being converted to kinetic energy.

The first law of thermodynamics is simply an expression of the conservation of energy principle, and it asserts that energy is a thermodynamic property.

Energy can cross the boundary of a closed system in two distinct forms: heat and work. It is important to distinguish between these two forms of energy.Therefore, they will be discussed first, to form a sound basis for the development of the first law of thermodynamics.

We can use the principle of conservation of energy to define a function U called the internal energy. When a closed system undergoes a process by which it passes from state A to state B, if the only interaction with its surroundings is in the form of transfer of heat Q to the system, or performance of work W on the system, the change in U will be

ΔU = UB – UA = Q + W                                                       2-1     

Note:

• In Equation 2-1 we have defined W as the work done on the system and Q is added to the system. If we had defined W as work done by the system, Equation 2-1 would become ΔU = Q- W.

• For an isolated system there is no heat or work transferred with the surroundings, thus, by definition W = Q = 0 and therefore ΔU = 0.

• The first law of thermodynamics states that this energy difference ΔU depends

only on the initial and final states, and not on the path followed between them.

Both Q and W have many possible values, depending on exactly how the system

passes from A to B, but Q + W = ΔU is invariable and independent of the path. If this were not true, it would be possible, by passing from A to B along one path and then returning from B to A along another, to obtain a net change in the energy of the closed system in contradiction to the principle of conservation of energy.

• For a differential change, Equation 2-1 becomes

                        dU = dQ +dW                                                 2-2

For a cyclic process, A→B→A, when the system returns to state A, it has the same U, thus

∫dU =0                                                             2-3

Next we will take a look separately at the heat transferred (dQ) and the work (dW) exchanged between the system and the surroundings.

Heat Transfer

Heat is defined as the form of energy that is transferred between two systems

(or a system and its surroundings) by virtue of a temperature difference. That is, an energy interaction is heat only if it takes place because of a temperature difference. Then it follows that there cannot be any heat transfer between two systems that are at the same temperature

 

Heat is energy in transition. It is recognized only as it crosses the boundary of a system. Consider the hot baked potato. The potato contains energy, but this energy is heat transfer only as it passes through the skin of the potato (the system boundary) to reach the air, as shown below.

Once in the surroundings, the transferred heat becomes part of the internal energy of the surroundings. Thus, in thermodynamics, the term heat simply means heat transfer.

• A process during which there is no heat transfer is called an adiabatic process. There are two ways a process can be adiabatic:

o Either the system is well insulated so that only a negligible amount of heat can pass through the boundary, or

o both the system and the surroundings are at the same temperature and therefore there is no driving force (temperature difference) for heat transfer.

• An adiabatic process should not be confused with an isothermal process. Even though there is no heat transfer during an adiabatic process, the energy content and thus the temperature of a system can still be changed by other means such as work.

The amount of heat transferred during the process between two states (states 1

and 2) is denoted by Q12, or just Q. Heat transfer per unit mass of a system is denoted q and is determined from

q = Q/m                                                           2-4

Sometimes it is desirable to know the rate of heat transfer (the amount of heat

transferred per unit time) instead of the total heat transferred over some time interval. The heat transfer rate is denoted Q

, where the overdot stands for the time derivative, or "per unit time." The heat transfer rate Q

has the unit kJ/s,

which is equivalent to kW. When Q varies with time, the amount of heat transfer during a process is determined by integrating Q over the time interval of the process

 

                                                                                                            2-5

When Q remains constant during the process, the relation reduces to

                                                                                   2-6

Heat is a directional (or vector) quantity; the universally accepted sign convention for

heat is as follows: Heat transfer to a system is positive, and heat transfer from a system is negative. That is, any heat transfer that increases the energy of a system is positive, and any heat transfer that decreases the energy of a system is negative.

Modes of Heat transfer

Heat can be transferred in three different ways: conduction,convection, and radiation. A detailed study of these heat transfer modes is given later. Below we will give a brief description of each mode to familiarize yourselves with the basic mechanisms of heat transfer. All modes of heat transfer require the existence of a temperature difference, and all modes of heat transfer are from the high-temperature medium to a lower-temperature one.

Conduction is the transfer of energy from the more energetic particles of asubstance to the adjacent less energetic ones as a result of interactionsbetween the particles. Conduction can take place in solids, liquids, or gases. Ingases and liquids, conduction is due to the collisions of the molecules duringtheir random motion. In solids, it is due to the combination of vibrations of the molecules in a lattice and the energy transport by free electrons. A cold canned drink in a warm room, for example, eventually warms up to the room temperature as a result of heat transfer from the room to the drink through the aluminum can by conduction.

 

It is observed that the rate of heat conduction Q cond through a layer of constant thickness Δx is proportional to the temperature difference ΔT across the layer and the area A normal to the direction of heat transfer, and is inversely proportional to the thickness of the layer. Therefore,

                                                  2-7

where the constant of proportionality k is the thermal conductivity of the material which is a measure of the ability of a material to conduct heat. Materials such as copper and silver that are good electric conductors are also good heat conductors: kcopper = 401 W/(m.K), and therefore have high k values. Materials such as rubber, wood, and styrofoam are poor conductors of heat (kurethane = 0.026), and therefore have low k values. Diamond has a very high thermal conductivity (k = 2300). In the limiting case of Δx→0, the equation above reduces to the differential form

                       

                                                                           2-8

which is known as Fourier's law of heat conduction. It indicates that the rate of heat conduction in a direction is proportional to the temperature gradient in that direction. Heat is conducted in the direction of decreasing temperature, and the temperature gradient becomes negative when temperature decreaseswith increasing x. Therefore, a negative sign is added in Eq. 2-8 to make heat transfer in the positive x direction a positive quantity.

Note: Temperature is a measure of the kinetic energies of the molecules. In a liquid or gas, the kinetic energy of the molecules is due to the random motion of the molecules as well as the vibrational and rotational motions. When two molecules possessing different kinetic energies collide, part of the kinetic energy of the more energetic (higher-temperature) molecule is transferred to the less energetic (lowertemperature) particle, in much the same way as when two elastic balls of the same mass at different velocitie  collide, part of the kinetic energy of the faster ball is transferred to the slower one. In solids, heat conduction is due to two effects: the lattice vibrational waves induced by the vibrational motions of the molecules positioned at relatively fixed positions in a periodic manner called a lattice, and the energy transported via the free flow of electrons in the solid. The thermal conductivity of a solid is obtained by adding the lattice and the electronic components. The thermal conductivity of pure metals is primarily due to the electronic component whereas the thermal conductivity of nonmetals is primarily due to the lattice component. The lattice component of thermal conductivity strongly depends on the way the molecules are arranged. For example, the thermal conductivity of diamond, which is a highly ordered crystalline solid, is much higher than the thermal conductivities of pure metals.

Convection is the mode of energy transfer between a solid surface and the adjacent liquid or gas which is in motion, and it involves the combined effects of conduction and fluid motion. The faster the fluid motion, the greater the convection heat transfer. In the absence of any bulk fluid motion, heat transfer between a solid surface and the adjacent fluid is by pure conduction The presence of bulk motion of the fluid enhances the heat transfer between the solid surface and the fluid, but it also complicates the determination of heat transfer rates.

·      Consider the cooling of a hot block by blowing of cool air over its top surface5-9 shown in figure below. The arrows in the figure indicate the velocity variation of air. Energy is first transferred to the air layer adjacent to the surface of the block by conduction. This energy is then carried away from the surface by convection; that is, by the combined effects of conduction within the air, which is due to random motion of air molecules, and the bulk or macroscopic motion of the air, which removes the heated air near the surface and replaces it by the cooler air.

·      Convection is called forced convection if the fluid is forced to flow in a tube or over a surface by external means such as a fan, pump, or the wind. In contrast, convection is called free (or natural) convection if the fluid motion is caused by buoyancy forces that are induced by density differences due to the variation of temperature in the fluid (see figure below).

Forced convection (left) and natural convection (right) For example, in the absence of a fan, heat transfer from the surface of the hot block will be by natural convection since any motion in the air in this case will be due to the rise of the warmer (and thus lighter) air near the surface and the fall of the cooler (and thus heavier) air to fill its place. Heat transfer between the block and the surrounding air will be by conduction if the temperature difference between the air and the block is not large enough to overcome the resistance of air to move and thus to initiate natural convection currents.

·      Heat transfer processes that involve change of phase of a fluid are also considered to be convection because of the fluid motion induced during the process such as the rise of the vapor bubbles during boiling or the fall of the liquid droplets during condensation.

·      The rate of heat transfer by convection Qconv  is determined from Newton's law of cooling, which is expressed as

Qconv = hA (Ts – Tf)                                      2-9

·      where h is the convection heat transfer coefficient, A is the surface area through which heat transfer takes place, Ts is the surface temperature, and Tf is bulk fluid temperature away from the surface. (At the surface, the fluid temperature equals the surface temperature of the solid.)

Note: The convection heat transfer coefficient h is not a property of the fluid. It is an experimentally determined parameter whose value depends on all the variables that influence convection such as the surface geometry, the nature of fluid motion, the properties of the fluid, and the bulk fluid velocity. Typical values of h, in W /(m2 . K), are 2-25 for the free convection of gases, 50-1000 for the free convection of liquids, 25-250 for the forced convection of gases, 50-20,000 for the forced convection of liquids, and 2500-100,000 for convection in boiling and condensation processes.

Radiation is the energy emitted by matter in the form of electromagnetic waves (or photons) as a result of the changes in the electronic configurations of the atoms or molecules. Unlike conduction and convection, the transfer of energy by radiation does not require the presence of an intervening medium. In fact, energy transfer by radiation is fastest (at the speed of light) and it suffers no attenuation in a vacuum. This is exactly how the energy of the sun reaches the earth.

·         In heat transfer studies, we are interested in thermal radiation, which is the form of radiation emitted by bodies because of their temperature. It differs from other forms of electromagnetic radiation such as X-rays, gamma rays, microwaves, radio waves, and television waves which are not related to temperature. All bodies at a temperature above absolute zero emit thermal radiation.

·         Radiation is a volumetric phenomena, and all solids, liquids, and gases emit, absorb, or transmit radiation to varying degrees. However, radiation is usually considered to be a surface phenomenon for solids that are opaque to thermal radiation such as metals, wood, and rocks since the radiation emitted by the interior regions of such material can never reach the surface, and the radiation incident on such bodies is usually absorbed within a few microns from the surface.

·         The maximum rate of radiation that can be emitted from a surface at an absolute temperature Ts is given by the Stefan-Boltzmann, law as

Qemit.max=σATs4                                         2-10

where A is the surface area and σ = 5.67 x 10-8 W / (m2 . K4) is the Stefan-Boltzmann constant. The idealized surface which emits radiation at this maximum rate is called a blackbody, and the radiation emitted by a blackbody is called blackbody radiation. The radiation emitted by all real surfaces is less than the radiation emitted by a blackbody at the same temperatures and is expressed as

Qemit.max=εσATs                                         4 2-11

where ε is the emissivity of the surface. The property emissivity, whose value is in the range 0 ≤ ε ≤ 1, is a measure of how closely a surface approximates a blackbody for which ε = 1. The human skin has an emissivity of 0.95, and aluminum foil 0.07.

·           Another important radiation property of a surface is its absorptivity, α, which is the fraction of the radiation energy incident on a surface that is absorbed by the surface. Like emissivity, its value is in the range 0 ≤ α ≤ 1. A blackbody absorbs the entire radiation incident on it. That is, a blackbody is a perfect absorber (α = 1) as well as a perfect emitter.

·           In general, both ε and α of a surface depend on the temperature and the wavelength of the radiation. Kirchhoff's law of radiation states that the emissivity and the absorptivity of a surface are equal at the same temperature and wavelength. In most practical applications, the dependence of ε and α on the temperature and wavelength is ignored, and the average absorptivity of a surface is taken to be equal to its average emissivity. The rate at which a surface absorbs radiation (Qabs) is determined from (see figure)

Qabs=αQinc

where Qinc is the rate at which radiation is incident on the surface and α is the absorptivity of the surface. For opaque (nontransparent) surfaces, the portion of  incident radiation that is not absorbed by the surface is reflected back.

·         The difference between the rates of radiation emitted by the surface and the radiation absorbed is the net radiation heat transfer. If the rate of radiation absorption is greater than the rate of radiation emission, the surface is said to be gaining energy by radiation. Otherwise, the surface is said to be losing energy by radiation. In general, the determination of the net rate of heat transfer by radiation between two surfaces is a complicated matter since it depends on the properties of the surfaces, their orientation relative to each other, and the interaction of the medium between the surfaces with radiation. However, in the special case of a relatively small surface of emissivity ε and surface area A at absolute temperature Ts that is completely enclosed by a much larger surface at absolute temperature Tsurr separated by a gas (such as air) that does not interact with radiation (i.e., the amount of radiation emitted, absorbed, or scattered by the medium is negligible), the net rate of radiation heat transfer between these two surfaces is determined from

Qrad =εσA(Ts⁴–T⁴ surr)

 Example 2-1: Consider a person standing in a breezy room at 20°C. Determine the total rate of heat transfer from this person if the exposed surface area and the average outer surface temperature of the person are 1.6 m2 and 29°C, respectively, and the convection heat transfer coefficient is 6 W/(m2.oC).

Solution  1. The heat transfer between the person and the air in the room will be by convection (instead of conduction) since it is conceivable that the air in the vicinity of the skin or clothing will warm up and rise as a result of heat transfer from the body,

initiating natural convection currents. It appears that the experimentally determined value for the rate of convection heat transfer in this case is 6 W per unit surface area (m2) per unit temperature difference (in K or oC) between the person and the air away from the person. Thus, the rate of convection heat transfer from the person to the air in the room is, from Eq. 2-9,

Qconv = hA (Ts – Tf) = 86.4 W

2. The person will also lose heat by radiation to the surrounding wall surfaces. We take the temperature of the surfaces of the walls, ceiling, and the floor to be equal to the air temperature in this case for simplicity, but we recognize that this does not need to be the case. These surfaces may be at a higher or lower temperature than the average temperature of the room air, depending on the outdoor conditions and the structure of the walls. Considering that air does not intervene with radiation and the person is completely enclosed by the surrounding surfaces, the net rate of radiation heat transfer from the person to the surrounding walls, ceiling, and the floor is, from Eq. 2-13

Qrad = εσA(Ts⁴ – T⁴surr) = 81.7 W

Note that we must use absolute temperatures in radiation calculations. Also note that we used the emissivity value for the skin and clothing at room temperature since the emissivity is not expected to change significantly at a slightly higher temperature. Then the rate of total heat transfer from the body is determined by adding thesetwo quantities to be

Qtotal = Qconv + Qrad = 168.1 W

Note: The heat transfer would be much higher if the person were not dressed since the exposed surface temperature would be higher. In the above calculations, heat transfer through the feet to the floor by conduction, which is usually very small, is neglected. Heat transfer from the skin by perspiration, which is the dominant mode of heat transfer in hot environments, is not considered here.

Work

Work, like heat, is an energy interaction between a system and its surroundings. As mentioned earlier, energy can cross the boundary of a closed system in the form of heat or work. Therefore, if the energy crossing the boundary of a closed system is not heat, it must be work.

·         Heat is easy to recognize: Its driving force is a temperature difference between the system and its surroundings. Then we can simply say that an energy interaction which is not caused by a temperature difference between a system and its surroundings is work. More specifically, work is the energy transfer associated with a force acting through a distance. A rising piston, a rotating shaft, and an electric wire crossing the system boundaries are all associated with work interactions.

·         The work done during a process between states 1 and 2 is denoted W12, or simply W. The work done per unit mass of a system is denoted w and is defined as

w = W/m 2-14

•         The work done per unit time is called power and is denoted W The unit of power is kJ/s, or kW.

•         The energy of a system decreases as it does work and increases as work is done on the system.

•         Heat transfer and work are interactions between a system and its surroundings, and there are many similarities between the two:

1)      Both are recognized at the boundaries of the system as they cross them. That is, both heat and work are boundary phenomena.

2)      Systems possess energy, but not heat or work. That is, heat and work are transfer phenomena.

3)      Both are associated with a process, not a state. Unlike properties, heat or work has no meaning at a state.

4)      Both are path functions (i.e., their magnitudes depend on the path followed during a process as well as the end states).

Path functions have inexact differentials designated by the symbol δ. Therefore, a differential amount of heat or work is represented by δ Q or δW, respectively, instead of dQ or dW. Properties, however, are point functions (i.e., they depend on the state only, and not on how a system reaches that state), and they have exact differentials designated by the symbol d. A small change in volume, for example, is represented by dV and the total volume change during a process between states 1 and 2 is

                                                                                          2

1∫dV =V −V1 = ΔV

That is, the volume change during process 1-2 is always the volume at state 2 minus the volume at state 1, regardless of the path followed (see figure below). The total work done during process 1-2, however, is                                                             2

1∫δW  = W12

That is, the total work is obtained by following the process path and adding the differential amounts of work (δW) done along the way. The integral of δW is not W2 – W1 (i.e., the work at state 2 minus work at state 1), which is meaningless since work is not a property and systems do not possess work at a state.

Example 2-4 A well-insulated electric oven is being heated through its heating element. If the entire oven, including the heating element, is taken to be the system, determine whether this is a heat or work interaction.

Solution For this problem, the interior surfaces of the oven form the system boundary. The energy content of the oven obviously increases during this process, as evidenced by a rise in temperature. This energy transfer to the oven is not caused by a temperature difference between the oven and the surrounding air. Instead, it is caused by the electrons crossing the system boundary and thus doing work. Therefore, this is a work interaction.

Example 2-5 Answer the question in Example 2-4 if the system is taken as only the air in the oven without the heating element.

5 -17

Solution This time, the system boundary will include the outer surface of the heating element and will not cut through it. Therefore, no electrons will be crossing the system boundary at any point. Instead, the energy generated in the interior of the heating element will be transferred to the air around it as a result of the temperature difference between the heating element and the air in the oven. Therefore, this is a heat transfer process. For both cases, the amount of energy transfer to the air is the same. These two examples show that the same interaction can be heat or work depending on how the system is selected.

Electrical Work

It was shown above that electrons crossing the system boundary do electrical work on the system. In an electric field, electrons in a wire move under the effect of electromotive forces, doing work. When N coulombs of electrons move through a potential difference V, the electrical work done is

We = VN

Which can also be expressed in the rate form

W e= VI                                                           2-14

where W e is the electrical power and I the electric current. ( e W=VI = I2R =V2/R) In general, both V and I vary with time, and the electrical work done during a time interval Δt is expressed as

If both V and I remain constant during the time interval Δt, this equation will reduce to

We = VIΔt                                                      2-16

Example 2-6 A small tank containing iced water at 0°C is placed in the middle of a large, wellinsulated tank filled with oil. The entire system is initially in thermal equilibrium at 0°C. The electric heater in the oil is now turned on, and 10 kJ of electrical work is done on the oil. After a while, it is noticed that the entire system is again at 0°C, but some ice in the small tank has melted. Considering the oil to be system A and the iced water to be system B, discuss the heat and work interactions for system A. system B, and the combined system (oil and iced water).

Solution  The boundaries of each system are indicated by dashed lines in the figure. Notice that the boundary of system B also forms the inner part of the boundary of system A.

System A: When the heater is turned on, electrons cross the outer boundary of system A, doing electrical work. This work is done on the system, and therefore WA = 10 kJ. Because of this added energy, the temperature of the oil will rise, creating a temperature gradient, which results in a heat flow process from the oil to the iced water through their common boundary. Since the oil is restored to its initial temperature of 0 oC, the energy lost as heat must equal the energy gained as work. Therefore, QA = 10 kJ (or QA,out = 10 kJ).

 System B: The only energy interaction at the boundaries of system B is the heat flow from system A. All the heat lost by the oil is gained by the iced water. Thus, WB = 0 and Qe = +10 kJ. Combined system: The outer boundary of system A forms the entire boundary of the combined system. The only energy interaction at this boundary is the electrical work. Since the tank is well insulated, no heat will cross this boundary. Therefore, Wcomb = 10 kJ and Qcomb = 0. Notice that the heat flow from the oil to the iced water is an internal process for the combined system and, therefore, is not recognized as heat. It is simply the redistribution of the internal energy.

Mechanical forms of Work

There are several different ways of doing work, each in some way related to a force acting through a distance. In elementary mechanics, the work done by a constant force F on a body that is displaced a distance s in the direction of the force is given by

W = Fs 2-17

If the force F is not constant, the work done is obtained by adding (i.e., integrating) the differential amounts of work (force times the differential displacement ds):

Obviously one needs to know how the force varies with displacement to perform this integration. Equations 2-17 and 2-18 give only the magnitude of the work. The sign is easily determined from physical considerations: The work done on a system by an external force acting in the direction of motion is positive, and work done by a system against an external force acting in the opposite direction to motion is negative.

 There are two requirements for a work interaction between a system and its surroundings to exist:

1. there must be a force acting on the boundary, and

2. the boundary must move. Therefore, the presence of forces on the boundary

without any displacement of the boundary does not constitute a work

interaction. Likewise, the displacement of the boundary without any force to

oppose or drive this motion (such as the expansion of a gas into an evacuated

space) is not a work interaction.

In many thermodynamic problems, mechanical work is the only form of work involved. It is associated with the movement of the boundary of a system or with the movement of the entire system as a whole .Some common forms of mechanical work are discussed below.

Moving Boundary Work

One form of mechanical work frequently encountered in practice is associated with the expansion or compression of a gas in a piston-cylinder device. During this process, part of the boundary (the inner face of the piston) moves back and forth. Therefore, the expansion and compression work is often called moving boundary work, or simply boundary work. Some prefer to call it the P dV work for reasons explained below. Moving boundary work is the primary form of work involved in automobile engines. During their expansion, the combustion gases force the piston to move, which in turn forces the crank shaft to rotate.

The moving boundary work associated with real engines or compressors cannot be determined exactly from a thermodynamic analysis alone because the piston usually moves at very high speeds, making it difficult for the gas inside to maintain equilibrium.

Then the states that the system passes through during the process cannot be specified, and no process path can be drawn. Work, being a path function, cannot be determined analytically without knowledge of the path. Therefore, the boundary work in real engines or compressors is determined by direct measurements.

In this section, we analyze the moving boundary work for a quasi-equilibrium process, a process during which the system remains in equilibrium at all times. A quasi-equilibrium process, also called a quasi-static process, is closely approximated by real engines, especially when the piston moves at low velocities. Under identical conditions, the work output of the engines is found to be a maximum, and the work input to the compressors to be a minimum, when quasi-equilibrium processes are used in place of non-quasiequilibrium processes. Below, the work associated with a moving boundary is evaluated for a quasi-equilibrium process.

Consider the gas enclosed in the piston-cylinder device shown below. 

The initial pressure of the gas is P, the total volume is V, and the cross-sectional area of the piston is A. If the piston is allowed to move a distance ds in a quasi-equilibrium manner, the differential work done during this process is

δW = Fds = PA ds = P dV                                        2-19

That is, the boundary work in the differential form is equal to the product of the absolute pressure P and the differential change in the volume dV of the system. This expression also explains why the moving boundary work is sometimes called the P dV work. In order to abide by the sign rule, for an expansion dV is positive, the pressure P is the absolute pressure which is always positive, thus, the work should be written as

δW = - P dV                                                               2-20

Thus, the boundary work is negative during an expansion process and positive during a compression process, which is consistent with the sign convention adopted for work. The total boundary work done during the entire process as the piston moves is obtained by adding all the differential works from the initial state to the final state:

This integral can be evaluated only if we know the functional relationship between P and V during the process. That is, P = f(V) should be available. Note that P = f(V) is simply the equation of the process path on a P-V diagram. The quasi-equilibrium expansion process described above is shown on a P-V diagram below. On this diagram, the differential area dA is equal to P dV, which is the differential work. The total area A under the process curve 1-2 is obtained by adding these

differential areas:

The area under the process curve on a P-V diagram is equal, in magnitude, to the

work done during a quasi-equilibrium expansion or compression process of a

closed system.

A gas can follow several different paths as it expands from state 1 to state 2. In general,each path will have a different area underneath it, and since this area represents the magnitude of the work, the work done will be different for each process.

This is expected, since work is a path function (i.e., it depends on the path followed as well as the end states). If work were not a path function, no cyclic devices (car engines, power plants) could operate as work-producing devices. The work produced by these  devices during one part of the cycle would have to be consumed during another part, and there would be no net work output.

Note: If the relationship between P and V during an expansion or a compression process is given in terms of experimental data instead of in a functional form, obviously we cannot perform the integration analytically. But we can always plot the P-V diagram of the process, using these data points, and calculate the area underneath graphically to determine the work done.

Example 2-7A frictionless piston-cylinder device contains 0.1 lb of water vapor at 20 psi and 320 oF. Heat is now added to the steam until the temperature reaches 400°F. If the piston is not attached to a shaft and its mass is constant, determine the work done by the steam during this process.

Hint: Even though it is not explicitly stated, the pressure of the steam within the cylinder remains constant during this process since both the atmospheric pressure and the weight of the piston remain constant. Therefore, this is a constant-pressure process.

Example 2-8 A piston-cylinder device initially contains 0.4 m3 of air at 100 kPa and 80°C. The air is now compressed to 0.1 m3 in such a way that the temperature inside the cylinder remains constant. Determine the work done during this process.

Solution  A sketch of the system and the P-V diagram of the process are shown below. At the specified conditions, air can be considered to be an ideal gas since it is at a high temperature and low pressure relative to its critical-point values (Tcr = -147°C, Pcr = 3390 kPa for nitrogen, the main constituent of air). For an ideal gas at constant temperature To,

                 

P = C / V

where C is a constant. Eq. 2-21 becomes:

where P1 V1 = P2 V2 .

Polytropic Process

During expansion and compression processes of real gases, pressure and volume are often related by P Vn = C, where n and C are constants. A process of this kind is called a polytropic process. Below we develop a general expression for the work done during a polytropic process.

A sketch of the system and the P-V diagram of the process are shown below

                      

The pressure for a polytropic process can be expressed as

n

The case for n = 1 is equivalent to the isothermal process already discussed.

Enthalpy

Four quantities called "thermodynamic potentials" are useful in the chemical thermodynamics of reactions and non-cyclic processes. They are internal energy, the enthalpy, the Helmholtz free energy and the Gibbs free energy. Enthalpy is defined by

H = U + PV

where P and V are the pressure and volume, and U is internal energy. Enthalpy is then a precisely measurable state variable, since it is defined in terms of three other precisely definable state variables. It is somewhat parallel to the first law of thermodynamics for a constant pressure system Q = ΔU + PΔV since in this case Q=ΔH

It is a useful quantity for tracking chemical reactions. If as a result of an exothermic reaction some energy is released to a system, it has to show up in some measurable form in terms of the state variables. An increase in the enthalpy H = U + PV might be associated with an increase in internal energy which could be measured by calorimetry, or with work done by the system, or a combination of the two.

The internal energy U might be thought of as the energy required to create a system in the absence of changes in temperature or volume. But if the process changes the volume, as in a chemical reaction which produces a gaseous product, then work must be done to produce the change in volume. For a constant pressure process the work you must do to produce a volume change ΔV is PΔV. Then the term PV can be interpreted as the work you must do to "create room" for the system if you presume it started at zero volume.

5. The Second Law Of Termodynamics

The second law of thermodynamics is an expression of the tendency that over time, differences in temperature, pressure, and chemical potential equilibrate in an isolated physical system. From the state of thermodynamic equilibrium, the law deduced the principle of the increase of entropy and explains the phenomenon of irreversibility in nature. The second law declares the impossibility of machines that generate usable energy from the abundant internal energy of nature by processes called perpetual motion of the second kind.

The second law may be expressed in many specific ways, but the first formulation is credited to the German scientist Rudolf Clausius. The law is usually stated in physical terms of impossible processes. In classical thermodynamics, the second law is a basic postulate applicable to any system involving measurable heat transfer, while in statistical thermodynamics, the second law is a consequence of unitarity in quantum theory. In classical thermodynamics, the second law defines the concept of thermodynamic entropy, while in statistical mechanics entropy is defined from information theory, known as the Shannon entropy.

Description

The first law of thermodynamics provides the basic definition of thermodynamic energy, also called internal energy, associated with all thermodynamic systems, but unknown in mechanics, and states the rule of conservation of energy in nature.

However, the concept of energy in the first law does not account for the observation that natural processes have a preferred direction of progress. For example, spontaneously, heat always flows to regions of lower temperature, never to regions of higher temperature without external work being performed on the system. The first law is completely symmetrical with respect to the initial and final states of an evolving system. The key concept for the explanation of this phenomenon through the second law of thermodynamics is the definition of a new physical property, the entropy.

A change in the entropy (S) of a system is the infinitesimal transfer of heat (Q) to a closed system driving a reversible process, divided by the equilibrium temperature (T) of the system.^[1]

The entropy of an isolated system that is in equilibrium is constant and has reached its maximum value.

Empirical temperature and its scale is usually defined on the principles of thermodynamics equilibrium by the zeroth law of thermodynamics.^[2] However, based on the entropy, the second law permits a definition of the absolute, thermodynamic temperature, which has its null point at absolute zero.^[3]

The second law of thermodynamics may be expressed in many specific ways,^[4] the most prominent classical statements^[3] being the original statement by Rudolph Clausius (1850), the formulation by Lord Kelvin (1851), and the definition in axiomatic thermodynamics by Constantin Carathéodory (1909). These statement cast the law in general physical terms citing the impossibility of certain processes. They have been shown to be equivalent.

Clausius statement

German scientist Rudolf Clausius is credited with the first formulation of the second law, now known as the Clausius statement:^[4]

No process is possible whose sole result is the transfer of heat from a body of lower temperature to a body of higher temperature.^{[note 1]}

Spontaneously, heat cannot flow from cold regions to hot regions without external work being performed on the system, which is evident from ordinary experience of refrigeration, for example. In a refrigerator, heat flows from cold to hot, but only when forced by an external agent, a compressor.

Kelvin statement

Lord Kelvin expressed the second law in another form. The Kelvin statement expresses it as follows:^[4]

No process is possible in which the sole result is the absorption of heat from a reservoir and its complete conversion into work.

This means it is impossible to extract energy by heat from a high-temperature energy source and then convert all of the energy into work. At least some of the energy must be passed on to heat a low-temperature energy sink. Thus, a heat engine with 100% efficiency is thermodynamically impossible. This also means that it is impossible to build solar panels that generate electricity solely from the infrared band of the electromagnetic spectrum without consideration of the temperature on the other side of the panel (as is the case with conventional solar panels that operate in the visible spectrum).

Note that it is possible to convert heat completely into work, such as the isothermal expansion of ideal gas. However, such a process has an additional result. In the case of the isothermal expansion, the volume of the gas increases and never goes back without outside interference.

Principle of Carathéodory

Constantin Carathéodory formulated thermodynamics on a purely mathematical axiomatic foundation. His statement of the second law is known as the Principle of Carathéodory, which may be formulated as follows:^[5]

In every neighborhood of any state S of an adiabatically isolated system there are states inaccessible from S.^[6]

With this formulation he described the concept of adiabatic accessibility for the first time and provided the foundation for a new subfield of classical thermodynamics, often called geometrical thermodynamics.

Equivalence of the statements

Derive Kelvin Statement from Clausius Statement

Suppose there is an engine violating the Kelvin statement: i.e.,one that drains heat and converts it completely into work in a cyclic fashion without any other result. Now pair it with a reversed Carnot engine as shown by the graph. The net and sole effect of this newly created engine consisting of the two engines mentioned is transferring heat from the cooler reservoir to the hotter one, which violates the Clausius statement. Thus the Clausius statement implies the Kelvin statement. We can prove in a similar manner that the Kelvin statement implies the Clausius statement, or, in a word, the two are equivalent.

Corollaries

Perpetual motion of the second kind

Prior to the establishment of the Second Law, many people who were interested in inventing a perpetual motion machine had tried to circumvent the restrictions of First Law of Thermodynamics by extracting the massive internal energy of the environment as the power of the machine. Such a machine is called a "perpetual motion machine of the second kind". The second law declared the impossibility of such machines.

Carnot theorem

Carnot's theorem is a principle that limits the maximum efficiency for any possible engine. The efficiency solely depends on the temperature difference between the hot and cold thermal reservoirs. Carnot's theorem states:

• All irreversible heat engines between two heat reservoirs are less efficient than a Carnot engine operating between the same reservoirs.
• All reversible heat engines between two heat reservoirs are equally efficient with a Carnot engine operating between the same reservoirs.

Historically, the principle was based on the invalid caloric theory and preceded the establishment of the second law;^[7] however, it has since been recognized as a result of the second law.

Clausius theorem

The Clausius theorem (1854) states that in a cyclic process

The equality holds in the reversible case^[8] and the '<' is in the irreversible case. The reversible case is used to introduce the state function entropy. This is because in cyclic processes the variation of a state function is zero.

Thermodynamic temperature

Main article: Thermodynamic temperature

For an arbitrary heat engine, the efficiency is:

where A is the work done per cycle. Thus the efficiency depends only on q_C/q_H.

Carnot's theorem states that all reversible engines operating between the same heat reservoirs are equally efficient. Thus, any reversible heat engine operating between temperatures T₁ and T₂ must have the same efficiency, that is to say, the effiency is the function of temperatures only:

In addition, a reversible heat engine operating between temperatures T₁ and T₃ must have the same efficiency as one consisting of two cycles, one between T₁ and another (intermediate) temperature T₂, and the second between T₂ andT₃. This can only be the case if

Now consider the case where T₁ is a fixed reference temperature: the temperature of the triple point of water. Then for any T₂ and T₃,

Therefore if thermodynamic temperature is defined by

then the function f, viewed as a function of thermodynamic temperature, is simply

and the reference temperature T₁ will have the value 273.16. (Of course any reference temperature and any positive numerical value could be used—the choice here corresponds to the Kelvin scale.)

Entropy

Main article: entropy (classical thermodynamics)

According to the Clausius equality, for a reversible process

That means the line integral is path independent.

So we can define a state function S called entropy, which satisfies

With this we can only obtain the difference of entropy by integrating the above formula. To obtain the absolute value, we need the Third Law of Thermodynamics, which states that S=0 at absolute zero for perfect crystals.

For any irreversible process, since entropy is a state function, we can always connect the initial and terminal status with an imaginary reversible process and integrating on that path to calculate the difference in entropy.

Now reverse the reversible process and combine it with the said irreversible process. Applying Clausius inequality on this loop,

Thus,

where the equality holds if the transformation is reversible.

Notice that if the process is an adiabatic process,then δQ = 0, so .

Available useful work

See also: Available useful work (thermodynamics)

An important and revealing idealized special case is to consider applying the Second Law to the scenario of an isolated system (called the total system or universe), made up of two parts: a sub-system of interest, and the sub-system's surroundings. These surroundings are imagined to be so large that they can be considered as an unlimited heat reservoir at temperature T_R and pressure P_R — so that no matter how much heat is transferred to (or from) the sub-system, the temperature of the surroundings will remain T_R; and no matter how much the volume of the sub-system expands (or contracts), the pressure of the surroundings will remain P_R.

Whatever changes to dS and dS_R occur in the entropies of the sub-system and the surroundings individually, according to the Second Law the entropy S_tot of the isolated total system must not decrease:

According to the First Law of Thermodynamics, the change dU in the internal energy of the sub-system is the sum of the heat δq added to the sub-system, less any work δw done by the sub-system, plus any net chemical energy entering the sub-system d ∑μ_iRN_i, so that:

where μ_iR are the chemical potentials of chemical species in the external surroundings.

Now the heat leaving the reservoir and entering the sub-system is

where we have first used the definition of entropy in classical thermodynamics (alternatively, in statistical thermodynamics, the relation between entropy change, temperature and absorbed heat can be derived); and then the Second Law inequality from above.

It therefore follows that any net work δw done by the sub-system must obey

It is useful to separate the work δw done by the subsystem into the useful work δw_u that can be done by the sub-system, over and beyond the work p_R dV done merely by the sub-system expanding against the surrounding external pressure, giving the following relation for the useful work that can be done:

It is convenient to define the right-hand-side as the exact derivative of a thermodynamic potential, called the availability or exergy X of the subsystem,

The Second Law therefore implies that for any process which can be considered as divided simply into a subsystem, and an unlimited temperature and pressure reservoir with which it is in contact,

i.e. the change in the subsystem's exergy plus the useful work done by the subsystem (or, the change in the subsystem's exergy less any work, additional to that done by the pressure reservoir, done on the system) must be less than or equal to zero.

In sum, if a proper infinite-reservoir-like reference state is chosen as the system surroundings in the real world, then the Second Law predicts a decrease in X for an irreversible process and no change for a reversible process.

Is equivalent to

This expression together with the associated reference state permits a design engineer working at the macroscopic scale (above the thermodynamic limit) to utilize the Second Law without directly measuring or considering entropy change in a total isolated system. (Also, see process engineer). Those changes have already been considered by the assumption that the system under consideration can reach equilibrium with the reference state without altering the reference state. An efficiency for a process or collection of processes that compares it to the reversible ideal may also be found (See second law efficiency.)

This approach to the Second Law is widely utilized in engineering practice, environmental accounting, systems ecology, and other disciplines.

The first theory of the conversion of heat into mechanical work is due to Nicolas Léonard Sadi Carnot in 1824. He was the first to realize correctly that the efficiency of this conversion depends on the difference of temperature between an engine and its environment.

Recognizing the significance of James Prescott Joule's work on the conservation of energy, Rudolf Clausius was the first to formulate the second law during 1850, in this form: heat does not flow spontaneously from cold to hot bodies. While common knowledge now, this was contrary to the caloric theory of heat popular at the time, which considered heat as a fluid. From there he was able to infer the principle of Sadi Carnot and the definition of entropy (1865).

Established during the 19th century, the Kelvin-Planck statement of the Second Law says, "It is impossible for any device that operates on a cycle to receive heat from a single reservoir and produce a net amount of work." This was shown to be equivalent to the statement of Clausius.

The ergodic hypothesis is also important for the Boltzmann approach. It says that, over long periods of time, the time spent in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e. that all accessible microstates are equally probable over a long period of time. Equivalently, it says that time average and average over the statistical ensemble are the same.

It has been shown that not only classical systems but also quantum mechanical ones tend to maximize their entropy over time. Thus the second law follows, given initial conditions with low entropy. More precisely, it has been shown that the local von Neumann entropy is at its maximum value with a very high probability.^[9] The result is valid for a large class of isolated quantum systems (e.g. a gas in a container). While the full system is pure and therefore does not have any entropy, the entanglement between gas and container gives rise to an increase of the local entropy of the gas. This result is one of the most important achievements of quantum thermodynamics^{[dubious – discuss]}.

Today, much effort in the field is attempting to understand why the initial conditions early in the universe were those of low entropy^[10][11], as this is seen as the origin of the second law (see below).

Informal descriptions

The second law can be stated in various succinct ways, including:

• It is impossible to produce work in the surroundings using a cyclic process connected to a single heat reservoir (Kelvin, 1851).
• It is impossible to carry out a cyclic process using an engine connected to two heat reservoirs that will have as its only effect the transfer of a quantity of heat from the low-temperature reservoir to the high-temperature reservoir (Clausius, 1854).
• If thermodynamic work is to be done at a finite rate, free energy must be expended.^[12]

Mathematical descriptions

In 1856, the German physicist Rudolf Clausius stated what he called the "second fundamental theorem in the mechanical theory of heat" in the following form:^[13]

where Q is heat, T is temperature and N is the "equivalence-value" of all uncompensated transformations involved in a cyclical process. Later, in 1865, Clausius would come to define "equivalence-value" as entropy. On the heels of this definition, that same year, the most famous version of the second law was read in a presentation at the Philosophical Society of Zurich on April 24, in which, in the end of his presentation, Clausius concludes:

The entropy of the universe tends to a maximum.

This statement is the best-known phrasing of the second law. Moreover, owing to the general broadness of the terminology used here, e.g. universe, as well as lack of specific conditions, e.g. open, closed, or isolated, to which this statement applies, many people take this simple statement to mean that the second law of thermodynamics applies virtually to every subject imaginable. This, of course, is not true; this statement is only a simplified version of a more complex description.

In terms of time variation, the mathematical statement of the second law for an isolated system undergoing an arbitrary transformation is:

where

S is the entropy and

t is time.

Statistical mechanics gives an explanation for the second law by postulating that a material is composed of atoms and molecules which are in constant motion. A particular set of positions and velocities for each particle in the system is called a microstate of the system and because of the constant motion, the system is constantly changing its microstate. Statistical mechanics postulates that, in equilibrium, each microstate that the system might be in is equally likely to occur, and when this assumption is made, it leads directly to the conclusion that the second law must hold in a statistical sense. That is, the second law will hold on average, with a statistical variation on the order of 1/√N where N is the number of particles in the system. For everyday (macroscopic) situations, the probability that the second law will be violated is practically zero. However, for systems with a small number of particles, thermodynamic parameters, including the entropy, may show significant statistical deviations from that predicted by the second law. Classical thermodynamic theory does not deal with these statistical variations.

Derivation from statistical mechanics

In statistical mechanics, the Second Law is not a postulate, rather it is a consequence of the fundamental postulate, also known as the equal prior probability postulate, so long as one is clear that simple probability arguments are applied only to the future, while for the past there are auxiliary sources of information which tell us that it was low entropy. The first part of the second law, which states that the entropy of a thermally isolated system can only increase is a trivial consequence of the equal prior probability postulate, if we restrict the notion of the entropy to systems in thermal equilibrium. The entropy of an isolated system in thermal equilibrium containing an amount of energy of E is:

where is the number of quantum states in a small interval between E and E + δE. Here δE is a macroscopically small energy interval that is kept fixed. Strictly speaking this means that the entropy depends on the choice of δE. However, in the thermodynamic limit (i.e. in the limit of infinitely large system size), the specific entropy (entropy per unit volume or per unit mass) does not depend on δE.

Suppose we have an isolated system whose macroscopic state is specified by a number of variables. These macroscopic variables can, e.g., refer to the total volume, the positions of pistons in the system, etc. Then Ω will depend on the values of these variables. If a variable is not fixed, (e.g. we do not clamp a piston in a certain position), then because all the accessible states are equally likely in equilibrium, the free variable in equilibrium will be such that Ω is maximized as that is the most probable situation in equilibrium.

If the variable was initially fixed to some value then upon release and when the new equilibrium has been reached, the fact the variable will adjust itself so that Ω is maximized, implies that that the entropy will have increased or it will have stayed the same (if the value at which the variable was fixed happened to be the equilibrium value).

The entropy of a system that is not in equilibrium can be defined as:

see here. Here the P_j is the probabilities for the system to be found in the states labeled by the subscript j. In thermal equilibrium the probabilities for states inside the energy interval δE are all equal to 1 / Ω, and in that case the general definition coincides with the previous definition of S that applies to the case of thermal equilibrium.

Suppose we start from an equilibrium situation and we suddenly remove a constraint on a variable. Then right after we do this, there are a number Ω of accessible microstates, but equilibrium has not yet been reached, so the actual probabilities of the system being in some accessible state are not yet equal to the prior probability of 1 / Ω. We have already seen that in the final equilibrium state, the entropy will have increased or have stayed the same relative to the previous equilibrium state. Boltzmann's H-theorem, however, proves that the entropy will increase continuously as a function of time during the intermediate out of equilibrium state.

Derivation of the entropy for reversible processes

The second part of the Second Law states that the entropy change of a system undergoing a reversible process is given by:

where the temperature is defined as:

See here for the justification for this definition. Suppose that the system has some external parameter, x, that can be changed. In general, the energy eigenstates of the system will depend on x. According to the adiabatic theorem of quantum mechanics, in the limit of an infinitely slow change of the system's Hamiltonian, the system will stay in the same energy eigenstate and thus change its energy according to the change in energy of the energy eigenstate it is in.

The generalized force, X, corresponding to the external variable x is defined such that Xdx is the work performed by the system if x is increased by an amount dx. E.g., if x is the volume, then X is the pressure. The generalized force for a system known to be in energy eigenstate E_r is given by:

Since the system can be in any energy eigenstate within an interval of δE, we define the generalized force for the system as the expectation value of the above expression:

To evaluate the average, we partition the energy eigenstates by counting how many of them have a value for within a range between Y and Y + δY. Calling this number , we have:

The average defining the generalized force can now be written:

We can relate this to the derivative of the entropy w.r.t. x at constant energy E as follows. Suppose we change x to x + dx. Then will change because the energy eigenstates depend on x, causing energy eigenstates to move into or out of the range between E and E + δE. Let's focus again on the energy eigenstates for which lies within the range between Y and Y + δY. Since these energy eigenstates increase in energy by Y dx, all such energy eigenstates that are in the interval ranging from E - Y dx to E move from below E to above E. There are

such energy eigenstates. If , all these energy eigenstates will move into the range between E and E + δE and contribute to an increase in Ω. The number of energy eigenstates that move from below E + δE to above E + δE is, of course, given by . The difference

is thus the net contribution to the increase in Ω. Note that if Y dx is larger than δE there will be the energy eigenstates that move from below E to above E + δE. They are counted in both and , therefore the above expression is also valid in that case.

Expressing the above expression as a derivative w.r.t. E and summing over Y yields the expression:

The logarithmic derivative of Ω w.r.t. x is thus given by:

The first term is intensive, i.e. it does not scale with system size. In contrast, the last term scales as the inverse system size and will thus vanishes in the thermodynamic limit. We have thus found that:

Combining this with

Gives:

Derivation for systems described by the canonical ensemble

If a system is in thermal contact with a heat bath at some temperature T then, in equilibrium, the probability distribution over the energy eigenvalues are given by the canonical ensemble:

Here Z is a factor that normalizes the sum of all the probabilities to 1, this function is known as the partition function. We now consider an infinitesimal reversible change in the temperature and in the external parameters on which the energy levels depend. It follows from the general formula for the entropy:

that

Inserting the formula for P_j for the canonical ensemble in here gives:

General derivation from unitarity of quantum mechanics

The time development operator in quantum theory is unitary, because the Hamiltonian is hermitian. Consequently the transition probability matrix is doubly stochastic, which implies the Second Law of Thermodynamics.^[14][15] This derivation is quite general, based on the Shannon entropy, and does not require any assumptions beyond unitarity, which is universally accepted. It is a consequence of the irreversibility or singular nature of the general transition matrix.

Non-equilibrium states

Statistically it is possible for a system to achieve moments of non-equilibrium. In such statistically unlikely events where hot particles "steal" the energy of cold particles enough that the cold side gets colder and the hot side gets hotter, for an instant. Such events have been observed at a small enough scale where the likelihood of such a thing happening is significant.^[16] The physics involved in such an event is described by the fluctuation theorem.

Controversies

Maxwell's demon

Main article: Maxwell's demon

Maxwell imagined one container divided into two parts, A and B. Both parts are filled with the same gas at equal temperatures and placed next to each other. Observing the molecules on both sides, an imaginary demon guards a trapdoor between the two parts. When a faster-than-average molecule from A flies towards the trapdoor, the demon opens it, and the molecule will fly from A to B. The average speed of the molecules in B will have increased while in A they will have slowed down on average. Since average molecular speed corresponds to temperature, the temperature decreases in A and increases in B, contrary to the second law of thermodynamics.

One of the most famous responses to this question was suggested in 1929 by Leó Szilárd and later by Léon Brillouin. Szilárd pointed out that a real-life Maxwell's demon would need to have some means of measuring molecular speed, and that the act of acquiring information would require an expenditure of energy. But later exceptions were found.

Loschmidt's paradox

Loschmidt's paradox, also known as the reversibility paradox, is the objection that it should not be possible to deduce an irreversible process from time-symmetric dynamics. This puts the time reversal symmetry of (almost) all known low-level fundamental physical processes at odds with any attempt to infer from them the second law of thermodynamics which describes the behavior of macroscopic systems. Both of these are well-accepted principles in physics, with sound observational and theoretical support, yet they seem to be in conflict; hence the paradox.

One approach to handling Loschmidt's paradox is the fluctuation theorem, proved by Denis Evans and Debra Searles, which gives a numerical estimate of the probability that a system away from equilibrium will have a certain change in entropy over a certain amount of time. The theorem is proved with the exact time reversible dynamical equations of motion and the Axiom of Causality. The fluctuation theorem is proved utilizing the fact that dynamics is time reversible. Quantitative predictions of this theorem have been confirmed in laboratory experiments at the Australian National University conducted by Edith M. Sevick et al. using optical tweezers apparatus.

Gibbs paradox

Main article: Gibbs paradox

In statistical mechanics, a simple derivation of the entropy of an ideal gas based on the Boltzmann distribution yields an expression for the entropy which is not extensive (is not proportional to the amount of gas in question). This leads to an apparent paradox known as the Gibbs paradox, allowing, for instance, the entropy of closed systems to decrease, violating the second law of thermodynamics.

The paradox is averted by recognizing that the identity of the particles does not influence the entropy. In the conventional explanation, this is associated with an indistinguishability of the particles associated with quantum mechanics. However, a growing number of papers now take the perspective that it is merely the definition of entropy that is changed to ignore particle permutation (and thereby avert the paradox). The resulting equation for the entropy (of a classical ideal gas) is extensive, and is known as the Sackur-Tetrode equation.

Poincaré recurrence theorem

The Poincaré recurrence theorem states that certain systems will, after a sufficiently long time, return to a state very close to the initial state. The Poincaré recurrence time is the length of time elapsed until the recurrence. The result applies to physical systems in which energy is conserved. The Recurrence theorem apparently contradicts the Second law of thermodynamics, which says that large dynamical systems evolve irreversibly towards the state with higher entropy, so that if one starts with a low-entropy state, the system will never return to it. There are many possible ways to resolve this paradox, but none of them is universally accepted^{[citation needed]}. The most typical argument is that for thermodynamical systems like an ideal gas in a box, recurrence time is so large that for all practical purposes it is infinite.

Heat death of the universe

Main article: Heat death of the universe

According to the second law the entropy of any isolated system, such as the entire universe, never decreases. If the entropy of the universe has a maximum upper bound then when this bound is reached the universe has no thermodynamic free energy to sustain motion or life, that is, the heat death is reached.

"The law that entropy always increases holds, I think, the supreme position among the laws of Nature. If someone points out to you that your pet theory of the universe is in disagreement with Maxwell's equations — then so much the worse for Maxwell's equations. If it is found to be contradicted by observation — well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation." — Sir Arthur Stanley Eddington, The Nature of the Physical World (1927)

The tendency for entropy to increase in isolated systems is expressed in the second law of thermodynamics — perhaps the most pessimistic and amoral formulation in all human thought. — Gregory Hill and Kerry Thornley, Principia Discordia (1965)

There are almost as many formulations of the second law as there have been discussions of it. — Philosopher / Physicist P.W. Bridgman, (1941)

http://en.wikipedia.org/wiki/Second_law_of_thermodynamics

6.Third law of thermodynamics

The third law of thermodynamics is a statistical law of nature regarding entropy and the impossibility of reaching absolute zero, the null point of the temperature scale. The most common enunciation of the third law of thermodynamics is: :As a system approaches absolute zero, all processes cease and the entropy of the system approaches a minimum value.

This minimum value, the residual entropy, is not necessarily zero, although it is almost always zero in a perfect, pure crystal

The third law was developed by the chemist Walther Nernst during the years 1906-1912, and is therefore often referred to as Nernst's theorem or Nernst's postulate. The third law of thermodynamics states that the entropy of a system at absolute zero is a well-defined constant. This is because a system at zero temperature exists in its ground state, so that its entropy is determined only by the degeneracy of the ground state. It means that

 "it is impossible by any procedure, no matter how idealised, to reduce any system to the absolute zero of temperature in a finite number of operations".

An alternative version of the third law of thermodynamics as stated by Gilbert N. Lewis and Merle Randall in 1923:

:If the entropy of each element in some (perfect) crystalline state be taken as zero at the absolute zero of temperature, every substance has a finite positive entropy; but at the absolute zero of temperature the entropy may become zero, and does so become in the case of perfect crystalline substances.

This version states not only ΔS will reach zero at 0 kelvins, but S itself will also reach zero, at least for perfect crystalline substances. (This statement is now known to have some rare exceptions.)

If we have sufficient heat capacity data (and the data on phase changes) we could write

                (1)

(If there is a phase change between 0 K and T we would have to add the entropy of the phase change.) If C_p were constant near T = 0 we would have,

which is undefined. Fortunately, experimentally C_p → 0 as T → 0. For nonmetals C_p is proportional to T ³ at low temperatures. For metals C_p is proportional to T ³ at low temperatures but shifts over to being proportional to T at extremely low temperatures. (The latter happens when the atomic motion "freezes out" and the heat capacity is due to the motion of the conduction electrons in the metal.)

Equation 1 could be used to calculate absolute entropies for substances if we knew what the entropy is at absolute zero. Experimentally it appears that the entropy at absolute zero is the same for all substances. The third law of thermodynamics codifies this observation and sets

for all elements and compounds in their most stable and perfect crystalline state at absolute zero and one atmosphere pressure. (All except for helium, which is a liquid at the lowest observable temperatures at one atmosphere.)

The advantage of this law is that it allows us to use experimental data to compute the absolute entropy of a substance. For example, suppose we want to calculate the absolute entropy of liquid water at 25^o C. We would need to know the C_p of ice from 0 K to 273.15 K and the C_p of liquid water from 273.15 K to 298.15 K. We also need the heat of fusion of water at its normal melting point. With all of this data, which can be obtained partly from theory and partly from experiment, we find

                (2)

Some substances may undergo several phase changes.

Entropy Changes in Chemical Reactions

We can use the third law entropies to calculate entropy changes for chemical reactions. For a typical reaction,

a A + b B → c C + d D.                (3)

the entropy change is

                (4)

Notice two things:

1.      We did not define or use an entropy of formation, Δ_fS ^o.

2.      2. S ^o_element is not zero.

As we have said before, Δ_rS ^o and Δ_rH ^o are independent of each other. They can not be calculated from each other. They must be calculated from Equation 4 and a comparable equation using heats of formation.

There is another way to calculate Δ_rS ^o,

               (5)

As we have seen before, the Δ_rG ^o and Δ_rH ^o can be calculated from free energies and heats of formation.

http://en.wikipedia.org/wiki/Third_law_of_thermodynamics

References

Cengel, Y.; Boles, M. (2006). Thermodynamics: An Engineering Approach (5th ed.). Boston: McGraw-Hill Higher Education

Chris Vuille; Serway, Raymond A.; Faughn, Jerry S. (2009). College physics. Belmont, CA: Brooks/Cole, Cengage Learning. pp.  

Reif, F. (1965). "Chapter 3: Statistical Thermodynamics". Fundamentals of Statistical and Thermal Physics. New York: McGraw-Hill. pp. 102.

http://en.wikipedia.org/wiki/Third_law_of_thermodynamics

http://en.wikipedia.org/wiki/Second_law_of_thermodynamics

http://www.chem1.com/acad/pdf/c1gas.pdf

http://en.wikipedia.org/wiki/Ideal_gas_law

http://en.wikipedia.org/wiki/Thermodynamics

http://www.chem.ufl.edu/~itl/2045/lectures/lec_e.html