In this article, the most important results of classical thermodynamics are concisely derived

- Introduction
- Partial Derivatives and Potentials
- Specific Heat and the TdS Equations
- Clapeyron's Equation
- Appendix
- References

From its Greek etymology, "thermodynamics" could be the study of the forces associated with beans, or with heat. We'll consider heat here, leaving beans for another article. Heat was long considered something like a substance, the Empedoclean element fire, with the ill-defined concept of temperature as a sort of driving force. Heat even got its own units, of which the calorie survives even today, and was the object of the practical science and technique of calorimetry. The heat substance was the subtle fluid *caloric*, which combined with matter in obscure ways, and could be latent (hidden) or evident. Relics of these ideas still survive in the terminology of thermodynamics.

In the early 19th century, it slowly became evident that heat was not a conserved quantity, as it should have been if it were material, but that work and heat were interconvertible. Count Rumford showed that limitless amounts of caloric were created in the boring of cannon, and Watt showed that heat could be converted into mechanical work. It is not an easy practical matter to show that the work done is proportional to the quantity of heat destroyed, but the converse is readily observed. It was a matter of experiment to find that 1 calorie of heat was the equivalent of 4.186 J of work, or 1 Btu the equivalent of 778 ft-lb. Heat and work were indentified as forms of energy, which was a conserved quantity. This was the fundamental concept of thermodynamics, the new science of heat and force. Things were not as simple as they seemed in the days of caloric.

The deductive science of classical thermodynamics rests on simple foundations, which were dignified as the Laws of Thermodynamics. Much effort has been rather uselessly expended in arguing over the best way of expressing these fundamental postulates. Let's restrict ourselves here to a pure substance whose state can be specified in terms of its pressure p, volume V and temperature T. Actual substances are extremely varied, and it would be tedious to attempt to handle all the variety in our discussion. All the important matters can be discussed on the basis of such a pure substance. The fixed amount of substance under consideration is called the *system*.

We assume, first of all, that the variables p, V and T satisfy an equation f(p,V,T) = 0 which we call the *equation of state*. The variables p and V are easy to define and measure, but the temperature T presents a great mystery. One way of proceeding would be to choose a definite substance, such as mercury, and arbitrarily define temperature as proportional to the volume of a certain amount of mercury at a standard pressure. Thermodynamics can limp along on such an unsatisfying definition, but it was found possible to make a better definition that does not depend on the properties of a substance. Carnot showed that such a definition could be based on a cyclical thermodynamic process in which Q/Q' = T/T', where Q,Q' are the amounts of heat transferred and T,T' are the corresponding *absolute temperatures*. This result did not depend on the particular substance used to execute the cycle. Carnot himself actually was working with caloric, but his results were interpreted later in this manner.

This shows that there is a natural zero of temperature, where Q = 0, and that absolute temperatures are determined up to a constant, which can be chosen to suit one's whim. If we decide that the absolute temperature of ice in equilibrium with air-saturated water at 1 atm is 273.15, we get the Kelvin scale, in which the difference in temperature of the steam and ice points is 100°, characteristic of the Celsius scale. If we take 9/5 of this, the difference is 180°, characteristic of the Fahrenheit scale (212 - 32 = 180), and the absolute temperatures are on the Rankine scale. Neither scale is any more metric or absolute than the other, but the Kelvin scale is standard.

The practical method of finding absolute temperature uses an ideal-gas thermometer. The equation of state for one gram-mole of an ideal gas is pV = RT, so that T = pV/R. This looks like basing the scale on a particular substance, but really is not, since the ideal gas is a theoretical concept that can be approached more or less closely by an actual gas, but is a universal concept independent of any particular substance. Taking an ideal gas through a Carnot cycle shows that T in the equation of state is the absolute temperature. In practice, the temperature is extrapolated to zero pressure, where the gas will be ideal in fact. Thermodynamics is much neater when the absolute temperature is used.

Now that the temperature T is properly defined, and the equation of state f(p,V,T) = 0 of a substance has been determined by experiment, thermodynamics postulates that two functions, the *internal energy* U = U(p,T) and the *entropy* S = S(p,T), exist. These are written as functions of p,T but really they are functions of p,V,T with the restriction f(p,V,T) = 0 that makes them functions of two independent variables. Any two variables can be used to specify U and S: p,T; p,V; or V,T. Moreover, U or S can be solved for either of the independent variables, giving p = p(U,T) or even S = S(U,V). It turns out that most of classical thermodynamics is merely the working out of mathematical relations between p,V,T,U,S and other functions defined in terms of them for special purposes. In particular, there are lots of the partial derivatives that are a familar feature of classical thermodynamics. The validity of these mathematical deductions gives classical thermodynamics an impressive accuracy as a physical theory.

Classical thermodynamics is haughtily independent of the structure of matter, as if it existed in an ideal Platonic realm of pure reason. This is, alas, only an illusion, and thermodynamics is a creature of the quantum-mechanical atomic properties of matter. The fundamental concepts of entropy and temperature are given a rigorous basis by considering the mechanics of matter, and rest on the huge numbers of quantum states available to a macroscopic system, among which the system continually and rapidly wanders. For an account of this, see Boltzmann's Factor. Classical thermodynamics is an expression of the atomic structure of matter, just as Chemistry is. The atomic theory of thermodynamics is called *Statistical Mechanics*, for historical reasons. A macroscopic system is one that we can see with our eyes, if only in a microscope, while a microscopic system consists of only a few atoms. There is no sharp distinction; as we go to smaller and smaller systems, fluctuations around the values specified by thermodynamics increase, and concepts such as entropy have less and less significance. Thermodyamics just gradually evaporates.

The most important result of statistical mechanics is that the entropy of an isolated system never decreases, and is a maximum at equilibrium. This condition, which can be expressed as dS > 0, is called the *Second Law of thermodynamics*. It applies to an *isolated* system. If two systems are brought into thermal contact, the entropy of one may decrease, and the entropy of the other increase, but the net entropy change will be zero or positive. The existence of the functions U (and S as well) which depend only on the state of the system, constitutes the *First Law*. Statistical mechanics shows that the change in internal energy of a system dU = TdS - pdV (called the Thermodynamic Identity), where TdS is the heat transferred to the system, and pdV the work done by the system, in a reversible process. A reversible process is one in which the net increase (system + surroundings) in entropy is zero. Of course, dS may be greater than or less than zero, depending on whether the system absorbs or gives up heat. Unlike the pressure, volume and temperature, the energy and entropy of a system cannot be directly measured experimentally.

Now that we have covered the fundamentals of thermodynamics, we can proceed to make some mathematical deductions from them. The notation for partial derivatives that appears in the reasoning may appear formidable, but the meaning is clear and simple, specifying the ratio of the change in one quantity due to a change in another, with what other variables are held constant. In (∂x/∂y)_{z}, x is considered as a function of y and z, x = x(y,z), and the partial derivative is the ratio of dx to dy when z is held constant. Most of the functions we consider are functions of two independent variables, so this notation is very apt. For example, if x = 2yz + z^{2}, then the above partial derivative is 2z, while (∂x/∂z)_{y} = 2y + 2z. Also, (∂y/∂x)_{z} = 1/(∂x/∂y)_{z}.

First of all, let's consider all the partial derivatives that flow from the equation of state, f(p,V,T) = 0. We can write df = (∂f/∂p)dp + (∂f/∂V)dV + (∂f/∂T)dT = 0, which tells us how the differentials of the three variables are interrelated. For simplicity, we write (∂f/∂p) = f_{p}, and so on. There are six partials between pairs of variables, which can be expressed in terms of the partials of f. For example, if dT = 0, then f_{p}dp + f_{V}dV = 0, or (∂V/∂p)_{T} = -f_{p}/f_{V}. From symmetry, the general pattern is easy to see.

If we multiply the above result by 1 = f_{T}/f_{T}, we find that (∂V/∂p)_{T} = -(f_{p}/f_{T})/(f_{V}/f_{T}) = -(∂V/∂T)_{p}/(∂p/∂T)_{V}. This is a relation between the three partials that is difficult to see by inspection, but which we shall find useful.

Since dU = TdS - pdV, (∂U/∂S)_{V} = T, and (∂U/∂V)_{S} = -p. The cross second partials must be equal, so (∂T/∂S)_{V} = -(∂p/∂V)_{S}. This is one of *Maxwell's Relations*, which can prove very useful. We also say that the "natural variables" of U are S and V. Also, in a process where dS = dV = 0, we have dU = 0, which is the condition for an extremum of U, either a maximum or a minimum. Therefore, the equation also gives a *criterion of equilibrium*.

The function H = U + pV, called the *enthalpy*, can be handled the same way. dH = dU + pdV + Vdp = TdS + Vdp. The first partials give (∂H/∂S)_{p} = T and (∂H/∂p)_{S} = V. Equality of the second partials gives a second Maxwell relation: (∂T/∂p)_{S} = (∂V/∂S)_{p}. In a process at constant entropy and pressure, the enthalpy is an extremum.

The *free energy* F = U - TS. dF = dU - TdS - SdT = -pdV - SdT. In a process at constant temperature, which describes many actual processes, the work done is the decrease in the free energy, which is the source of the name. At constant temperature, the free energy F is a minimum at equilibrium, not the energy U. This is a very important result with many applications. As for partials, (∂F/∂V)_{T} = -p, and (∂F/∂T)_{V} = -S. The Maxwell relation is (∂p/∂T)_{V} = (∂S/∂V)_{T}.

Finally, the *Gibbs function* G = U - TS + pV, and dG = -SdT + Vdp. In a process at constant temperature and pressure, the Gibbs function is constant. When temperature and pressure are held constant, the Gibbs function is a minimum at equilibrium. If g is the Gibbs function per mole for the reactants in a chemical reaction, and g' the Gibbs function per mole for the products, then at equilibrium g = g'. A similar relation also holds for a liquid and its vapor in equilibrium at a temperature T and a pressure equal to the vapor pressure. (∂G/∂T)_{p} = -S and (∂G/∂p)_{T} = V. The Maxwell relation is (∂S/∂p)_{T} = -(∂V/∂T)_{p}.

The internal energy, enthalpy, free energy and Gibbs function are known as *thermodynamic potentials*, because their derivatives are of so much use. The free energy is also called the Helmholtz function, and the symbol A is often used in place of F. The Gibbs function per mole is called the *chemical potential*, which is of great importance. The meaning of the chemical potential is greatly clarified in statistical mechanics.

Although the internal energy cannot be measured directly, heat transfers affecting it can be. At constant volume for an ideal gas, dU = TdS = C_{V}dT, which defines the *specific heat* at constant volume, C_{V}. Specific heat is also called *heat capacity*. It is proportional to the amount of substance, so is usually specified per gram or per mole, in cal/g/K or J/mol/K. As a partial derivative, it is C_{V} = T(∂S/∂T)_{V}.

At constant pressure, we have dH = TdS = C_{p}dT, or C_{p} = T(∂S/∂T)_{p}. C_{p} is the specific heat at constant pressure, much easier to measure experimentally than the specific heat at constant volume. We can find U and H at any temperature T by integrating the corresponding specific heat from T = 0 to T. The knowledge of specific heats at low temperatures is very useful in the calculation of thermodynamic functions.

The entropy is a particularly important function, so it is useful to look at how it depends on the specific heats. Let's begin by considering S a function of T and V, S = S(T,V). Then, dS = (∂S/∂T)_{V}dT + (∂S/∂V)_{T}. The first partial is C_{V}/T, as we have just seen, and a Maxwell relation says the second is (∂p/∂T)_{V}. This gives TdS = C_{V}dT + T(∂p/∂T)_{V}dV. Now dS is given in terms of experimentally measurable quantities only. This is called the *First TdS Equation*.

If S is now considered as a function of p and T, S = S(p,T), we find dS = (∂S/∂T)_{p}dT + (∂S/∂p)_{T}dp. The first partial is C_{p}/T, while a Maxwell relation gives the second as -(∂V/∂T)_{p}. Therefore, TdS = C_{p}dT - T(∂V/∂T)_{p}dp. This is the *Second TdS Equation*. The TdS equations relate the specific heats to the entropy and to derivatives of the equation of state.

If we subtract the first TdS equation from the second, we get 0 = (C_{p} - C_{V})dT - T(∂V/∂T)_{p}dp - T(∂p/∂T)_{V}dV. If this is solved for dT, we find that dT = T[(∂V/∂T)_{p}/(C_{p} - C_{V})]dp + T[(∂p/∂T)_{V}/(C_{p} - C_{V})]dV. Now this is a relation between the three differentials dT, dp and dV, which we know are related by the equation of state. Considering T as a function of p and V, we have (∂T/∂p)_{V}dp + (∂T/∂V)_{p}dV. The coefficients of dp and dV in the two expressions for dT must be equal. The coefficients of dp give C_{p} - C_{V} = T(∂V/∂T)_{p}/(∂T/∂p)_{V} = T(∂V/∂T)_{p}(∂p/∂T)_{V}. At the top of the preceding section, we showed that (∂p/∂T)_{V} = -(∂V/∂T)_{p}(∂p/∂V)_{T}, so our result can be cast into the form C_{p} - C_{V} = -T(∂V/∂T)_{p}^{2}(∂p/∂V)_{T}. Since (∂p/∂V)_{T} for all substances is negative, this shows that C_{p} is always greater than C_{V}. The difference is expressed entirely in derivatives of the equation of state.

This equation has several interesting consequences. At T = 0, C_{p} = C_{V}, and they are also equal when (∂V/∂T)_{p} = 0, when there is a minimum or maximum of the density, as with water at 4°C. For an ideal gas, (∂V/∂T)_{p} = R/p and (∂p/∂V)_{t} = -RT/V^{2}, so that C_{p} - C_{V} = -T(R/p)^{2}(-RT/V^{2}) = R.

If we set dS = 0, then C_{p}/C_{V} = -[(∂V/∂T)_{p}/(∂p/∂T)_{V})] (∂p/∂V)_{S}. The last partial is just the ratio of dp to dV in this case, when dS = 0. Again we use our relation between three partials of the equation of state to find that C_{p}/C_{V} = (∂V/∂p)_{T}(∂p/∂V)_{S}. Now, the bulk modulus k = V(∂p/∂V), where isothermal or adiabatic condtions must be specified. Denoting C_{p}/C_{V} by γ, we have γk_{T} = k_{S}. The ratio of C_{p} to C_{V} is the ratio of the adiabatic (isentropic) bulk modulus to the isothermal bulk modulus. From this, we also conclude that the adiabatic bulk modulus is always greater than the isothermal bulk modulus, and that they are equal at T = 0 or extrema of the density as a function of temperature. The reciprocal of the bulk modulus, the *compressibility*, is often used instead.

These results for the difference and the ratio of the specific heats at constant pressure and constant volume are very important and useful. They are borne out to great precision by experiment, and show the power of the mathematical arguments of classical thermodynamics.

The volume thermal expansivity of a substance is β = (1/V)(∂V/∂T)_{p}. For an ideal gas, it is simply 1/T, so that absolute zero can be determined by measuring the thermal expansivity of a gas. At 0°C, β = 1/273.15, as we know. The isothermal bulk modulus of an ideal gas is the pressure. If you wish to try the equations on a non-gaseous substance, for mercury at T = 273K, C_{p} = 6.69 cal/gmol-K, specific volume 14.72 cm^{3}/gmol, β = 1.81 x 10^{-4}K^{-1} and k_{T} = 2.58 x 10^{11} dyne/cm^{2}. From this data, it is easy to find that C_{V} = 5.88 cal/gmol/K, and γ = 1.14. A gmol of mercury is 200.59 g. In terms of β and k, C_{p} - C_{V} = -TVβ^{2}k. If we are dealing with specific heats per mole, then V is the volume per mole.

Water is an unusual liquid, with a high specific heat of C_{p} = 18 cal/mol/K, and V = 18 cm^{3}/mol. At 0°C, β = -6.814 x 10^{-5} per K, but at 20°C it is +2.066 x 10^{-4} per K. At about 4°C, it is zero. k decreases steadily from 1.99 x 10^{10} dy/cm^{2} at 0°C to 2.21 x 10^{10} at 20°C. At 0°C, it is close to 2.04 x 10^{10}. At 20°C, C_{p} - C_{V} = 0.111 cal/mol/K, so γ = 1.006. The speed of sound in a liquid is c = √(γk/ρ). In water, this amounts to 1491 m/s at 20°C. In mercury, the figure is 1474 m/s.

Iron, or steel, has density 7.86 g/cc, molecular weight 55.847, so its specific volume is 7.11 cc/mol. β is three times the linear expansion coefficient, or 3.6 x 10^{-5} per K. k, calculated from Young's modulus E and the shear modulus μ by k = Eμ/(9μ - 3E), is 1.51 x 10^{12} dyne/cm^{2}. C_{p} is 0.1075 cal/g/K or 6.00 cal/mol/K. From these figures, C_{p} - C_{V} = 0.097 cal/mol/K. Therefore, C_{V} = 5.90 cal/mol/K and γ = 1.017. For solids and liquids, we see that the difference in the specific heats is small, and γ is about 1.0.

You may have noticed that C_{V} for mercury or for iron is not far from 6 cal/mol/K (that is, 3R). This is the rule of Dulong and Petit, reflecting the fact that a mole contains the same number of atoms of any substance. For a gram-mole this number is Avogadro's number, N_{A} = 6.02 x 10^{23}. N atoms have 6N vibrational degrees of freedom (3N for kinetic energy and 3N for potential energy), and at sufficiently high temperatures equipartition of energy gives each degree of freedom kT/2 of energy, or 3N_{A} kT = 3RT in all, which means a specific heat of 3R. This is evidence of the atomic structure of matter from classical thermodynamics, and is of the same kind as similar evidence in classical Chemistry. A more detailed investigation, such as Debye's theory, gives a better result and allows the behavior of the specific heat at low temperatures to be described, where it is proportional to T^{3}.

The change of ice to water, and water to steam, as a sample of H_{2}O is heated, provides a familiar example of *phase change*. The molecules are arranged differently in the three different manifestations, although the chemical composition remains the same. Ice is a rigid crystalline lattice, really a large molecule. Water is a transient association of molecular clusters locally with dodecahedral symmetry. Steam is a gas of mostly independent H_{2}O molecules, and perhaps some dimers. There are many ways to arrange the molecules, but these are the most stable of each general type. All are bound largely by strong hydrogen bonds between the protruding protons and the unbonded pairs of electrons on the oxygens. The strength of the hydrogen bonds makes H_{2}O an atypical substance. There needs to be a word for all appearances of H_{2}O, so that ice, water and steam can be restricted to the individual structures. Each of the distinct structures is a *phase*, with distinctive thermodynamic properties.

When water is heated at atmospheric pressure, it *boils* at about 100°C. At this temperature, the *vapor pressure* of water is equal to the atmospheric pressure, so bubbles of steam can form and the water can rapidly turn into steam as quickly as sufficient energy can be supplied. The water has been turning into steam all along, which has been leaving the surface of the water and diffusing into the atmosphere. What makes boiling distinct is the formation of the bubbles when the vapor pressure equals the atmospheric pressure. If the atmospheric pressure is changed, then so is the boiling point. In Denver, where the atmospheric pressure is about 850 mb, water boils around 95°C. It is noticed that the temperature remains constant at 100°C or whatever as long as there is any water left. Water can be superheated, and then may burst explosively into steam, and there are various kinetic phenomena associated with boiling, but when treated gently, it behaves pretty much as described.

If water is heated in a closed vessel in the absence of air, it does not boil in this way. At 0°C, say, the water lies at the bottom of the vessel, and above it is vapor (usually not called steam below 100°C) at a pressure of 4.579 mmHg. This is about the atmospheric pressure on Mars, so any water there would boil. If we increase the temperature to 100°C, the vapor pressure rises to 760 mmHg, roughly exponentially. Nothing particular happens here, and when we increase the temperature to 200°C, the pressure has risen to 11 659 mmHg, or 225 psi. That is, for every temperature there is a corresponding pressure, the vapor pressure, at which the vapor is in equilibrium with the liquid. If we allow some of the steam to escape, it is quickly replaced by more by evaporation of water to maintain the appropriate temperature. If the vessel is a cylinder fitted with a piston, and we press on the steam as if to increase its pressure, this does not happen, but some of the steam condenses to water so the pressure does not increase.

It is easily perceived that if some water changes to steam, heat must be added to maintain the temperature. At 100°C, the amount of heat is about 540 cal/g or 9720 cal/mol. This supplies the energy to break the hydrogen bonds and to give the liberated molecules a kinetic energy appropriate to the temperature (3kT/2 each). When heat was thought to be the fluid caloric, this represented caloric that was hidden in the water but suddenly became evident in the steam, where it lubricated the water into a gas. It was called the *latent heat* L, and the term is still used. A phase change of this type, where there is a latent heat, is called a *first-order* phase change. Since the temperature remains constant during the change, the entropy changes by ΔS = L/T. An abrupt change in entropy is characteristic of a first-order phase transition. An abrupt change in the density, or volume V, practically always accompanies the change in entropy. In the case of water at 100°C, the specific volume of the water is 1.0435 cc/g, and of the steam, 1673 cc/g. The entropy changes from 1.3071 J/g/K to 7.3546 J/g/K.

The phase change from water to steam is an excellent example of the competition between energy and entropy to determine the state of a substance. Energy says be water, since water molecules attract one another. Entropy says be steam, since steam molecules have so much more freedom. It is the minimum of F = U - TS that determines what happens (at constant V and T). At constant pressure, which is the case for the equilibrium between water and vapor, it is the Gibbs function that governs, and what the pressure is for any temperature is the point at which g_{s} = g_{w}, where g is the Gibbs function per gram or mole, and "s" refers to steam, "w" to water. If a small amount of water dm evaporates, the net change in Gibbs function will be g_{s}dm - g_{w}dm = 0.

If the temperature is changed by dT, then the Gibbs functions will change by amounts dG = -SdT + Vdp. For equilibrium, then, dg_{s} = dg_{w}, or -S_{s}dT + V_{s}dp = -S_{w}dT + V_{w}dp. Of course, dT and dp are the same for the water and the steam. Therefore, (S_{w} - S_{s})dT = (V_{w} - V_{s})dp, and dp/dT = (S_{s} - S_{w})/(V_{s} - V_{w}), or dp/dT = L/(TΔV), which is Clapeyron's equation. Clausius later demonstrated the role of the entropy, so the equation is also known as the Clausius-Clapeyron equation. This equation gives the slope of the curve of vapor pressure versus temperature.

For water at 100°C, we have dp/dT = (7.3546 - 1.3071)/(1673 - 1.0435) J/cc/K = 3.617 x 10^{4} (dy/cm^{2})/K = 27.13 mmHg/K. Experiment gives 787.57 mmHg at 101°C and 733.24 mmHg at 99°C, or dp/dT = 27.165 mmHg/K, which is good agreement. The reciprocal of this is 0.0368 K/mmHg, from which we can calculate the boiling point at Denver when the pressure is 850 mb or 638 mmHg. The result is 100 - (760 - 638)(0.0368) = 100 - 4.49 = 95.51°C.

The ideal gas law gives 1701 cc/g as the specific volume of water vapor at 100°C, which is not too far from 1673 cc/g. If the ideal gas law is used in Clapeyron's equation, we find dp/dT = MLp/RT^{2} if L is per gram, and dp/dT = Lp/RT^{2} if per mole. *Trouton's Rule* says that the entropy change at the normal boiling point at 1 atm is about 20 cal/mol/K, or 4.65 J/g/K, which is at least of the same order of magnitude as the actual 6.05 for water (water does not obey Trouton's Rule very well). The approximate equation dp/dT = 10p/T^{2} then results. At 1 atm and 373K, it gives 0.055 mmHg/K, which is not too bad. It should work for any liquid near its boiling point as a rough estimate.

Clapeyron's equation is useful in many situations. When applied to the ice-water transition, the fact that water is denser than ice means that dp/dT is negative, so as the pressure increases, the transition temperature decreases. This is an unusual situation, which occurs rarely, though antimony acts in the same way. Expansion on solidification is advantageous in casting type, since the solid must fill the mold completely. The heat of fusion of ice at 0°C is 1436 cal/mol or 79.72 cal/g. The specific volume of water is 1.000160 cc/g and the specific volume of ice is 1.09086 cc/g. This is common ice, hexagonal ice or ice I, where the oxygens are tetrahedrally stacked and there are tunnels through the crystal in the c-direction. The properties of ice are not listed in the Handbook of Chemistry and Physics, a strange thing for such an important substance. Ice also sublimates directly into vapor. At -10°C, the vapor pressure of ice is 1.950 mmHg, and at -20°C it is 0.776 mmHg. Of course, dp/dT is positive for the sublimation of ice. The entropy of sublimation is 44.674 cal/mol/K at 0°C, corresponding to a latent heat of 678 cal/g. The coefficient of cubical expansion, β is about 2 x 10^{-4} per K at 0°. At -15°C, the coefficients of linear expansion are 46 perpendicular to the c-axis, and 63 parallel to it, with β = 156, all times 10^{-6} per K. The adiabatic bulk modulus at about the same temperature is 7.81 x 10^{10} dyne/cm^{2}. The specific heat C_{p} of ice at 0°C is 9.09 cal/mol/K, or 0.505 cal/g/K, half that of water.

Another interesting example of the competition between energy and entropy is the existence of lattice vacancies in crystals above 0K. It requires energy to make a vacancy, but the entropy increase is so favorable (because the atoms can now be shuffled) that some vacancies always exist at positive temperatures. It is good to remember that this competition is taken care of by reasoning with the free energy F instead of with the energy U.

Water vapor and air form a nearly ideal gas mixture under normal conditions, with each gas acting as if the other did not exist. If the partial pressure of water vapor at 20°C is 17.535 mmHg, then it will be in equilibrium with water of that temperature. If we cool the air slightly, some water vapor must condense so that the pressure is appropriate to the lower temperature. This will appear as a fog, or as condensation on surfaces, which is dew. Therefore, the temperature at which the water vapor will be in equilibrium with liquid water is called the *dew-point temperature*. The *relative humidity* of a sample of air is the actual pressure of water vapor divided by the equilibrium vapor pressure at that temperature, time 100%. Air that has a relative humidity of 100% at 0°C, when brought into the house and heated to 20°C, will have a relative humidity of 4.579/17.535 x 100 = 26%.

The fact that dew forms when moist air is cooled is sometimes faultily expressed as "cool air holds less moisture than warm air." Well, air doesn't "hold" any moisture at all--it behaves quite independently of the water vapor. It's just that the equilibrium vapor pressure of water decreases with temperature.

Energy units, joule or erg, can be used for both heat and work, avoiding any unnecessary conversions or consideration of the "energy equivalent of heat." 1 J, a newton-metre, is 10^{7} erg. However, special heat units are still very often used, and can be quite convenient in practice. The calorie is the heat required to heat one gram of water by one kelvin near 20°C, about 4.186 J. The dieter's calorie is the Cal: the kcal or kilocalorie. The British thermal unit, or Btu, is the heat required to heat one pound of water (453.59 g) by one degree Rankine near 68°F, an energy of 778 ft-lb, or 252 cal, or 1055 J. If you want more accuracy, you get into a welter of slightly different definitions. The use of ft-lb for heat in engineering is rare. These figures are good enough for general purposes. Water has an unusually high specific heat, a factor that should be taken into consideration in estimates.

M. W. Zemansky, *Heat and Thermodynamics*, 4th ed. (New York: McGraw-Hill, 1957). Chapter 13, especially.

D. Eisenberg and W. Kauzmann, *The Structure and Properties of Water* (Oxford: Clarendon Press, 1969). This is a good reference on water substance. An extensive Internet search failed to turn up any useful information on the elastic properties of ice or on its strength. The Internet is a very spotty source of information; there are some wonderful things, like satellite photos and webcams, but rather few numbers.

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Composed by J. B. Calvert

Created 5 April 2003

Last revised