# V 1 To T 2 example essay topic

Substituting equation (1-2) in equation (1-3), we obtain (1-4) dH = T dS + V dP. From the defining the Helmholtz function A we obtain (1-5) dA = dU - d (TS) = dU - T dS - S dT. Substituting equation (1-2) in equation (1-5) (1-6) dA = -S dT - P dV. From the Gibbs Free Energy equation and equation (1.4) (1.7) dG = -S dT + V dP. We have in equations (1-2), (1-4), (1-6), and (1-7) expressed dU, dH dA, and dG in terms of P, V, T, and S. We know that thermodynamic properties have exact differentials. If a property M is a function of x and y, (1.7 a) M = M (x, y) then a differential change in M, dM, is the sum of the amount that M changes in the interval dx, with y held constant, plus the amount that M changes in the interval dy, with x held constant (see figure 1.1), or (1.8) dM = (M / X) y dx + (M / Y) x dy.

The terms (M / X) y and (M / Y) x are called partial derivatives of M and dM is called total differential. Equation (6-8) can be written (6.9) dM = B dx + C dy, where B and C represent (M / X) y and (M / Y) x respectively. Now equations (1.2), (1.4), (1.6), and (1.7) are total differentials, and have the same form as equation (1.9). This equation is called the Euler criterion for integrability or exactness.

Thus if CV is known as a function of T and V and the equation of state is known, then the entropy change can be calculated by integration. For change in state from T 1, V 1 to T 2, V 2, the entropy change S 2-S 1 is (1-23) S 2 - S 1 = T 1∫ T 2 CV d ln T + V 1∫ V 2 pα V dV, the first integration being carried out at constant volume of V 1 and the second integration at a constant temperature of T 2 (or the second integral being evaluated first at T 1 and then the first integral evaluated at V 2). Actually, it is not necessary to have complete knowledge of CV as a function of T and V if the equation of state is known. Next, before I shall proceed to show how dependence of CV on V can be calculated from the equation of state, I should point out that the method used in this first example is applicable to the calculation of the change of any state function, and is not restricted to the calculation of entropy changes. Furthermore, the independent variables do not have to be T and V as in this example, but can be T and p or p and V. (b) Dependence of Heat Capacities on p and V The Maxwell relation can also be used to relate the change in heat capacity with pressure or volume to the elastic properties of the system. Equation (1-26) and (1-27) state that if we know CV or Cp at one point on an isotherm, we can obtain their values at any other point on the isotherm if equation of state data are available.

For instance, we can always integrate equation (1-26), by numerical methods if necessary, (1.28) CV 1 - CV 2 = TV 1∫ V 2 (∂ 2 p/∂ T 2) V dV (T constant) a similar result holding for Cp. (c) Calculations of Temperature Changes The Maxwell relation are also useful in the calculation of temperature changes accompanying a change in state. I shall illustrate two types of processes: (1) constant entropy process, and (2) constant energy process. The general calculation procedure is the same for both cases: first differential expression is written for the quantity held constant, and this is equated to zero. The resulting equation is then obtained algebraically for dT, and a Maxwell relation is used to obtain the final expression involving only heat capacities and the equation of state. (1) Constant Entropy Processes. An experimental example of a constant entropy process is a reversible adiabatic change in state.

Clearly T will always be one of the independent variables chosen, but whether the second variable is V or p depends on the problem. An adiabatic process in which no work is performed is an example of a constant energy process. I choose the second independent variable to be V, because V is a natural independent variable for E (1-32) dE = (∂ E/∂ T) V dT + (∂ E/∂ V) T dV = 0 Solving for the ratio (∂ T/∂ V) E, which is called the Joule coefficient and denoted by η , we obtain (1-33) η ≡ (∂ T/∂ V) E = -1/CV (∂ E/∂ V) T. where we have used CV = (∂ E/∂ T) V. So far I have used only the first law, but to express (∂ E/∂ V) V in terms of quantities obtainable form an equation of state I must use the second law as well. Since all the quantities in this additional term are positive, we see that the temperature drop is greater in the isentropic process. We would expect this, since the work which appeared in the surroundings did so at the expense of the internal energy of the system. (b) Relationship between Heat Capacities As a further example of the use of Maxwell's relation I shall show how one heat capacity can be determined from another heat capacity plus equation of state data, in connection with the definition of the enthalpy which is (1-36) Cp - CV = [p + (∂ E/∂ V) T] (∂ V/∂ T) V. Notice that he left-hand side of this equation involves only heat quantities and the right-hand side involves work quantities (per temperature increase dT at constant p).

The term in p is the work involved in pushing back the external atmosphere when the volume increased by dV; the term in (∂ E/∂ V) T can be interpreted as a sort if internal work. If (∂ E/∂ V) T is zero, then this equation can be used to calculate the mechanical equivalent of heat from data on the specific heats and equation of state of gases. This was first done by J.R. Mayer some years before Joule's paddle wheel experiment, but the indirect calculation made little impression on contemporary physicist. The formulation of the first law had to await the direct, painstaking work of Joule to be convincing. The same result could have been obtained a little more directly by using the second law from the beginning and writing (1-38) dS = (∂ S/∂ T) V dT + (∂ S/∂ V) T dV. Use the Maxwell relation (1-17) then gives equation (1-37).

The only slight danger in starting a derivation immediately with the second law is that a desired result may not really depend on the second law at all. The result will of course be correct, but an erroneous idea of the basis if the result will be obtained. This concludes by report on Thermodynamics Maxwell relation, where I showed hoe to derive them and also gave several examples of how to illustrate the Maxwell relation in Thermodynamics calculations.

Bibliography

An introduction to thermodynamics with some new derivation based on real irreversible process by R.S. Silver. Cambridge (Eng.) University Press, 1971.

Thermal Physics by Phillip M. Morse. Edition 2d ed. New York, W.A. Benjamin, 1969.

Introduction to thermodynamics: classical and statistical by Richard E. Sonntag and Gordon J. Van Wyle n. New York, Wiley 1971.