Maxwell relations entropy derivation with

To evaluate the derivative (dU/dV)T, we start from expression (5.46a) for dU in terms of its natural variables S, V, differentiate with respect to V at constant T, and use the Maxwell relation (5.49c) to replace the entropy derivative,... 

James Clerk Maxwell first conceived the statistical basis of the second law of thermodynamics in 1871, and he is considered the founder of statistical thermodynamics. He was also famous for his electromagnetic wave theory and kinetic theory of gases. Maxwell derived the Maxwell Relations which were mathematical formulations used for advanced study of thermodynamics. Ludwig Boltzmann continued with Maxwell s kinetic theory of gases and arrived at important conclusions regarding dissipation of energy and increase in entropy. 

The second derivative of the four measures of energy, i.e., internal energy, U, enthalpy, H, Gibbs free energy, G, and Helmholz free energy. A, can be obtained with respect to two independent variables from among temperatnre, pressnre, volnme, and entropy. The order of differentiation does not matter as long as the function is analytic. This property is used to derive the Maxwell relations as follows ... 

Equations (5.8.5) may be used to obtain 24 Maxwell relations by cross-differentiation these are listed in Table 5.8.1. Several of these arise as trivial modifications of those specified in Section 1.12. The new expressions involve partial derivatives of either fyd rP or fyd rM with respect to independent variables. The order of differentiation and integration may not be interchanged unless the volume is kept fixed. If Vis variable, the machinery of Section 1.3 and/or an adaptation of Eq. (1.3.6) must be employed to evaluate the pertinent derivatives some examples are provided below. A number of interrelations are useful for starting further derivations they show, for example, how Pm varies with q or Tfo under a variety of fixed conditions (lines h, 1, t, and x), or how the entropy depends on electromagnetic fields (lines a, e, g, and k). Still others pertain to an interrelation between electric or magnetic polarizations in applied fields under various constraints (lines c, i, o, and u). Other interrelations specify either electrostrictive or magnetostrictive effects (lines b, f, n, and r). We shall later reexamine some of these phenomena. 

The above relations are known as Maxwell Equations. Eqs. (1.13.10) and (1.13.12) are particularly useful if the equations of state is known in the form P = P(V,T) or y = V(P, T) for any given material then its entropy may be determined by integration of the partial differential equation with respect to P or V More generally, these expressions are used to eliminate partial derivatives of the entropy in favor of temperature derivatives of pressure or volume, quantities that are directly accessible by experiment. 

The above 12 relations are known as Maxwell equations. Again, the question arises what good are they Here, Eqs. [7] and [8] are particularly useful because entropy meters are not available so, Eqs. [7] and [8] are used to eliminate the unknown partial derivatives of entropy with respect to pressure or volume—which raise their ugly heads in many of our subsequent operations—in favor of the known partial derivatives of the equation of state. This also addresses the awkward situation of Eqs. (1.12.1), where the two state functions E and H are expressed in terms of another state function, S. The actual procedure will be illustrated shortly. It should be obvious how [5], [6] or [11], [12], coupled with the equation of state, may be employed to determine the chemical potential under a variety of different experimental conditions. Knowledge of this quantity will come in handy later on. Further, [9] and [10]... 

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