CLASSICAL RATIONALE FOR POSTULATORY APPROACH

B. CLASSICAL RATIONALE FOR POSTULATORY APPROACH

We have seen that a Stable Equilibrium State (SES) is achieved when the entropy reaches a maximum value for fixed values of U, V and m (or for fixed number of moles N 1 , N 2 , …). The internal energy of an open system that exchanges mass with its surroundings is represented by the relation

(1) which is also known as the energy fundamental equation. The internal energy is a single valued

U = U(S, V, N 1 ,N 2 ,N 3 , ...),

function of S,V, N 1 ,N 2 ,N 3 , …, since there is a single stable equilibrium state for a specified set of conditions. Upon differentiating Eq. (1) we obtain the relation

fixed

Figure 1: Plot of U vs. S at specified values of V.

V fixed

Figure 2: A plot of U vs. S plot at specified values of V showing similar temperatures (slopes) at two different values of S.

dU = TdS – PdV + Σµ k dN k , where (2) ∂U/∂S = F = T is the affinity T or force driving heat transfer (cf. Figure 1 ), ∂U/∂V = F P = –P is

the affinity driving mechanical work, and ∂U/∂N k = F m,k = µ k is the affinity that drives mass transfer (say, during a chemical reaction or a phase transition). In general, the partial derivative ∂U/∂ξ j represents the force driving a parameter ξ j . For instance, if ξ i = N i , i.e., the number of moles N i of the i–th species in the system, then ∂U/∂N i = µ i represents the chemical potential of that species.

Rewriting Eq.(2), dS = (1/T)dU + (P/T)dV – Σ(µ i /T)dN i , or S = S(U, V, N 1 ,N 2 , ...).

(3) Equation (3) is called the entropy fundamental equation. It implies that the equilibrium states

are described by the extensive set of properties (U, V, N 1 , ..., N n ), which is also known as Postulate I. We have seen that the entropy is a single valued function (for prescribed values of U,

V, and m), since there is a unique stable equilibrium state. The fundamental equation (cf. Eq. (3)) is written in terms of extensive parameters. Each sub–system in a composite system can be described by the fundamental equation. However, the equation cannot be applied to the com- posite system itself. (This is also called Postulate II.) Equations (1) and (3) are, respectively, the energy and entropy representations of the fundamental equation, and these are valid only in

a positive coordinate system in which the values of the variables U, V, N 1 , ... , S > 0. We can define ( ∂S/∂U) V, N i = F T = 1/T, T( ∂S/∂V) U, N i =F P = P, etc. for any system that changes from

one equilibrium state to another. This definition fails for systems that do not exist in equilib- rium states.

It is possible to generalize Eq. (3) in the rate form as dS/dt = Σ(dS/dx k )(dx k /dt) = ΣF k J k .

(4) where x k ‘s are U, V, N 1 ,N 2 etc.. For a single component system, Eq. (4) yields S = S(U,V,N)

or S = S(U,V,M) so that s = s(u,v), i.e., 1/T = ( ∂s/∂u) v , and P/T = ( ∂s/∂v) u .

(5) These relations are valid only in the octant where (u,v > 0), and 1/T and P/T represent tangents

to the S–surface in the (u,v) plane. It is possible to have identical values of P/T and 1/T for various combinations of values of s, u and v. Examples of this include the saturated liquid and saturated vapor states for a specified pressure. Even though these states correspond to identical P and T, the values of s, u and v differ for the two states (cf. Figure 2 ).

1. Simple Compressible Substance

Since Eq.(1) is a first order homogeneous equation, the application of Euler equation (cf. Chapter 1) yields

S +V

(6) ∂ S

=U .

Using the partial derivatives described by Eqs.(5), TS – PV + Σµ i N i = U.

(7) The total differentiation of Eq. (7) and use of Eq.(2) yields the Gibbs–Duhem (G–D) equation,

namely,

(8) Equation (8) gives the intensive equation of state and it is apparent that

SdT – VdP + ΣN i d µ i =0

(9) Dividing Eq.(8) by N and solving for d µ (for, say, species 1), x 1 d µ 1 = – (S/N) dT + (V/N)dP – x 2 d µ 2 –x 3 d µ 3 –…–x n d µ n .

T = T(P, µ 1 , µ 2 , ...).

(10) Since x 1 =1– x 2 –x 3 –…–x n , this results in an intensive equation of state that is the zeroth

order homogeneous function

(11) As before, we see that (n+1) intensive properties describe the intensive state of an

µ 1 = f(T, P, x 2 ,x 3 ,x 4 , ..., x n ).

n–component simple compressible substance.