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.
Parts
» COMPUTATIONAL MECHANICS and APPLIED ANALYSIS
» Explicit and Implicit Functions and Total Differentiation
» Exact (Perfect) and Inexact (Imperfect) Differentials
» Intermolecular Forces and Potential Energy
» Internal Energy, Temperature, Collision Number and Mean Free Path
» Vector or Cross Product r The area A due to a vector product
» First Law for a Closed System
» First Law For an Open System
» STATEMENTS OF THE SECOND LAW
» Cyclical Integral for a Reversible Heat Engine
» Irreversibility and Entropy of an Isolated System
» Degradation and Quality of Energy
» SINGLE–COMPONENT INCOMPRESSIBLE FLUIDS
» Evaluation of Entropy for a Control Volume
» Internally Reversible Work for an Open System
» MAXIMUM ENTROPY AND MINIMUM ENERGY
» Generalized Derivation for a Single Phase
» LaGrange Multiplier Method for Equilibrium
» Absolute and Relative Availability Under Interactions with Ambient
» Irreversibility or Lost Work
» Applications of the Availability Balance Equation
» Closed System (Non–Flow Systems)
» Heat Pumps and Refrigerators
» Work Producing and Consumption Devices
» Graphical Illustration of Lost, Isentropic, and Optimum Work
» Flow Processes or Heat Exchangers
» CLASSICAL RATIONALE FOR POSTULATORY APPROACH
» Generalized Legendre Transform
» Van der Waals (VW) Equation of State
» Other Two–Parameter Equations of State
» Compressibility Charts (Principle of Corresponding States)
» Boyle Temperature and Boyle Curves
» Three Parameter Equations of State
» Empirical Equations Of State
» State Equations for Liquids/Solids
» Internal Energy (du) Relation
» EXPERIMENTS TO MEASURE (u O – u)
» Vapor Pressure and the Clapeyron Equation
» Saturation Relations with Surface Tension Effects
» Temperature Change During Throttling
» Throttling in Closed Systems
» Procedure for Determining Thermodynamic Properties
» Euler and Gibbs–Duhem Equations
» Relationship Between Molal and Pure Properties
» Relations between Partial Molal and Pure Properties
» Mixing Rules for Equations of State
» Partial Molal Properties Using Mixture State Equations
» Ideal Solution and Raoult’s Law
» Completely Miscible Mixtures
» DEVIATIONS FROM RAOULT’S LAW
» Mathematical Criterion for Stability
» APPLICATION TO BOILING AND CONDENSATION
» Physical Processes and Stability
» Constant Temperature and Volume
» Equivalence Ratio, Stoichiometric Ratio
» Entropy, Gibbs Function, and Gibbs Function of Formation
» Entropy Generated During an Adiabatic Chemical Reaction
» MASS CONSERVATION AND MOLE BALANCE EQUATIONS
» Evaluation of Properties During an Irreversible Chemical Reaction
» Criteria in Terms of Chemical Force Potential
» Generalized Relation for the Chemical Potential
» Nonideal Mixtures and Solutions
» Gas, Liquid and Solid Mixtures
» Availability Balance Equation
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