# chapter4 - Chapter 4 Equilibrium and Phase Stability in...

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Chapter 4 Equilibrium and Phase Stability in One-Component Systems The purpose of this chapter is to provide answers to the following questions: ( i ) under what conditions are the two phases of a pure substance in equilibrium with each other?, ( ii ) at a given temperature and pressure, in which form does a pure substance exist - solid, liquid, or gas? 4.1 EQUILIBRIUM CRITERIA For a closed system, the second law of thermodynamics, Eq. (1.6-8), states that TdS δQ 0 (4.1-1) From the f rst law of thermodynamics dU = δQ PdV (4.1-2) Elimination of δQ between Eqs. (4.1-1) and (4.1-2) leads to TdS dU PdV 0 (4.1-3) 4.1.1 Maximum Entropy Under the conditions of constant internal energy and volume, Eq. (4.1-3) simpli f es to dS U,V 0 (4.1-4) where the subscripts U and V imply that these quantities are kept constant during a process. Under these conditions, any spontaneous process tends to increase the entropy of a system as shown in Figure 4.1. When the system is at equilibrium, entropy reaches its maximum value and dS U,V =0 At equilibrium (4.1-5) Direction of a spontaneous process 0 = dS State of System Equilibrium Constant U & V Entropy Figure 4.1 The equilibrium condition for constant internal energy and volume. 77

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4.1.2 Minimum Internal Energy When entropy and volume are kept constant, Eq. (4.1-3) simpli f es to dU S,V 0 (4.1-6) Therefore, the internal energy of a system decreases as a result of a spontaneous process under the conditions of constant entropy and volume. This phenomenon can be explained as follows. Spontaneous processes lead to an increase in entropy. In order to keep entropy constant, heat must be transferred to the surroundings with a concomitant decrease in internal energy. When the system is at equilibrium, internal energy reaches its minimum value and dU S,V =0 At equilibrium (4.1-7) 4.1.3 Minimum Helmholtz Energy In di f erential form, Helmholtz energy is given by dA = d ( U TS )= dU SdT TdS (4.1-8) The use of Eq. (4.1-8) in Eq. (4.1-3) gives dA + SdT + PdV 0 (4.1-9) When temperature and volume are kept constant, Eq. (4.1-9) simpli f es to dA T,V 0 (4.1-10) Therefore, the Helmholtz energy of a system decreases as a result of a spontaneous process under the conditions of constant temperature and volume. When the system is at equilibrium, Helmholtz energy reaches its minimum value and dA T,V =0 At equilibrium (4.1-11) 4.1.4 Minimum Gibbs Energy In di f erential form, Gibbs energy is given by dG = d ( H TS )= d ( U + PV TS ) = dU + PdV + VdP TdS SdT (4.1-12) The use of Eq. (4.1-12) in Eq. (4.1-3) gives dG + SdT VdP 0 (4.1-13) When temperature and pressure are kept constant, Eq. (4.1-13) simpli f es to dG T,P 0 (4.1-14) Under these conditions, any spontaneous process tends to decrease the Gibbs energy of a system as shown in Figure 4.2. When the system is at equilibrium, Gibbs energy reaches its minimum value and dG T,P =0 At equilibrium (4.1-15) 78
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chapter4 - Chapter 4 Equilibrium and Phase Stability in...

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