第五章 Auxiliary Functions.pdf - Chapter 5 Auxiliary...

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Chapter 5 Auxiliary Functions

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5.1 Introduction Combination of the first and second laws of Thermodynamics leads to the derivation of dU = TdS PdV Combination of the First and Second Law also provides the criteria for equilibrium that 1. In a system of constant internal energy and constant volume, the entropy has its maximum value, and 2. In a system of constant entropy and constant volume, the internal energy has its minimum value. In this chapter the thermodynamic functions A (the Helmholtz free energy), G (the Gibbs free energy), and i (the chemical potential of the species i) are introduced, and their properties and interrelationships are examined. The functions A and G are defined as A= U - TS (5.1) and G = U + PV - TS= H -TS (5.2)
5.2 The Enthalpy H p p p q H H H PV U H q V P U V P U V V P q U U 1 2 1 1 1 2 2 2 1 2 1 2 ) ( ) ( ) (

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5.3 The Helmholtz Free Energy A For a system undergoing a change of state from state 1 to state 2 (A 2 - A 1 ) = (U 2 - U 1 ) - (T 2 S 2 T 1 S 1 ) and, if the system is closed, (U 2 - U 1 ) = q -w in which case (A 2 - A 1 ) = q- w - (T 2 S 2 T 1 S 1 ) If the process is isothermal, that is, T 2 = T 1 = T, the temperature of the heat reservoir which supplies or withdraws heat during the process, then, from the Second Law, q (T 2 S 2 T 1 S 1 ) q - (T 2 S 2 T 1 S 1 ) ≤ 0→ (T 2 S 2 T 1 S 1 ) q ≥ 0 (A 2 - A 1 ) + (T 2 S 2 T 1 S 1 ) q = - w → (A 2 - A 1 ) ≤ - w (A 2 - A 1 ) + T S irr = - w irr irr rev irr irr rev S T q q dS T q T q
5.3 The Helmholtz Free Energy A (A 2 - A 1 ) + T S irr = - w during a reversible isothermal process, for which S irr is zero, the amount of work done by the system w max is equal to the decrease in the value of the Helmholtz free energy. Furthermore, for an isothermal process conducted at constant volume. (A 2 - A 1 ) + T S irr = 0 → dA + TdS irr = 0 → dA ≤ 0 (dS irr 0) Thus in a closed system held at constant T and V, the Helmholtz free energy can only decrease or remain constant, and equilibrium is attained in such a system when A achieves its minimum value. The Helmholtz free energy thus provides a criterion for equilibrium in a system at constant temperature and constant volume.

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5.3 The Helmholtz Free Energy A Consider n atoms of some element occurring in both a solid crystalline phase and a vapor phase contained in a constant-volume vessel, which, in turn, is immersed in a constant-temperature heat reservoir. The two extreme states of existence which are available to the system are 1. That in which all of the atoms exist in the solid crystalline phase and none occurs in the vapor phase, and 2. That in which all of the atoms exist in the vapor phase and none occurs in the solid phase. solid crystalline phase vapor phase
5.3 The Helmholtz Free Energy A q q U min , S min U max , S max U , S U ↑↑ , S ↑↑ If an atom is to be removed from the surface of a solid and placed in a gas phase, work must be done against the attractive forces operating between the atom and its neighbors. For the

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