Thermodynamics

806 | ThermodynamicsSolving Eqs. (1), (2), (3), and (4) simultaneously for the four unknowns x, y,z, and w yieldsx 0.271 y 0.213z 1.849w 1.032Therefore, the equilibrium composition of 1 kmol H 2 O and 2 kmol O 2 at1 atm and 4000 K is0.271H 2 O 0.213H 2 1.849O 2 1.032OHDiscussion We could also solve this problem by using the K P relation for thestoichiometric reaction O 2 ∆ 2O as one of the two equations.Solving a system of simultaneous nonlinear equations is extremely tediousand time-consuming if it is done by hand. Thus it is often necessary to solvethese kinds of problems by using an equation solver such as EES.16–5 ■ VARIATION OF K P WITH TEMPERATUREIt was shown in Section 16–2 that the equilibrium constant K P of an idealgas depends on temperature only, and it is related to the standard-stateGibbs function change ∆G*(T) through the relation (Eq. 16–14)¢G* 1T2ln K P R u TIn this section we develop a relation for the variation of K P with temperaturein terms of other properties.Substituting ∆G*(T) ∆H*(T) T ∆S*(T) into the above relation anddifferentiating with respect to temperature, we getd 1ln K p 2 ¢H* 1T2 d 3 ¢H* 1T24 d 3 ¢S* 1T24 dT R u T 2 R u T dT R u dTAt constant pressure, the second Tdsrelation, Tds dh v dP, reduces toTds dh. Also, T d(∆S*) d(∆H*) since ∆S* and ∆H* consist of entropyand enthalpy terms of the reactants and the products. Therefore, the last twoterms in the above relation cancel, and it reduces tod 1ln K p 2dT¢H* 1T2 h R 1T2R u T 2 R u T 2(16–17)where h R 1T2 is the enthalpy of reaction at temperature T. Notice that wedropped the superscript * (which indicates a constant pressure of 1 atm)from ∆H(T), since the enthalpy of an ideal gas depends on temperature onlyand is independent of pressure. Equation 16–17 is an expression of the variationof K P with temperature in terms of h R 1T2, and it is known as the van’tHoff equation. To integrate it, we need to know how h varies with T. ForRsmall temperature intervals, h R can be treated as a constant and Eq. 16–17can be integrated to yieldln K P 2 h Ra 1 1 bK P1R u T 1 T 2(16–18)