Thermodynamics

796 | ThermodynamicsH 2 → 2H0.1H 2 → 0.2H0.01H 2 → 0.02H0.001H 2 → 0.002Hn = 1 H 2nH = 2FIGURE 16–6The changes in the number of molesof the components during a chemicalreaction are proportional to thestoichiometric coefficients regardlessof the extent of the reaction.mixtures because the changes in the number of moles of the components areproportional to the stoichiometric coefficients (Fig. 16–6). That is,dN A en A dN C en C(16–8)dN B en B dN D en Dwhere e is the proportionality constant and represents the extent of a reaction.A minus sign is added to the first two terms because the number ofmoles of the reactants A and B decreases as the reaction progresses.For example, if the reactants are C 2 H 6 and O 2 and the products are CO 2and H 2 O, the reaction of 1 mmol (10 6 mol) of C 2 H 6 results in a 2-mmolincrease in CO 2 ,a 3-mmol increase in H 2 O, and a 3.5-mmol decrease in O 2in accordance with the stoichiometric equationC 2 H 6 3.5O 2 S2CO 2 3H 2 OThat is, the change in the number of moles of a component is one-millionth(e 10 6 ) of the stoichiometric coefficient of that component in this case.Substituting the relations in Eq. 16–8 into Eq. 16–6 and canceling e, weobtainn C g C n D g D n A g A n B g B 0(16–9)This equation involves the stoichiometric coefficients and the molar Gibbsfunctions of the reactants and the products, and it is known as the criterionfor chemical equilibrium. It is valid for any chemical reaction regardlessof the phases involved.Equation 16–9 is developed for a chemical reaction that involves tworeactants and two products for simplicity, but it can easily be modified tohandle chemical reactions with any number of reactants and products. Nextwe analyze the equilibrium criterion for ideal-gas mixtures.16–2 ■ THE EQUILIBRIUM CONSTANTFOR IDEAL-GAS MIXTURESConsider a mixture of ideal gases that exists in equilibrium at a specifiedtemperature and pressure. Like entropy, the Gibbs function of an ideal gasdepends on both the temperature and the pressure. The Gibbs function valuesare usually listed versus temperature at a fixed reference pressure P 0 ,which is taken to be 1 atm. The variation of the Gibbs function of an idealgas with pressure at a fixed temperature is determined by using the definitionof the Gibbs functionand the entropy-change relationfor isothermal processes 3 ¢ 1g h Ts 2s R u ln 1P 2 >P 1 24. It yields1¢g 2 T ¢h →0 T 1¢ s 2 T T 1¢ s 2 T R u T ln P 2Thus the Gibbs function of component i of an ideal-gas mixture at its partialpressure P i and mixture temperature T can be expressed asP 1gi 1T, P i 2 g * i1T2 R u T ln P i(16–10)