Callen H.B - Thermodynamics And An Introduction To Thermostatistics · Problem 6.4-1

MAXWELL RELATIONS

Physics — Statistical Mechanics · gibbs-potential-chemical-reaction-equilibrium

Problem

One half mole of

\mathrm{H_2S}

\frac{3}{4}

mole of

\mathrm{H_2O}

, 2 moles of

\mathrm{H_2}

, and 1 mole of

\mathrm{SO_2}

are allowed to react in a vessel maintained at a temperature of 300 K and a pressure of

10^5

Pa. The components can react by the chemical reaction

3\mathrm{H_2} + \mathrm{SO_2} \rightleftharpoons \mathrm{H_2S} + 2\mathrm{H_2O}

a

) Write the condition of equilibrium in terms of the partial molar Gibbs potentials.

b

) Show that

N_{\mathrm{H_2}} = 2 - 3\Delta\tilde{N}

and similarly for the other components. For what value of

\Delta\tilde{N}

does each

N_j

vanish?

c

) Show that

\Delta\tilde{N}_{\max} = \frac{2}{3}

and

\Delta\tilde{N}_{\min} = -\frac{3}{8}

. Which components are depleted in each of these cases?

d

) Assume that the nominal solution of the equilibrium condition gives

\Delta\tilde{N} = \frac{1}{4}

. What is the degree of reaction

\varepsilon

? What are the mole fractions of each of the components in the equilibrium mixture?

e

) Assume that the pressure is raised and that the nominal solution of the equilibrium condition now yields the value

\Delta\tilde{N} = 0.8

. What is the degree of reaction? What are the mole fractions of each of the components in the final state? Answers:

c

)

\mathrm{H_2}

and

\mathrm{H_2O}

depleted

d

)

\varepsilon = \frac{3}{5}

x_{\mathrm{H_2O}} = \frac{5}{16}

e

)

\Delta\tilde{N} = \frac{2}{3}

x_{\mathrm{H_2O}} = .59