Over a limited range of temperatures we can often assume

Question

Free Energy and Chemical Thermodynamics Figure.

Experimental phase diagram for nitrogen and oxygen at atmospheric pressure. Data from International Critical Tables (volume 3), with endpoints adjusted to values in Lide (1994).

1. In this problem you will derive approximate formulas for the shapes of the phase boundary curves in diagrams such as Figures 5.31 and 5.32, assuming that both phases behave as ideal mixtures. For definiteness, suppose that the phases are liquid and gas.

(a) Show that in an ideal mixture of A and B, the chemical potential of species A can be written

μA = μ^o A + kT In(1 - x),

where μ^o A is the chemical potential of pure A (at the same temperature and pressure) and x = NB/(NA + NB). Derive a similar formula for the chemical potential of species B. Note that both formulas can be written for either the liquid phase or the gas phase.

(b) At any given temperature T, let xl and x9 be the compositions of the liquid and gas phases that are in equilibrium with each other. By setting the appropriate chemical potentials equal to each other, show that xl and xg obey the equations

(1-x1)/(1-xg) = e^ΔGoA/RT   annd x1/ xg = e^ΔGoA/RT

where ΔG^o represents the change in G for the pure substance undergoing the phase change at temperature T.

(c) Over a limited range of temperatures, we can often assume that the main temperature dependence of ΔG^o = ΔH^o - T ΔS^o comes from the explicit T; both ΔH^o and ΔS^o are approximately constant. With this simplification, rewrite the results of part (b) entirely in terms of ΔH_A^o,ΔH_B^o, TA, and TB (eliminating ΔG and ΔS). Solve for xl and xg as functions of T.

(d) Plot your results for the nitrogen-oxygen system. The latent heats of the pure substances are ΔH^o_N2 = 5570 J/mol and ΔH^o_O2 = 6820 J/mol. Com-pare to the experimental diagram, Figure 5.31.