The - phase diagram of a pure substance Q is specified by the following readings

PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · Interacting systems, real gases and phase equilibria · Interacting systems, real gases and phase equilibria

Thermal Physics, Ch Phase coexistence and the Clausius-Clapeyron relation

Problem

The

p

T

phase diagram of a pure substance Q is specified by the following readings. Triple point:

210.0\,\mathrm{K}

8.00\,\mathrm{kPa}

. Critical point:

560\,\mathrm{K}

5.00\,\mathrm{MPa}

. Solid-liquid line: a straight line of slope

\mathrm{d}p/\mathrm{d}T = +2.00\,\mathrm{MPa\,K^{-1}}

through the triple point. Liquid-vapour line: passes through

101.3\,\mathrm{kPa}

320.0\,\mathrm{K}

. Latent heat of vaporisation at

320.0\,\mathrm{K}

L = 27.0\,\mathrm{kJ\,mol^{-1}}

. A separate temperature-composition diagram, at fixed pressure, is given for solid-liquid coexistence of Q with a second substance R, whose mole fraction is

x

. At one temperature the tie line runs from

x_L = 0.25

in the liquid to

x_S = 0.62

in the solid. ((a)) A sealed sample of pure Q is warmed slowly at a constant pressure of

101.3\,\mathrm{kPa}

from

150\,\mathrm{K}

500\,\mathrm{K}

. Give, to the nearest

0.1\,\mathrm{K}

, the two temperatures at which it changes phase (melting first, then boiling). ((b)) On this isobar the molar Gibbs functions

g_{\rm liq}(T)

and

g_{\rm vap}(T)

cross at the boiling point. Using

(\partial g/\partial T)_p = -s

, find the magnitude of the difference between the slopes of the two curves at the crossing. ((c)) A mixture of Q and R of overall composition

x_0 = 0.40

sits on the given tie line. Find the fraction of the material that is solid.

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