Gallium melts at under

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

Gallium melts at 302.9 K302.9\,\mathrm{K} under 101.3 kPa101.3\,\mathrm{kPa}, with latent heat of fusion Lfus=5.59 kJ mol−1L_{\rm fus} = 5.59\,\mathrm{kJ\,mol^{-1}}. Its molar mass is 69.72 g mol−169.72\,\mathrm{g\,mol^{-1}}; at the melting point the solid's density is 5910 kg m−35910\,\mathrm{kg\,m^{-3}} and the liquid's is 6095 kg m−36095\,\mathrm{kg\,m^{-3}}. Both phases may be treated as incompressible over the pressure range used here. Solid and liquid coexist where their molar chemical potentials are equal. By stepping along that line with dμ=−s dT+v dp\mathrm{d}\mu = -s\,\mathrm{d}T + v\,\mathrm{d}p for each phase, derive the slope of the melting line, and then find: ((a)) the change of molar volume on melting, Δv=vliq−vsol\Delta v = v_{\rm liq} - v_{\rm sol}, in cm3 mol−1\mathrm{cm^3\,mol^{-1}}; ((b)) the slope dp/dT\mathrm{d}p/\mathrm{d}T of gallium's melting line, in MPa K−1\mathrm{MPa\,K^{-1}}, with its sign; ((c)) the pressure rise, in MPa\mathrm{MPa}, needed to lower gallium's melting point by 1.00 K1.00\,\mathrm{K}.

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