Calculation of the Fugacity of Ice from Density Data
Compute the fugacity of ice at −5°C and pressures of 0.1 MPa, 10 MPa, and 100 MPa.
Question 7.4.8: Calculation of the Fugacity of Ice from Density Data Compute...
We assume that ice is incompressible over the range from 0.4 kPa to 100 MPa, so that its molar volume over this pressure range is
\underline{V}=\frac{18.015 \mathrm{~g} / \mathrm{mol}}{0.915 \mathrm{~g} / \mathrm{cc}}=19.69 \mathrm{cc} / \mathrm{mol}=1.969 \times 10^{-5} \mathrm{~m}^{3} / \mathrm{mol}
From Eq. 7.4-24b, we then have
f^{\mathrm{S}}(T, P)=P^{\mathrm{sat}}(T) \exp \left[\frac{\underline{V}^{\mathrm{S}}\left(P-P^{\text {sat }}(T)\right)}{R T}\right] (7.4-24b)
\begin{aligned}f_{\text {ice }}\left(-5^{\circ} \mathrm{C}, 0.1 \mathrm{MPa}\right) &=0.4 \mathrm{kPa} \exp \left[\frac{1.969 \times 10^{-5} \frac{\mathrm{m}^{3}}{\mathrm{~mol}} \times(0.1-0.0004) \mathrm{MPa}}{268.15 \mathrm{~K} \times 8.314 \times 10^{-6} \frac{\mathrm{MPa} \mathrm{m}^{3}}{\mathrm{~mol} \mathrm{~K}}}\right] \\\\&=0.4 \exp \left[8.797 \times 10^{-4}\right] \mathrm{kPa} \\\\&=0.4 \times 1.00088 \mathrm{kPa} \\\\&=0.4004 \mathrm{kPa}\end{aligned}
Similarly,
f_{\text {ice }}\left(-5^{\circ} \mathrm{C}, 10 \mathrm{MPa}\right)=0.4 \mathrm{kPa} \mathrm{exp}\left[\frac{1.969 \times 10^{-5} \frac{\mathrm{m}^{3}}{\mathrm{~mol}} \times(10-0.0004) \mathrm{MPa}}{268.15 \mathrm{~K} \times 8.314 \times 10^{-6} \frac{\mathrm{MPa} \mathrm{m}^{3}}{\mathrm{~mol} \mathrm{~K}}}\right]
= 0.4369 kPa
and
f_{\text {ice }}\left(-5^{\circ} \mathrm{C}, 10 \mathrm{MPa}\right)=0.4 \mathrm{kPa} \mathrm{exp}\left[\frac{1.969 \times 10^{-5} \frac{\mathrm{m}^{3}}{\mathrm{~mol}} \times(100-0.0004) \mathrm{MPa}}{268.15 \mathrm{~K} \times 8.314 \times 10^{-6} \frac{\mathrm{MPa} \mathrm{m}^{3}}{\mathrm{~mol} \mathrm{~K}}}\right]
= 0.9674 kPa
COMMENT
Generally, the change in fugacity of a condensed species (liquid or solid) with small pressure changes is small. Here we find that the fugacity of ice increases by 10 percent for an increase in pressure from the sublimation pressure to 100 bar, and by a factor of 2.5 as the pressure on ice increases to 1000 bar.
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