Calculation of the Fugacity of Naphthalene from Density Data
The saturation pressure of solid naphthalene can be represented by
\log _{10} P^{\text {sat }}(\mathrm{bar})=8.722-\frac{3783}{T} (T in K)
The density of the solid is 1.025 g/cm^3, and the molecular weight of naphthalene is 128.19.Estimate the fugacity of solid naphthalene at 35°C at its vapor pressure, and also at 1, 20,30, 40, 50, and 60 bar. (This information is used in Sec. 12.1 in a study of supercritical phase behavior.)
We start with Eq. 7.4-23 for a single solid phase with the assumption that the solid is incompressible:
f^{\mathrm{S}}(T, P)=P^{\mathrm{sat}}(T)\left(\frac{f}{P}\right)_{\mathrm{sat}, T} \exp \left[\frac{1}{R T} \sum_{J=1} \int_{P^{\mathrm{J}}}^{P^{\mathrm{J}+1}} \underline{V}^{\mathrm{J}} d P\right] (7.4.23)
f^{\mathrm{S}}(T, P)=P^{\mathrm{sat}}(T)\left(\frac{f}{P}\right)_{\mathrm{sat}, T} \exp \left[\frac{\underline{V}\left(P-P^{\text {sat }}(T)\right)}{R T}\right]
Using the data in the problem statement,
P^{\text {sat }}\left(35^{\circ} \mathrm{C}\right)=10^{[8.722-(3783 /(273.15+35))]}=2.789 \times 10^{-4} \mathrm{bar}
Since the sublimation pressure is so low, we can use (see Fig. 7.4-1)
\left(\frac{f}{P}\right)_{\mathrm{sat}}=1
Therefore,
f^{\mathrm{S}}\left(T=35^{\circ} \mathrm{C}, P^{\mathrm{sat}}\right)=2.789 \times 10^{-4} \mathrm{bar}
At any higher pressure, we have
\begin{aligned}&f^{\mathrm{S}}\left(T=35^{\circ} \mathrm{C}, P\right)=2.789 \times 10^{-4} \text { (bar) }\\\\&\times \exp \left[\frac{\frac{128.19}{1.025} \frac{\mathrm{cm}^{3}}{\mathrm{~mol}} \times\left(P-2.879 \times 10^{-4}\right) \text { bar } \times 10^{-6} \frac{\mathrm{m}^{3}}{\mathrm{~cm}^{3}}}{8.314 \times 10^{-5} \frac{\mathrm{bar} \mathrm{m}}{\mathrm{mol} \mathrm{K}} \times 308.15 \mathrm{~K}}\right]\end{aligned}
The result is
| P (bar) | f^s(3°C, P) (bar) |
| 2.789 × 10^{−4} | 2.789 × 10^{−4} |
| 1 | 2.803 × 10^{−4} |
| 10 | 2.933 × 10^{−4} |
| 20 | 3.083 × 10^{−4} |
| 30 | 3.241 × 10^{−4} |
| 40 | 3.408 × 10^{−4} |
| 50 | 3.582 × 10^{−4} |
| 60 | 3.766 × 10^{−4} |
