Question 9.2: Calculate the specific heat at constant volume (J/K · mol) f...
Calculate the specific heat at constant volume (J/K · mol) for monatomic nitrogen at 3000 K.
From Eq. (9.7), the translational contribution to the specific heat at constant volume for any ideal gas is
\left(\frac{c_{\nu}}{R}\right)_{tr} = \left[\frac{\partial}{\partial T}T^{2}\left(\frac{\partial\ln Z_{tr}}{\partial T}\right) \right]_{V} = \frac{3}{2}. (9.7)
\left(\frac{c_{\nu}}{R}\right)_{tr} = \frac{3}{2},
whereas the electronic contribution, from Eq. (9.19), is
\left(\frac{c_{\nu}}{R}\right)_{el} = \left(\frac{c_{p}}{R}\right)_{el} = \frac{Z^{\prime\prime}_{el}}{Z_{el}}-\left(\frac{Z^{\prime }_{el}}{Z_{el}}\right)^{2}. (9.19)
\left(\frac{c_{\nu}}{R}\right)_{el} = \frac{Z^{\prime\prime}_{el}}{Z_{el}}-\left(\frac{Z^{\prime }_{el}}{Z_{el}}\right)^{2}.
The electronic contribution to any thermodynamic property can be easily determined by setting up the following table, where from Eq. (8.1)
\frac{\varepsilon}{k} = \frac{hc}{k}\widetilde{\varepsilon} = (1.4387_\ cm-K)\widetilde{\varepsilon}, (8.1)
\frac{\varepsilon_{j}}{kT} = \frac{hc}{k}\left(\frac{\widetilde{\varepsilon}_{j}}{T}\right) = (1.4387_\ cm-K)\frac{\widetilde{\varepsilon}_{j}}{T}.
| \widetilde{\varepsilon}_{j}(cm^{-1}) | g_{j} | \varepsilon_{j}/kT | g_{j}e^{-\varepsilon_{j}/kT} | g_{j}(\varepsilon_{j}/kT)e^{-\varepsilon_{j}/kT} | g_{j}(\varepsilon_{j}/kT)^{2}e^{-\varepsilon_{j}/kT} |
| 0 | 4 | 0 | 4 | 0 | 0 |
| 19,229 | 10 | 9.2216 | 9.8880×10^{-4} | 9.1183×10^{-3} | 8.4086×10^{-2} |
| 28,839 | 6 | 13.830 | 5.9137×10^{-6} | 8.1787×10^{-5} | 1.1311×10^{-3} |
| 4.0010 | 9.2001×10^{-3} | 8.5217×10^{-2} |
Here, we consider the first three electronic energy levels of atomic nitrogen, based on the listed term symbols and energies of Appendix J.1. The energy corresponding to the fourth level is 83,322 cm^{-1}, which proves much too high to produce any further influence on thermodynamic properties at the given temperature of 3000 K. The final row of the table contains sums for the last three columns, which conveniently represent Z_{el}, Z^{\prime}_{el}, and Z^{\prime\prime}_{el}, respectively. Employing the calculated data from this table, the electronic contribution to the specific heat at constant volume becomes
\left(\frac{c_{\nu}}{R}\right)_{el} = \frac{Z^{\prime\prime}_{el}}{Z_{el}}-\left(\frac{Z^{\prime}_{el}}{Z_{el}}\right)^{2} = \frac{8.5217\times10^{-2}}{4.0010}-\left(\frac{9.2001×10^{-3}}{4.0010}\right)^{2} = 0.02129.
Hence, summing the translational and electronic contributions, the dimensionless specific heat at constant volume for monatomic nitrogen at 3000 K is
\frac{c_{\nu}}{R} = \left(\frac{c_{\nu}}{R}\right)_{tr}+\left(\frac{c_{\nu}}{R}\right)_{el} = 1.5000 + 0.0213 = 1.5213,
so that
c_{\nu} = 1.5213 R = 1.5213(8.3145 J/K · mol) = 12.649 J/K · mol .