Question 9.5: Employing the semirigorous model, calculate the Gibbs free e...

From Eq. (4.39), the specific Gibbs free energy can be determined from

\frac{g}{RT} = -\ln\left(\frac{Z}{N}\right);

thus, from Eq. (9.71), we require the semirigorous evaluation,

Z = Z_{tr}Z_{el}Z^{\circ}_{R-V}Z_{corr},                (9.71)

\frac{Z}{N} = \frac{Z_{tr}}{N}Z_{el}Z^{\circ}_{R-V}Z_{corr}.

The translational contribution to the molecular partition function can be obtained most directly from the result found in Problem 4.1, i.e.,

The electronic contribution can be determined from the data in Appendix K.2, as evaluated in the following table. We again convert from the T_{e}-formulation to the T_{\circ}-formulation to account for our common zero of energy. From Fig. 9.1, we recognize that, for the complex model, the conversion can be made via

T_{\circ}=T_{e}+\frac{1}{2}\left(\omega_{e}^{\prime}-\omega_{e}^{\prime \prime}\right)-\frac{1}{4}\left(\omega_{e}^{\prime} x_{e}^{\prime}-\omega_{e}^{\prime \prime} x_{e}^{\prime \prime}\right).

Nevertheless, because the energy corresponding to the first excited electronic level is very high, the electronic contribution to the partition function for N_{2} is simply given by its ground-state degeneracy. In other words, despite a temperature of 3000 K, essentially all nitrogen molecules can be associated with the ground electronic level. Hence, only mode parameters for the ground electronic state are needed to determine contributions from the combined rotational and vibrational modes.

The rovibronic contribution based on the semirigorous model can be evaluated from Eqs. (9.68–9.70). The required rotational temperature is, from Appendix K (K.1, K.2, K.3), θ_{r} = (1.4387)(1.9982) = 2.8748 K. Similarly, the vibrational temperature becomes θ_{v} = (1.4387)(2358.57) = 3393.3 K. Given these characteristic temperatures, the corrected rigid-rotor/harmonic-oscillator parameters for the semirigorous model are
Z_{R-V}=Z_{R-V}^{\circ} Z_{c o r r},                    (9.68)

t=\frac{\omega^{*}}{\omega_{e}}\left(\frac{\theta_{v}}{T}\right)=\frac{\left(\omega_{e}-2 \omega_{e} x_{e}\right)}{\omega_{e}}\left(\frac{\theta_{v}}{T}\right)=\left(\frac{2329.92}{2358.57}\right)\left(\frac{3393.3}{3000}\right)=1.1174.

Hence, from Eq. (9.69), the corrected simplex contribution to the semirigorous model becomes

Z_{R-V}^{\circ}=\frac{1}{\sigma} \frac{1}{y\left(1-e^{-t}\right)}=\left[2\left(9.5414 \times 10^{-4}\right)(0.67287)\right]^{-1}=778.80.

To evaluate the second-order correction term, we must first determine the required correction parameters from Eqs. (9.59), (9.65) and (9.66); i.e.,

\delta=\frac{\alpha_{e}}{B^{*}}=\frac{0.0173}{1.9896}=8.695 \times 10^{-3}.

Consequently, from Eq. (9.70), the second-order correction term is

Z_{ corr }=1+\frac{y}{3}+\frac{y^{2}}{15}+\frac{2 \gamma}{y}+\frac{\delta}{e^{t}-1}+\frac{2 x^{*} t}{\left(e^{t}-1\right)^{2}}                              (9.70)

=1+13.872 \times 10^{-3}=1.0139.

As a result, the total partition function for the semirigorous model can now be expressed as

\frac{Z}{N}=\frac{Z_{t r}}{N} Z_{R-V}^{\circ} Z_{\text {corr }} Z_{e l}=\left(1.8965 \times 10^{9}\right)(778.80)(1.0139)(1.0000)=1.4975 \times 10^{12}.

Therefore, the dimensionless Gibbs free energy becomes

\frac{g}{R T}=-\ln \left(\frac{Z}{N}\right)=-\ln \left(1.4975 \times 10^{12}\right)=-28.035,

so that

g = −28.035 (8.3145 J/K · mol) (3000 K) = −699.29 kJ/mol.