Question 4.1: We evaluate α∗ for the N2 ⇔ N + N dissociation–recombination...
We evaluate \alpha ^{\divideontimes } for the N_{2} ⇔ N + N dissociation_ recombination system.
m_{a}= {14}/{6.023} \times 10^{26} kg =2.32\times 10^{- 26} kg (4.19)
Q^{i}_{tr} = V \left(\frac{2\pi m_{i}kT }{h^{2} } \right)^{{3}/{2 }} (4.20)
Q_{rot} = \frac{1}{2} \frac{T}{\theta _{rot} } (homonuclear,\theta _{rot} = 2.9 K) (4.21)
Q_{vib} = \left( 1-exp\left(-\frac{\theta _{vib} }{t} \right) \right)^{-1} (\theta _{vib}=3390 K ) (4.22)
Recall the law of mass action:
\frac{(\alpha ^{*} )^{2} }{1-\alpha ^{*}}= \frac{exp(-\theta _{d} / T )}{\rho} \left\{m_{a} \left(\frac{\pi m_{a} k }{h^{2} }\right) ^{{3}/{2}} \theta _{rot} \sqrt{T} \left[1-exp\left(-\frac{\theta_{vib} }{T} \right) \right] \frac{(Q^{a}_{el})^{2}}{Q^{aa}_{el}} \right\} (4.23)
Now, from our discussions in statistical mechanics:
{Q^{N}_{el}}\approx 4 (4.24)
{Q^{N_{2} }_{el}}\approx 1 (4.25)
⇒\frac{(\alpha ^{*} )^{2} }{1-\alpha ^{*} }= \frac{exp(-\frac{\theta _{d} }{T} )}{\rho} \left\{3700\sqrt{T} \left[1-exp(-\frac{Q_{vib} }{T} )\right] \right\} (4.26)
The term in { } is sometimes called the characteristic density for dissociation \rho _{d}v, and is plotted in Fig. 4.3 for nitrogen.
We now evaluate the degree of dissociation as a function of pressure, by making use of the ideal gas law, and temperature: note that Q_{d} = 113,000 K for N_{2} and θ_{d} = 59,500 K for O_{2}. The results are shown in Fig. 4.4. At the conditions considered, we see in general that
• For fixed pressure, higher temperature gives higher \alpha ^{* } that occurs because there is more energy available for dissociation.
• For fixed temperature, higher pressure gives lower \alpha ^{* } that is explained by the fact that recombination is a three-body process that occurs more frequently at higher pressure. The effect can be seen directly from the presence of density in the denominator on the right hand side of Eq. 4.26. At a given temperature, a higher value of density will give a higher pressure and a lower value of \alpha ^{*}.
Also shown in Fig. 4.4 are data for oxygen under the same conditions. The lower characteristic temperature for dissociation of oxygen, because of its weaker chemical bond, means that oxygen dissociates at lower temperatures than nitrogen.
