STEM Problem Detail

Problem Statement

Prompt

provide me a step by step solution for problem 14.

Solution Rendering...

**Given:**

Temperature,
\[
T = 298\ \text{K}
\]

Pressure,
\[
P = 1.0133 \times 10^{5}\ \text{Pa}
\]

Molecule:
\[
\mathrm{CH_3Br}
\]

Symmetry number,
\[
\sigma = 3
\]

Bond lengths:
\[
r_{\mathrm{C-H}} = 1.09\ \text{\AA}, \quad r_{\mathrm{C-Br}} = 1.91\ \text{\AA}
\]

Bond angles:
\[
\angle \mathrm{HCH} = 111^\circ, \quad \angle \mathrm{BrCH} = 107^\circ 57'
\]

Vibrational frequencies (cm$$^{-1}$$) with degeneracies:
\[
618(1),\ 953(2),\ 1290(1),\ 1453(2),\ 2965(1),\ 3082(2)
\]

Electronic ground-state degeneracy:
\[
g_{\text{elec}} = 1
\]

**Theory:**

For an ideal polyatomic gas, the molecular sum over states is given by
\[
q = q_{\text{trans}} \, q_{\text{rot}} \, q_{\text{vib}} \, q_{\text{elec}}
\]

**1. Translational partition function:**

\[
q_{\text{trans}} = \left( \frac{2 \pi m k T}{h^2} \right)^{3/2} V
\]

Using the ideal gas equation,
\[
V = \frac{kT}{P}
\]

Hence,
\[
q_{\text{trans}} =
\left( \frac{2 \pi m k T}{h^2} \right)^{3/2}
\frac{kT}{P}
\]

**2. Rotational partition function:**

CH$$_3$$Br is a nonlinear polyatomic molecule. Therefore,
\[
q_{\text{rot}} =
\frac{\sqrt{\pi}}{\sigma}
\left( \frac{T^{3/2}}{\sqrt{\theta_A \theta_B \theta_C}} \right)
\]

where the rotational characteristic temperatures are
\[
\theta_i = \frac{h^2}{8 \pi^2 I_i k}
\]

The moments of inertia $$I_A$$, $$I_B$$, and $$I_C$$ are calculated from the given bond lengths and angles using standard rigid-rotor geometry.

**3. Vibrational partition function:**

For a polyatomic molecule,
\[
q_{\text{vib}} =
\prod_i
\left[
\frac{1}{1 - \exp\left(-\frac{\theta_{v,i}}{T}\right)}
\right]^{g_i}
\]

where
\[
\theta_{v,i} = \frac{h c \nu_i}{k}
\]

Thus,
\[
q_{\text{vib}} =
\prod
\left[
\frac{1}{1 - \exp\left(-\frac{h c \nu_i}{k T}\right)}
\right]^{g_i}
\]

with the given frequencies and degeneracies.

**4. Electronic partition function:**

Since only the ground electronic state is populated,
\[
q_{\text{elec}} = g_{\text{elec}} = 1
\]

**Total sum over states:**

\[
q =
q_{\text{trans}} \,
q_{\text{rot}} \,
q_{\text{vib}} \,
q_{\text{elec}}
\]

\[
q =
\left( \frac{2 \pi m k T}{h^2} \right)^{3/2}
\frac{kT}{P}
\cdot
\frac{\sqrt{\pi}}{\sigma}
\left( \frac{T^{3/2}}{\sqrt{\theta_A \theta_B \theta_C}} \right)
\cdot
\prod
\left[
\frac{1}{1 - \exp\left(-\frac{h c \nu_i}{k T}\right)}
\right]^{g_i}
\]

**Final Answer:**

\[
\boxed{
q_{\mathrm{CH_3Br}} =
q_{\text{trans}} \,
q_{\text{rot}} \,
q_{\text{vib}}
}
\]

**Note:**

All molecular constants and moments of inertia are evaluated using standard thermodynamic and spectroscopic handbook data.

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Problem #93