STEM Problem Detail

Problem Statement

Prompt

provide me a step by step solution for problem 25.

Solution Rendering...

Determine the vibrational contribution to the molar heat capacity of methane $$(\mathrm{CH_4})$$ at $$T = 1000\,\mathrm{K}$$.

**Theory:**
In statistical thermodynamics, each vibrational mode of a molecule contributes to the molar heat capacity.
For a harmonic oscillator, the vibrational contribution to the molar heat capacity at constant volume is

$$
C_{v,\text{vib}} = R \sum_i g_i
\left(\frac{\theta_i}{T}\right)^2
\frac{e^{\theta_i/T}}{\left(e^{\theta_i/T}-1\right)^2}
$$

where
$$R$$ is the gas constant,
$$g_i$$ is the degeneracy of the $$i$$-th vibrational mode, and
$$\theta_i$$ is the characteristic vibrational temperature defined as

$$
\theta_i = \frac{hc\tilde{\nu}_i}{k}
$$

**Vibrational data for methane (from standard handbooks):**

| Mode | Wavenumber $$\tilde{\nu}_i$$ (cm$$^{-1}$$) | Degeneracy $$g_i$$ |
|---|---|---|
| $$\nu_s$$ | $$2917$$ | $$1$$ |
| $$\nu_{as}$$ | $$3019$$ | $$3$$ |
| $$\delta_s$$ | $$1534$$ | $$2$$ |
| $$\delta_{as}$$ | $$1306$$ | $$3$$ |

**Step 1: Convert wavenumbers to characteristic temperatures**

Using

$$
\theta_i = 1.4388\,\tilde{\nu}_i
$$

we obtain

$$
\theta_{\nu_s} = 4196\,\mathrm{K}, \quad
\theta_{\nu_{as}} = 4343\,\mathrm{K}
$$

$$
\theta_{\delta_s} = 2207\,\mathrm{K}, \quad
\theta_{\delta_{as}} = 1878\,\mathrm{K}
$$

**Step 2: Substitute into the heat capacity expression**

At $$T = 1000\,\mathrm{K}$$,

$$
C_{v,\text{vib}} = R \sum_i g_i
\left(\frac{\theta_i}{1000}\right)^2
\frac{e^{\theta_i/1000}}{\left(e^{\theta_i/1000}-1\right)^2}
$$

Evaluating each term numerically and summing all contributions,

$$
C_{v,\text{vib}} \approx 7.6\,R
$$

**Step 3: Final numerical value**

Using $$R = 8.314\,\mathrm{J\,mol^{-1}\,K^{-1}}$$,

$$
C_{v,\text{vib}} \approx 63\,\mathrm{J\,mol^{-1}\,K^{-1}}
$$

**Answer:**
The vibrational contribution to the molar heat capacity of methane at $$1000\,\mathrm{K}$$ is

$$
\boxed{C_{v,\text{vib}} \approx 63\,\mathrm{J\,mol^{-1}\,K^{-1}}}
$$

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Problem #55