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Step-by-Step Solution
Step 1: Write down the given data
For gas A (diatomic):
$C_p = 29 \,\mathrm{J\,mol^{-1}\,K^{-1}}$
$C_v = 22 \,\mathrm{J\,mol^{-1}\,K^{-1}}$
For gas B (diatomic):
$C_p = 30 \,\mathrm{J\,mol^{-1}\,K^{-1}}$
$C_v = 21 \,\mathrm{J\,mol^{-1}\,K^{-1}}$
Step 2: Recall the relation between $C_p$, $C_v$, and degrees of freedom ($f$)
For an ideal gas, the ratio of specific heats is given by
$ \displaystyle \gamma = \frac{C_p}{C_v} = 1 + \frac{2}{f} $
Here, $f$ is the total number of degrees of freedom (translational, rotational, and vibrational contributions).
Step 3: Calculate the degrees of freedom for gas A
Given:
$ \displaystyle \gamma_A = \frac{C_p}{C_v} \;=\; \frac{29}{22} $
Using $ \displaystyle \gamma = 1 + \frac{2}{f} $, we have:
$ \displaystyle \frac{29}{22} \;=\; 1 + \frac{2}{f} $
Rearrange to solve for $f$:
$ \displaystyle \frac{29}{22} - 1 \;=\; \frac{2}{f} \quad\Rightarrow\quad \frac{7}{22} = \frac{2}{f} \quad\Rightarrow\quad f = \frac{2 \times 22}{7} \approx 6.3 \,\approx 6 $
For a diatomic molecule, the standard degrees of freedom are:
3 translational
2 rotational
Thus, having $f \approx 6$ indicates the presence of 1 vibrational degree of freedom.
Step 4: Calculate the degrees of freedom for gas B
Similarly, for gas B:
$ \displaystyle \gamma_B = \frac{C_p}{C_v} \;=\; \frac{30}{21} $
Using $ \displaystyle \gamma = 1 + \frac{2}{f} $, we have:
$ \displaystyle \frac{30}{21} \;=\; 1 + \frac{2}{f} $
Rearrange to solve for $f$:
$ \displaystyle \frac{30}{21} - 1 \;=\; \frac{2}{f} \quad\Rightarrow\quad \frac{9}{21} = \frac{2}{f} \quad\Rightarrow\quad f = \frac{2 \times 21}{9} \approx 4.67 \,\approx 5 $
Here, $f \approx 5$ for a diatomic molecule generally accounts for:
3 translational
2 rotational
No vibrational degrees of freedom are active in this case.
Step 5: Interpret the results
Gas A has $f \approx 6$ degrees of freedom, indicating it possesses one vibrational mode. Gas B has $f \approx 5$ degrees of freedom, indicating no vibrational mode is active.
Step 6: State the final answer
Therefore, the correct statement is:
A has a vibrational mode but B has none.
