Question

Consider two ideal diatomic gases $A$ and $B$ at some temperature $T$. Molecules of the gas A are rigid , and have an mass $m$. Molecules of the gas $B$ have an additional vibrational mode, and have a mass $\frac{m}{4}.$ The ratio of the specific heats $(C^{A}_{V}$ and $C^{B}_{V})$ of gas $A$ and $B$, respectively is :

• A. 5:09
• B. 7:09
• C. 3:05
• D. 5:07

Answer 1

The Correct Option is D

To solve this problem, we need to find the ratio of the specific heat capacities at constant volume \( C_V \) for two ideal diatomic gases \( A \) and \( B \).

First, let's understand the properties of these gases:

1. Gas \( A \) is a diatomic gas with rigid molecules.
2. Gas \( B \) is also diatomic but has an additional vibrational mode.

The specific heat capacity at constant volume \( C_V \) for a diatomic gas can be calculated using the degrees of freedom \( f \). The formula is given by:

C_V = \frac{f}{2} R

where \( R \) is the ideal gas constant.

For a diatomic gas without vibrational modes, the degrees of freedom \( f \) are 5 (3 translational and 2 rotational).

Therefore, for gas \( A \):

C^A_V = \frac{5}{2} R

For a diatomic gas with an additional vibrational mode, there is 1 additional degree of freedom.

Thus, for gas \( B \), the degrees of freedom \( f \) are 7 (5 from the previous modes + 2 from vibrational mode).

Therefore, for gas \( B \):

C^B_V = \frac{7}{2} R

Now, calculate the ratio \( \frac{C^A_V}{C^B_V} \):

\frac{C^A_V}{C^B_V} = \frac{\frac{5}{2} R}{\frac{7}{2} R} = \frac{5}{7}

Thus, the ratio of specific heats \( \frac{C^A_V}{C^B_V} \) is \frac{5}{7}.

Therefore, the correct answer is: 5:07