Question

The ratio of specific heat capacities ($C_p/C_v$) of a diatomic gas is ________.

• A. 1.4
• B. 1.6
• C. 1.7
• D. 1.8
• E. 1.5

Hint:

Monatomic: 1.67, Diatomic: 1.40, Triatomic: 1.33.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
The question asks for the value of the adiabatic index or heat capacity ratio, denoted by \(\gamma\), for a diatomic gas. This ratio is defined as \(\gamma = \frac{C_P}{C_V}\), where \(C_P\) is the molar specific heat at constant pressure and \(C_V\) is the molar specific heat at constant volume. The value of \(\gamma\) depends on the degrees of freedom of the gas molecules.
Step 2: Key Formula or Approach
According to the equipartition theorem, the molar specific heat at constant volume is given by:
\[ C_V = \frac{f}{2}R \] where \(f\) is the number of degrees of freedom and R is the ideal gas constant.
The molar specific heat at constant pressure is related to \(C_V\) by Mayer's relation:
\[ C_P = C_V + R \] The ratio \(\gamma\) is then:
\[ \gamma = \frac{C_P}{C_V} = \frac{C_V + R}{C_V} = 1 + \frac{R}{C_V} = 1 + \frac{R}{(f/2)R} = 1 + \frac{2}{f} \] Step 3: Detailed Explanation
1. Determine the degrees of freedom (f) for a diatomic gas.
A diatomic molecule (like \(O_2\) or \(N_2\)) can be modeled as a rigid dumbbell. It has:
- 3 translational degrees of freedom (motion along x, y, and z axes).
- 2 rotational degrees of freedom (rotation about two axes perpendicular to the bond). Rotation along the bond axis is negligible.
At ordinary temperatures, vibrational modes are not excited.
So, the total number of degrees of freedom is \(f = 3 + 2 = 5\).
2. Calculate \(\gamma\).
Using the formula \(\gamma = 1 + \frac{2}{f}\):
\[ \gamma = 1 + \frac{2}{5} \] \[ \gamma = 1 + 0.4 = 1.4 \] Step 4: Final Answer
The ratio of specific heat capacities for a diatomic gas is 1.4.