Question

If the molar specific heat of a rigid diatomic gas at constant pressure is \(C\), then the molar specific heat of a monoatomic gas at constant volume is

• A. \(\dfrac{2C}{5}\)
• B. \(\dfrac{3C}{5}\)
• C. \(\dfrac{5C}{7}\)
• D. \(\dfrac{3C}{7}\)

Hint:

For ideal gases, \[ \boxed{ \begin{aligned} \text{Monoatomic:}\quad & c_V=\frac32R,\qquad C_P=\frac52R, \text{Rigid diatomic:}\quad & c_V=\frac52R,\qquad C_P=\frac72R. \end{aligned} } \]

Answer 1

The Correct Option is D

Step 1: Write the molar specific heat of a diatomic gas. For a rigid diatomic gas, \[ C_P = \frac72R. \] Given, \[ C_P=C. \] Hence, \[ R=\frac{2C}{7}. \]

Step 2: Find the molar specific heat of a monoatomic gas at constant volume. For a monoatomic gas, \[ C_V = \frac32R. \] Substituting \[ R=\frac{2C}{7}, \] we obtain \[ C_V = \frac32\cdot\frac{2C}{7} = \frac{3C}{7}. \]

Step 3: Write the answer. Hence, \[ \boxed{\frac{3C}{7}}. \] Thus, \[ \boxed{(D)} \] is the correct answer.