Question

A diatomic gas does a work of \( \frac{Q}{4} \) when a heat of \( Q \) is supplied to it. Then the molar heat capacity of the gas is:

• A. \( C = 3R \)
• B. \( C = \frac{5}{3}R \)
• C. \( C = \frac{10}{3}R \)
• D. \( C = \frac{5}{2}R \)

Hint:

In thermodynamics, always apply the first law and carefully consider the work done and heat supplied to find the change in internal energy.

Answer 1

The Correct Option is C

We know from the first law of thermodynamics:

\[ \Delta Q = \Delta U + \Delta W \] Where: 
- \( \Delta Q \) is the heat supplied to the gas, 
- \( \Delta U \) is the change in internal energy, 
- \( \Delta W \) is the work done by the gas. 
The work done is: \[ \Delta W = \frac{Q}{4} \] The change in internal energy for a diatomic gas is: \[ \Delta U = \frac{5}{2} n R \Delta T \] The heat supplied is: \[ \Delta Q = n C \Delta T \] Combining these expressions, we get: \[ n C \Delta T = \frac{5}{2} n R \Delta T + \frac{Q}{4} \] Simplifying: \[ n C = \frac{5}{2} n R + \frac{Q}{4 \Delta T} \] Solving for the molar heat capacity: \[ C = \frac{10}{3} R \] Thus, the molar heat capacity is: \[ C = \frac{10}{3} R \]