Question

One mole of a diatomic gas does a work $\frac{Q{3}$, when the amount of heat supplied is $Q$. In this process, the molar heat capacity of the gas is}

• A. $\frac{15R}{4}$
• B. $\frac{9R}{4}$
• C. $\frac{7R}{4}$
• D. $\frac{3R}{4}$

Hint:

For thermodynamic processes, use the first law of thermodynamics to relate heat, work, and change in internal energy.

Answer 1

The Correct Option is A

Step 1: Relation for molar heat capacity.
The work done by the gas is related to the heat supplied by: \[ W = \frac{Q}{3} \] The first law of thermodynamics gives: \[ \Delta U = Q - W \] For a diatomic gas, the change in internal energy is: \[ \Delta U = \frac{5}{2} nR \Delta T \]
Step 2: Using the given information.
The total heat supplied $Q = n C_p \Delta T$. For one mole of gas, this becomes: \[ Q = C_p \Delta T \] Substitute into the first law: \[ C_p \Delta T = \frac{5}{2}R \Delta T + \frac{Q}{3} \]
Step 3: Solving for $C_p$.
By solving the equation, we find that: \[ C_p = \frac{15R}{4} \]
Step 4: Conclusion.
The molar heat capacity is $\frac{15R}{4}$.