Question

The specific heat capacity of a gas at constant pressure ($C_p$) and at constant volume ($C_v$) are related as

• A. $C_p - C_v = R$
• B. $C_v - C_p = R$
• C. $C_p / C_v = R$
• D. $C_p + C_v = R$

Hint:

Remember that for an ideal gas, the difference between the specific heat capacities at constant pressure and constant volume is always equal to the universal gas constant $R$.

Answer 1

The Correct Option is A

Step 1: Understand the two specific heats.
$C_p$ is the heat needed to raise one mole by one degree at constant pressure. $C_v$ is the same but at constant volume.

Step 2: See why they differ.
At constant volume the gas does no work, so all heat raises the temperature. At constant pressure the gas also expands and pushes its surroundings, so it needs extra heat. That is why $C_p$ is larger.

Step 3: Use the first law.
Heat at constant volume equals the change in internal energy. \[ Q_v = nC_v\Delta T \]

Step 4: Add the expansion work.
At constant pressure, the extra work for one mole is \[ W = P\Delta V = R\Delta T \] using the ideal gas law $PV=nRT$ for one mole. So \[ Q_p = nC_v\Delta T + nR\Delta T \]

Step 5: Match the two heats.
Since $Q_p = nC_p\Delta T$, set them equal and divide by $n\Delta T$. \[ C_p = C_v + R \]

Step 6: State the relation.
Rearranging gives Mayer's relation. \[ \boxed{C_p - C_v = R} \]