Question

Which one of the following is the correct relation between $C_p$ and $C_V$ for one mole of an ideal gas? (R is molar gas constant)

• A. $C_p = C_V - R$
• B. $C_p = C_V + R$
• C. $C_p = R - C_V$
• D. $C_p = C_V \times R$
• E. $C_p = C_V / R$

Hint:

Always remember $C_p$ is greater than $C_V$ because work is done during constant pressure heating.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
$C_p$ is molar heat capacity at constant pressure, and $C_V$ is molar heat capacity at constant volume. For an ideal gas, they are related via the gas constant $R$.
Step 2: Key Formula or Approach:
The relation is known as Mayer’s relation: \[ C_p - C_V = R \] Step 3: Detailed Explanation:
Heat added at constant volume increases internal energy only.
Heat added at constant pressure increases internal energy AND does expansion work ($P \Delta V$).
For 1 mole of an ideal gas, $P \Delta V = R \Delta T$.
Thus, $C_p \Delta T = C_V \Delta T + R \Delta T$.
Dividing by $\Delta T$ gives $C_p = C_V + R$.
Step 4: Final Answer:
The correct relation is $C_p = C_V + R$.