Question

For a monoatomic gas, the work done at constant pressure is $W$. The heat supplied at constant volume for the same rise in temperature of the gas is

• A. $2W$
• B. $W$
• C. $\frac{W}{2}$
• D. $\frac{3W}{2}$

Hint:

For an ideal gas, work done at constant pressure is always $nR\Delta T$, while the internal energy change (which equals heat at constant volume) is $C_v$ scaled by $n\Delta T$. The ratio of $Q_v$ to $W$ is simply the ratio $\frac{C_v}{R}$. For a monoatomic gas, this ratio is automatically $\frac{3}{2}$.

Answer 1

The Correct Option is D

Step 1: Express everything through $nR\Delta T$.
Both the isobaric work and the isochoric heat are multiples of the common group $nR\Delta T$, so finding that group lets us relate them instantly.
Step 2: Work at constant pressure.
$W = P\Delta V = nR\Delta T$, so the building block $nR\Delta T$ equals $W$.
Step 3: Heat at constant volume.
$Q_v = n C_v \Delta T$, where $C_v$ is the molar heat at constant volume.
Step 4: Use the monoatomic $C_v$.
For a monoatomic gas, $C_v = \dfrac{3}{2}R$, so $Q_v = \dfrac{3}{2} n R \Delta T$.
Step 5: Substitute the building block.
Replacing $nR\Delta T$ with $W$ gives $Q_v = \dfrac{3}{2}W$.
Step 6: Conclude.
The heat supplied at constant volume is $\dfrac{3W}{2}$. \[ \boxed{Q_v = \frac{3W}{2}} \]