Question

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

• A. W
• B. $\frac{5W}{2}$
• C. $\frac{W}{2}$
• D. $\frac{3W}{2}$

Hint:

For any ideal gas, the ratio of work done at constant pressure to heat at constant volume is matching the degree factor: $\frac{W}{Q_v} = \frac{R}{C_v}$. Since $C_v = 1.5R$ for a monoatomic gas, the work is roughly $\frac{2}{3}$ of the internal energy, making $Q_v = 1.5W$ instantly!

Answer 1

The Correct Option is D

Step 1: Express the work done.
At constant pressure, $W=nR\Delta T$.

Step 2: Express the heat at constant volume.
At constant volume the heat is $Q_v=nC_v\Delta T$. For a monoatomic gas $C_v=\frac{3}{2}R$.

Step 3: Combine.
\[ Q_v=n\cdot\frac{3}{2}R\cdot\Delta T=\frac{3}{2}(nR\Delta T)=\frac{3W}{2} \]
\[ \boxed{\frac{3W}{2}} \]