1. Use heat capacities from the given processes

For a fixed mass of an ideal gas:

• At constant pressure: $Q = n C_p \Delta T$
• At constant volume: $Q = n C_v \Delta T$

We are given:

• Heating by $50^\circ C$ at constant pressure requires $160$ cal: $nC_p \cdot 50 = 160$ $nC_p = \frac{160}{50} = 3.2\ \text{cal/}^\circ C$

• Cooling by $100^\circ C$ at constant volume releases $240$ cal:

Magnitude-wise, $nC_v \cdot 100 = 240$ $nC_v = \frac{240}{100} = 2.4\ \text{cal/}^\circ C$

2. Find the ratio $\gamma = \dfrac{C_p}{C_v}$

Since the same mass of gas is used, the factor $n$ cancels: $\gamma = \frac{C_p}{C_v} = \frac{nC_p}{nC_v} = \frac{3.2}{2.4} = \frac{4}{3}$

3. Relate $\gamma$ to degrees of freedom

For an ideal gas with $f$ degrees of freedom: $C_v = \frac{f}{2}R, \qquad C_p = C_v + R = \frac{f+2}{2}R$

So, $\gamma = \frac{C_p}{C_v} = \frac{f+2}{f}$

Given $\gamma = \frac{4}{3}$, $\frac{f+2}{f} = \frac{4}{3}$

Cross-multiplying: $3(f+2) = 4f$ $3f + 6 = 4f$ $f = 6$

4. Check options

• A: $6$ ✅
• B: $7$ ❌
• C: $5$ ❌
• D: $3$ ❌

Therefore, the gas molecules have 6 degrees of freedom.