**Question:**

To raise the temperature of a certain mass of gas by $50^{\circ} \mathrm{C}$ at a constant pressure, 160 calories of heat is required. When the same mass of gas is cooled by $100^{\circ} \mathrm{C}$ at constant volume, 240 calories of heat is released. How many degrees of freedom does each molecule of this gas have (assume gas to be ideal)?

Correct Option: , 2

**Solution:**

(2) Let $C_{p}$ and $C_{v}$ be the specific heat capacity of the gas

at constant pressure and volume.

At constant pressure, heat required

$\Delta Q_{1}=n C_{p} \Delta T$

$\Rightarrow 160=n C_{p} \cdot 50$ ....(1)

At constant volume, heat required

$\Delta Q_{2}=n C_{v} \Delta T$

$\Rightarrow 240=n C_{v} \cdot 100$ ...(2)

Dividing (i) by (ii), we get

$\frac{160}{240}=\frac{C_{p}}{C_{v}} \cdot \frac{50}{100} \Rightarrow \frac{C_{p}}{C_{v}}=\frac{4}{3}$

$\gamma=\frac{C_{p}}{C_{v}}=\frac{4}{3}=1+\frac{2}{f}$ (Here, $f=$ degree of freedom)

$\Rightarrow f=6 .$