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
(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)
$\Rightarrow f=6 .$
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