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  4. To raise the temperature of a certain mass of gas by 50° C at a constant pressure, 160 calories of heat is required. When the same mass of gas is cooled by 100° 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)?

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Q. To raise the temperature of a certain mass of gas by $50^{\circ} C$ at a constant pressure, $160$ calories of heat is required. When the same mass of gas is cooled by $100^{\circ} C$ 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)?

JEE MainJEE Main 2020Kinetic Theory

Solution:

$nC _{ p }(50)=160$
$nC _{ v }(100)=240$
$\Rightarrow \frac{ C _{ p }}{2 C _{ v }}=\frac{160}{240}=\frac{\gamma}{2}$
$\therefore \gamma=\frac{4}{3}$ and $f =\frac{2}{\gamma-1}=6$