Single Choice

The average degree of freedom per molecule for a gas are $$6$$. The gas performs $$25 J$$ of work when it expands at a constant pressure. The heat absorbed by gas is

A$$75 \ J$$
B$$100 \ J$$
Correct Answer
C$$150\ J$$
D$$125 \ J$$

Solution

For a gas with 'n' degrees of freedom:
$$\gamma = 1 + \dfrac{2}{n} = 1 + \dfrac{2}{6} = \dfrac{4}{3}$$
$$C_{p} = \dfrac{\gamma R}{\gamma - 1} = 4R$$
$$C_{v} = \dfrac{R}{\gamma - 1} = 3R$$

Heat supplied for constant pressure process is $$nC_{p}\Delta T$$

Change in internal energy $$nC_{v} \Delta T$$
$$\dfrac{\Delta U}{Q} = \dfrac{C_{v}}{C_{p}} = \dfrac{1}{\gamma} = \dfrac{3}{4}$$

Hence $$\dfrac{W}{Q} = 1 - \dfrac{\Delta U}{Q} = \dfrac{1}{4}$$
$$\dfrac{W}{Q}=1-\dfrac{3}{4}=\dfrac{1}{4}$$
$$\implies Q = 100J$$

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