The adiabatic bulk modulus of a perfect gas a | vedclass

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(D) For an adiabatic process,the relationship between pressure $P$ and volume $V$ is given by $PV^{\gamma} = 	ext{constant}$,where $\gamma$ is the ad

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$\gamma P$

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(D) For an adiabatic process,the relationship between pressure $P$ and volume $V$ is given by $PV^{\gamma} = 	ext{constant}$,where $\gamma$ is the adiabatic index.Differentiating both sides with respect to $V$,we get: $P(\gamma V^{\gamma-1}) + V^{\gamma} \frac{dP}{dV} = 0$.Rearranging the terms: $V^{\gamma} \frac{dP}{dV} = -\gamma P V^{\gamma-1}$.Dividing by $V^{\gamma-1}$,we get: $V \frac{dP}{dV} = -\gamma P$.The bulk modulus $B$ is defined as $B = -V \frac{dP}{dV}$.Substituting the expression,we get $B = -(-\gamma P) = \gamma P$.Therefore,the adiabatic bulk modulus is $\gamma P$.

The adiabatic bulk modulus of a perfect gas at pressure $P$ is given by:

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