The adiabatic bulk modulus of a gas at a pres | vedclass

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(B) The bulk modulus $K$ is defined as $K = -V \frac{dP}{dV}$.For an adiabatic process,the relation between pressure $P$ and volume $V$ is given by $P

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$\gamma P$

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

The adiabatic bulk modulus of a gas at a pressure $P$ is (where $\gamma$ is the ratio of specific heat capacities of the gas).

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