Under isothermal condition, the pressure of a gas is given by where is a constant and is the volume of the gas. The bulk modulus at constant temperature is equal to [2023]

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Solution

Q.
Under isothermal condition, the pressure of a gas is given by $P = a V^{- 3},$ where $a$ is a constant and $V$ is the volume of the gas. The bulk modulus at constant temperature is equal to [2023]

1 $\frac{P}{2}$

2 $3 P$

3 $2 P$

4 $P$

Ans.
(2)

$Bulk modulus, B = - \frac{d P}{\frac{d v}{v}}$

$Given: P v^{3} = a$

$Differentiating with respect to pressure:$

$v^{3} + 3 P v^{2} \frac{d v}{d P} = 0$ $\Rightarrow v = - 3 \frac{P d v}{d P}$

$Hence, \frac{d P v}{d v} = - 3 P$

$B = - (\frac{d P v}{d v}) = - (- 3 P) = 3 P$

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