The pressure P and volume V of a gas are connected by


The pressure P and volume V of a gas are connected by the relation PV1/4 = constant. The percentage increase in the pressure corresponding to a deminition of 1/2 % in the volume is

(a) $\frac{1}{2} \%$

(b) $\frac{1}{4} \%$

(c) $\frac{1}{8} \%$


(d) none of these


(C) $\frac{1}{8} \%$

We have

$\frac{\Delta V}{V}=\frac{-1}{2} \%$

$P V^{\frac{1}{4}}=$ constant $=k \quad$ (say)

Taking $\log$ on both sides, we get

$\log \left(P V^{\frac{1}{4}}\right)=\log k$

Differentiating both sides w.r.t. $x$, we get

$\frac{1}{P} \frac{d P}{d V}+\frac{1}{4 V}=0$

$\Rightarrow \frac{d P}{P}=-\frac{d V}{4 V}=-\frac{1}{4} \times \frac{-1}{2}=\frac{1}{8}$

Hence, the increase in the pressure is $\frac{1}{8} \%$.

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