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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 - CBSE (Science) Class 12 - Mathematics

Question

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

• $\frac{1}{2} \%$

• $\frac{1}{4} \%$

• $\frac{1}{8} \%$

• none of these

Solution

$\frac{1} {8}$%

We have

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

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

$\text { Taking log on both sides, we get }$

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

$\Rightarrow \log P + \frac{1}{4}\log V = \log k$

$\text { Differentiating both sides w . r . t . x, we get }$

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

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

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

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Solution 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 Concept: Approximations.
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