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Question

For a reversible adiabatic expansion of an ideal gas $$dp/p$$ equal to:


A
γdvv
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B
dvv
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C
(γγ1)dvv
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D
γdvv
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Solution

The correct option is D $$-\gamma \dfrac{dv}{v}$$
Solution:- (D) $$-\gamma \cfrac{dv}{v}$$
In an adiabatic process,
$$p{v}^{\gamma} = K \left( \text{constant} \right)$$
Differentiating the above equation, we have
$$\left( 1. dp \right) {v}^{\gamma} + p. \left( \gamma {v}^{\gamma - 1} dv \right) = 0$$
$$dp {v}^{\gamma} = - \gamma p {v}^{\gamma - 1} dv$$
$$\cfrac{dp}{p} = \cfrac{-\gamma {v}^{\gamma -1}}{{v}^{\gamma}} dv$$
$$\cfrac{dp}{p} = -\gamma \cfrac{dv}{v}$$
Here $$\cfrac{dp}{p}$$ and $$\cfrac{dv}{v}$$ are the fractional change in pressure and volume respectively.

Chemistry

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