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Problem

Adiabatic-expansion

Suppose the volume $ V $ and the pressure $ P $ of a gas undergoing adiabatic compression is modelled by $ {PV}^{1.2} = C $, where $ C $ is constant. When the volume is 64 L, the pressure is 20 kPa.

If at a moment the pressure is decreasing at the rate of 1 kPa per second, find the instantaneous rate of change of the volume.

Solution

What is known?

$ \frac{dP}{dt} = -1 \: kPa/s $ at $ P = 20 \: kPa $ and $ V = 64 \: L $.

Derive the function with respect to time.

$ \frac{d}{dt} \left(PV^{1.2} \right) = \frac{d}{dt} (C) $

Since $ C $ is constant,

$ \frac{dP}{dt} V^{1.2} + P(1.2) V^{0.2} \frac{dV}{dt} = 0 $
$ (-1){(64)}^{1.2} + (20)(1.2){(64)}^{0.2} \frac{dV}{dt} = 0 $
$ \frac{dV}{dt} = \frac{64}{24} $
$ \frac{dV}{dt} = \frac{8}{3} \: L/s $
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