Temperature and heat-thermal properties of matter

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Question

In an adiabatic process, the pressure of a monoatomic ideal gas increases by 0.5%. The volume will decrease by___%

Moderate

Solution

In an adiabatic process ${PV}^{γ} = constant (k)$ .
Taking logarithm,
$\log P + γlog V = \log k$
Differentiating,
$\begin{array}{l} \frac{ΔP}{P} + γ \frac{ΔV}{V} = 0 \\ \Rightarrow \frac{ΔV}{V} = - \frac{1}{γ} \frac{ΔP}{P} \end{array}$
For a monoatomic gas, $γ = \frac{5}{3}$ . Therefore $\frac{ΔV}{V} = - \frac{3}{5} \times 0 .5 % = - 0 .3 %$
The negative sign shows that the volume decreases.

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