Question

An ideal gas is expanding such that $P{T}^2=$constant. The coefficient of volume expansion of the gas is $\frac{A}{T}$, where $T$ is temperature in $kelvin$ and $P$ is pressure. Find the value of $A.$

Answer 1

Given, $P{T}^2=C$

Let the coefficient of volume expansion be $\gamma ,$

$\gamma =\frac{dv}{vdr}$

From ideal gas equation,

$pV=nRT\Rightarrow P=\frac{nRT}{V}$

$\Rightarrow \frac{T}{V}{T}^2=$ Constant

$\Rightarrow \frac{T^3}{V}=$ Constant

By partial differentiation on both sides, we get

$\therefore \frac{1}{V}.3dT.T^2-\frac{T^{3}dV}{V^2}=0$

$\Rightarrow \frac{dV}{VdT}=\frac{3}{T}=\gamma$

Final answer: $\left(3\right)$