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Q.
If $\alpha_{v}$ and $T$ are the coefficient of volume expansion and temperature for an ideal gas respectively, then
TS EAMCET 2020
Solution:
Change in volume due to increase of temperature $(\Delta T)$ is given by
$\Delta V=\alpha_{V} V \Delta T \ldots$ (i)
where, $\alpha_{V}=$ coefficient of volume expansion
and $V=$ initial volume of gas.
For an ideal gas, we know that
$p V=n R T \ldots$ (ii)
$\Rightarrow p \Delta V=n R \Delta T$
$\Rightarrow \Delta V=\frac{n R \Delta T}{p} \ldots$ (iii)
From Eqs. (i) and (iii), we get
$\alpha_{V} \cdot V \Delta T=\frac{n R \Delta T}{p}$
$\Rightarrow \alpha_{V}=\frac{n R}{p V}=\frac{n R}{n R T}[$ from Eq. (ii)]
$\Rightarrow \alpha_{V}=\frac{1}{T}$