Question

If ${\alpha }_v$ and $T$ are the coefficient of volume expansion and temperature for an ideal gas respectively, then

• A. ${\alpha }_v=\frac{1}{T}$
• B. ${\alpha }_v=\sqrt{T}$
• C. ${\alpha }_v=\frac{1}{\sqrt{T}}$
• D. ${\alpha }_v=\frac{1}{T^2}$

Answer 1

Answer: (A)

Change in volume due to increase of temperature $(\Delta T)$ is given by
$\Delta V={\alpha }_{V}V\Delta T\dots$ (i)
where, ${\alpha }_V=$ coefficient of volume expansion
and $V=$ initial volume of gas.
For an ideal gas, we know that
$pV=nRT\dots$ (ii)
$\Rightarrow p\Delta V=nR\Delta T$
$\Rightarrow \Delta V=\frac{nR\Delta T}{p}\dots$ (iii)
From Eqs. (i) and (iii), we get
${\alpha }_V\cdot V\Delta T=\frac{nR\Delta T}{p}$
$\Rightarrow {\alpha }_V=\frac{nR}{pV}=\frac{nR}{nRT}[$ from Eq. (ii)]
$\Rightarrow {\alpha }_V=\frac{1}{T}$