Question

If T represents the absolute temperature of an ideal gas, the volume coefficient of thermal expansion at constant pressure is:
A. T
B. ${{T}^{2}}$
C. $\dfrac{1}{T}$
D. $\dfrac{1}{{{T}^{2}}}$

Answer 1

Hint: As a very first step, we could read the question properly and then note down the key point that is the pressure is constant here. Now let us recall the ideal gas equation and then differentiate on both sides. We could now compare the thus obtained expression with the expression for volume expansion and hence find the volume coefficient of thermal expansion.
Formula used:
Ideal gas equation,
$PV=nRT$
For volume expansion,
$dV=\gamma VdT$

Complete answer:
In the question, we are supposed to find the volume coefficient of thermal expansion at constant pressure for absolute temperature T of an ideal gas.
We know that for n mole of an ideal gas with P pressure, T absolute temperature and V volume, the ideal gas equation can be given by,
$PV=nRT$………………………………………. (1)
Where, R is the ideal gas constant.
We could now differentiate on both sides keeping in mind that the pressure remains constant.
$PdV=nRdT$
$\Rightarrow dV=\left( \dfrac{nR}{P} \right)dT$
From (1) we have,
$dV=\left( \dfrac{V}{T} \right)dT$ ………………………………. (2)
Let $\gamma $ be the coefficient of volume expansion then we have the relation,
$dV=\gamma VdT$ ………………………………… (3)
Now, we could compare equations (2) and (3) to get,
$\therefore \gamma =\dfrac{1}{T}={{T}^{-1}}$
Therefore, we found the volume coefficient of thermal expansion at constant pressure to be${{T}^{-1}}$.

Option C is correct.

Note:
While reading the question and looking at the options we realize that the volume coefficient of expansion is to be obtained in terms of the absolute temperature of the given ideal gas. The coefficient of thermal expansion basically describes how the change in size of an object takes place with temperature.