Question

A given ideal gas with $\gamma =\dfrac{{{C}_{P}}}{{{C}_{V}}}=1.5$ is at a temperature T. If the gas is compressed adiabatically to one-fourth of its initial volume, the final temperature will be :-
 A. 2T
 B. 8T
$\text{C}\text{. }2\sqrt{2}T$
 D. 4T

Answer 1

Hint: Use the equation of state for an adiabatic process formed on a gas. You will find that the relation between the temperature and volume of the gas is constant. Then substitute the given values and find the final temperature.

Formula used:
$P{{V}^{a}}=k$
PV = nRT

Complete step by step answer:
For any process performed on an ideal gas, the pressure (P) and the volume (V) of the gas are related as $P{{V}^{a}}=k$, where a and k are constants. This is called an equation of state.
The value of ‘a’ depends on the process that is performed on the gas.
For an adiabatic process, a = $\gamma $.
Therefore,
 $\Rightarrow P{{V}^{\gamma }}=k$ ….. (i).
From the ideal gas equation we know that PV = nRT,
where n is the number of moles of the gas, R is the universal gas constant and T is the temperature of the gas.
$\Rightarrow P=\dfrac{nRT}{V}$.
Substitute the value of P in equation (i).
$\Rightarrow \left( \dfrac{nRT}{V} \right){{V}^{\gamma }}=k$
$\Rightarrow nRT{{V}^{\gamma -1}}=k$
$\Rightarrow T{{V}^{\gamma -1}}=\dfrac{k}{nR}$.
For a given gas n is constant and R is also a constant. Therefore, $\dfrac{k}{nR}$ is a constant value.
This means that $T{{V}^{\gamma -1}}$ is a constant value.
It is given that the initial temperature T and let its initial volume be V.
The gas is compressed to a volume equal to $\dfrac{V}{4}$ and let the temperature of the gas at this time be T’.
 Since $T{{V}^{\gamma -1}}$ is constant,
$T{{V}^{\gamma -1}}=T'{{\left( \dfrac{V}{4} \right)}^{\gamma -1}}$ …. (ii).
Substitute the given value of $\gamma $=1.5 in equation (ii).
$\Rightarrow T{{V}^{1.5-1}}=T'{{\left( \dfrac{V}{4} \right)}^{1.5-1}}$
$\Rightarrow T{{V}^{1.5-1}}=T'\left( \dfrac{{{V}^{1.5-1}}}{{{4}^{1.5-1}}} \right)$
$\Rightarrow T=\dfrac{T'}{{{4}^{0.5}}}$
$\Rightarrow T'={{4}^{0.5}}T=2T$
This means that the final temperature of the gas is 2T.
Hence, the correct option is A.

Note:
The equation of state, i.e. $P{{V}^{a}}=k$ is applicable for all types of processes. However, it is applicable for an ideal gas only.
For an isothermal process, a = 1.
For an isochoric process, a = $\infty $.
For an isobaric process, a = 0.