Question

1 mole of an ideal gas (${C_v} = 12.55J{K^{ - 1}}mo{l^{ - 1}}$) at 300 K compressed adiabatically and reversibly to one fourth of its original volume. What is the final temperature of the gas?
(A) 750 K
(B) 752 K
(C) 754 K
(D) 756 K

Answer 1

Hint: In this question, the relation between molar heat capacity at constant pressure and molar heat capacity at constant volume is applied as the value of molar heat capacity at constant volume is given. The relation is given as ${C_p} - {C_v} = R$.

Complete step by step answer:
Given that,
Number of moles is 1 mole.
${C_v}$ is 12.55 J${K^{ - 1}}$ $mo{l^{ - 1}}$
Initial Temperature is 300 k
The relation between ${C_p}$ and ${C_v}$is shown below.
${C_p} - {C_v} = R$
Where,
${C_p}$is molar heat capacity when pressure is constant.
${C_v}$is the molar heat capacity when volume is constant.
R is the rate constant.
Rearrange the above equation to derive the formula.
${C_p} = {C_v} + R$
The value of R is 8.314 J${K^{ - 1}}$$mo{l^{ - 1}}$
To calculate the${C_p}$, substitute the values in the above equation.
$\Rightarrow {C_p} = (12.55 + 8.314)J{K^{ - 1}}$$mo{l^{ - 1}}$
$\Rightarrow {C_p} = 20.864J{K^{ - 1}}mo{l^{ - 1}}$
Given that 1 mole of an ideal gas at 300 K compressed adiabatically and reversibly to one fourth of its original volume
Let ${V_1} = V$, ${V_2} = \dfrac{V}{4}$
$T{V^{\gamma - 1}} = cons\tan t$
When the system is adiabatically and reversible, then the relation between temperature and volume is given as shown below.
$({T_1}){({V_1})^{\gamma - 1}} = ({T_2}){({V_2})^{\gamma - 1}}$
Substitute the values in the above equation.
$\Rightarrow (300){(V)^{\gamma - 1}} = ({T_2}){\left( {\dfrac{V}{4}} \right)^{\gamma - 1}}$
$\Rightarrow {T_2} = 300 \times {4^{0.66}}$
$\Rightarrow {T_2} = 748.99K$
The formula for calculating the $\gamma$ is shown below.
$\gamma = \dfrac{{{C_P}}}{{{C_v}}}$
To calculate the $\gamma$, substitute the values in the above equation.
$\Rightarrow \gamma = \dfrac{{20.864}}{{12.55}}$
$\Rightarrow \gamma = 1.66$
Thus, the final temperature of the gas is 748.99 K $\approx$ 750 K.
Therefore, the correct option is A.

Note:
The relation between molar heat capacity at constant pressure and molar heat capacity at constant volume ${C_P} - {C_v} = R$ is known as Meyer’s temperature. In adiabatic compression, the final temperature is always higher than the initial temperature.