Question

In an adiabatic process $1{\mathrm{m}}^3$ of an ideal monatomic gas expands to $8{\mathrm{m}}^3$. What is the ratio of final to initial temperature?

• A. $1/4$
• B. $1/8$
• C. $1$
• D. $8$

Answer 1

The correct option is A

$1/4$

The explanation for the correct option

Option (A): $\frac{1}{4}$

Step 1: Given data

${\mathrm{V}}_1=\mathrm{V};{\mathrm{V}}_2=8\mathrm{V}$

Step 2: Equation for an adiabatic process

For adiabatic process,

${\mathrm{P}}_1{\mathrm{V}}_1^{\mathrm{\gamma }}={\mathrm{P}}_2{\mathrm{V}}_2^{\mathrm{\gamma }}$

For monoatomic gas $\mathrm{\gamma }=3/5$

Step 4: Calculation

​$\Rightarrow {\mathrm{P}}_2/{\mathrm{P}}_1={({\mathrm{V}}_1/{\mathrm{V}}_2)}^{\mathrm{\gamma }}$

$\Rightarrow {\mathrm{P}}_2/{\mathrm{P}}_1={(\mathrm{V}/8\mathrm{V})}^{5/3}$

$\Rightarrow {\mathrm{P}}_2/{\mathrm{P}}_1={(1/8)}^{5/3}$

$\Rightarrow ({\mathrm{P}}_2\times {\mathrm{V}}_2)/({\mathrm{P}}_1\times {\mathrm{V}}_1)={\mathrm{T}}_2/{\mathrm{T}}_1$

$\Rightarrow {\mathrm{T}}_2/{\mathrm{T}}_1=(1/32)\times 8=1/4$

Hence, A is the correct option.