Question

An ideal gas undergoes an adiabatic process obeying the relation ${PV}^{4/3}=constant$. If its initial temperature is 300K and then its pressure is increased up to four times its initial value, then the final temperature is:

• A. $300\sqrt{2}$
• B. $3003\sqrt{2}$
• C. $600$
• D. $1200$

Answer 1

Answer: (A)

$300\sqrt{2}$

Let $P_1,V_1$ and $T_1$ be the initial pressure, initial volume and inital temprrature respectively and $P_2,V_2$ and $T_2$ be the final pressure, final volume and final temprrature respectively.

${PV}^{4/3}=constant$

$V_2=V_1{\left(\frac{P_1}{P_2}\right)}^{\frac{3}{4}}$.

But $P_2=4P_1$.

$V_2=V_1{\left(\frac{1}{4}\right)}^{\frac{3}{4}}$

Also, $PV=nRT$

Hence, $4P_1{V}_1{\left(\frac{1}{4}\right)}^{\frac{3}{4}}=nR{T}_2$

$T_2=4{\left(\frac{1}{2}\right)}^{\frac{3}{2}}=nR{T}_2$

Thus, $T_2=\sqrt{2}{T}_1=300\sqrt{2}$.

Hence, the correct answer is option $\text{A}$.