Question

Consider a hypothetical gas temperature of which increases to $\sqrt{2}$ times when compressed adiabatically to half the volume. Its equation can be written as

• A. $P{V}^{5/3}=constant$
• B. $P{V}^{7/5}=constant$
• C. $P{V}^{3/2}=constant$
• D. $P{V}^{4/3}=constant$

Answer 1

Answer: (C)

$P{V}^{3/2}=constant$

The relation between T & V in adiabatic process is given by

$T{V}^{\gamma -1}=constant$

Given that $T_2=\sqrt{2}{T}_1$

$V_2=\frac{V_1}{2}$

$T_1{V}_1^{\gamma -1}=T_2{V}_2^{\gamma -1}$

$\frac{T_1}{T_2}={\left(\frac{V_2}{V_1}\right)}^{\gamma -1}$

$\frac{1}{\sqrt{2}}={\left(\frac{1}{2}\right)}^{\gamma -1}$

${\left(\frac{1}{2}\right)}^{\frac{1}{2}}={\left(\frac{1}{2}\right)}^{\gamma -1}$

Equating on both sides,

$\gamma -1=\frac{1}{2}$

$\gamma =\frac{3}{2}$

The relation between P & V in an adiabatic process is given by $P{V}^{\gamma }$=constant

$\because P{V}^{\frac{3}{2}}=constant$