Question

An ideal gas at pressure $P$ is adiabatically compressed so that its density becomes $n$ times the initial value. The final pressure of the gas will be $\left(\gamma =\frac{C_P}{C_V}\right)$

• A. $n^{(\gamma )}P$
• B. $\left(n-\gamma \right)P$
• C. $n\left(\gamma -1\right)P$
• D. $n\left(1-\gamma \right)P$

Answer 1

Answer: (A)

For an adiabatic process, $P{V}^{\gamma }=$ constant
$\therefore P_1{V}_1^{\gamma }=P_2{V}_2^{\gamma }$
where subscripts $1$ and $2$ represent the initial and final states respectively
or $\frac{P_2}{P_1}={\left(\frac{V_1}{V_2}\right)}^{\gamma }={\left(\frac{{\rho }_2}{{\rho }_1}\right)}^{\gamma }={\left(\frac{n{\rho }_1}{{\rho }_1}\right)}^{\gamma }$
or $P_2=P_1{n}^{\gamma }=n^{\gamma }P\left(\because P_1=P\right)$