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)

$n^({\gamma})P$

Answer 2

For an adiabatic process, $PV^{\gamma} =$ 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 $P_{2} = P_{1}n^{\gamma} = n^{\gamma} P \quad\left(\because P_{1}=P\right)$