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, $ 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) $