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:

Let m be the mass of the gas and Y its initial volume. Its initial density is ρi=m/V. Let Pf and Vf be the final pressure and volume of the gas respectively after adiabatic compression. The final density ρf=nρi. For adiabatic process, PVγ=PfVfγ∴ Pm/ρiγ=Pfm/ρfγ=Pfm/nρiγ or  P=Pf1nγ  or  Pf=Pnγ