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:

Easy

Solution

Let m be the mass of the gas and Y its initial volume. Its initial density is $ρ_{i} = m / V$ . Let P_f and V_f be the final pressure and volume of the gas respectively after adiabatic compression. The final density $ρ_{f} = {nρ}_{i}$ . For adiabatic process, ${PV}^{γ} = P_{f} V_{f}^{γ}$
$∴ P {(m / ρ_{i})}^{γ} = P_{f} {(m / ρ_{f})}^{γ} = P_{f} {(m / {nρ}_{i})}^{γ}$
$or P = P_{f} {(\frac{1}{n})}^{γ} or P_{f} = {Pn}^{γ}$

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