Question 900: Thermodynamics & KTG

Question

An ideal gas with pressure $$\mathrm{P}$$, volume $$\mathrm{V}$$ and temperature $$\mathrm{T}$$ is expanded isothermally to a volume $$2 \mathrm{~V}$$ and a final pressure $$\mathrm{P}_{\mathrm{i}}$$. The same gas is expanded adiabatically to a volume $$2 \mathrm{~V}$$, the final pressure is $$\mathrm{P}_{\mathrm{a}}$$. In terms of the ratio of the two specific heats for the gas '$$\gamma$$', the ratio $$\frac{P_i}{P_a}$$ is

Accepted Answer

For isothermal expansion we have
$$\begin{aligned}
& P_1 V_1=P_2 V_2 \
	herefore \quad & P_2=P_1 \frac{V_1}{V_2}=P_1 	imes \frac{1}{2}=\frac{P}{2} \
	herefore & P_i=\frac{P}{2}
\end{aligned}$$
For adiabatic process :
$$\begin{aligned}
& P_1 V_1^\gamma=P_2 V_2^\gamma \
	herefore \quad & P_2=P_1\left(\frac{V_1}{V_2}ight)^\gamma=P_1\left(\frac{1}{2}ight)^\gamma=\frac{P}{2^\gamma} \
	herefore \quad & P_a=\frac{P}{2^\gamma} \
	herefore \quad & \frac{P_i}{P_a}=\frac{2^\gamma}{2}=2^{\gamma-1}
\end{aligned}$$

Answer

$$2^{\gamma+1}$$

Answer

$$2^{\gamma-1}$$

Answer

$$2^{1-\gamma}$$

Answer

$$2^\gamma$$

Thermodynamics & KTG

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