In the first part, the gas expands isothermally. This means the temperature $T$ remains constant. During expansion, volume $V$ increases and pressure $P$ decreases.

For an isothermal process,

$PV = \text{constant}$ or $P \propto \dfrac{1}{V}$.

In the second part, the gas expands adiabatically. This means no heat is exchanged, so $q = 0$. The gas does work by using its internal energy, so its temperature decreases. Again, volume $V$ increases and pressure $P$ decreases.

For an adiabatic process,

$PV^{\gamma} = \text{constant}$ and $T^{\gamma} P^{1-\gamma} = \text{constant}$.

Diagram (i): $P$ vs $V$

For the isothermal expansion, $P$ decreases as $V$ increases. This gives a rectangular hyperbola, so the solid curve is correct.

For the adiabatic expansion, the curve should fall more steeply than the isothermal curve because

$\dfrac{dP}{dV} = -\gamma \dfrac{P}{V}$ for adiabatic process, while $\dfrac{dP}{dV} = -\dfrac{P}{V}$ for isothermal process.

Since $\gamma > 1$, the adiabatic curve is steeper. So diagram (i) is correct.

Diagram (ii): $T$ vs $V$

During isothermal expansion, temperature remains constant, so the solid horizontal line is correct.

But during adiabatic expansion, temperature should decrease as volume increases. The dashed line in the diagram shows temperature increasing, which is wrong. So diagram (ii) is not correct.

Diagram (iii): $P$ vs $T$

During isothermal expansion, $T$ remains constant while $P$ decreases. So the solid vertical line going downward is correct.

For adiabatic expansion, both $T$ and $P$ decrease. Also,

$P \propto T^{\dfrac{\gamma}{\gamma-1}}$.

Since $\gamma > 1$, the pressure decreases with temperature in the way shown by the dashed curve. So diagram (iii) is correct.

Diagram (iv): $P$ vs $1/V$

During expansion, $V$ increases, so $1/V$ decreases. For an isothermal process,

$P = nRT(1/V)$.

So the graph should move towards smaller values of $1/V$ and smaller values of $P$. The given diagram does not show the expansion properly. So diagram (iv) is not correct.

So, the correct diagrams are (i) and (iii) only.

Thermodynamics

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