An ideal gas is compressed from a common initial state $I$ to a final volume $V_f$ via three different processes as shown in the $P-V$ diagram. The processes are known to be isothermal, adiabatic, and isobaric. Which curve correctly represents the adiabatic compression?

Question

An ideal gas is compressed from a common initial state

I

to a final volume

V_f

via three different processes as shown in the

P-V

diagram. The processes are known to be isothermal, adiabatic, and isobaric. Which curve correctly represents the adiabatic compression?

Quantrex Academy · Accepted Answer

Correct answer: A. Curve 1

For an ideal gas, the equations for the given processes are:
Isobaric:

P = \text{constant}

Isothermal:

PV = \text{constant}

Adiabatic:

PV^\gamma = \text{constant}

Curve 3 is a horizontal line (

P = \text{constant}

), so it represents the isobaric process.

The slope of the

P-V

curve is given by

\dfrac{dP}{dV}

.
For an isothermal process:

\dfrac{dP}{dV} = -\dfrac{P}{V}

For an adiabatic process:

\dfrac{dP}{dV} = -\gamma \dfrac{P}{V}

Since the heat capacity ratio

\gamma > 1

, the magnitude of the slope for the adiabatic curve is greater than that for the isothermal curve. This means the adiabatic curve is steeper.

During compression, the volume decreases. Starting from a common initial state

I

, the steeper curve will rise to a higher final pressure for the same final volume

V_f

.

From the given graph, at

V = V_f

, the pressure for Curve 1 is greater than the pressure for Curve 2. Therefore, Curve 1 represents the adiabatic compression and Curve 2 represents the isothermal compression.

Answer: Curve 1

An ideal gas is compressed from a common initial state I to a final...

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