Animated Solution for Physics - Thermodynamics: Which of the following is an equivalent cyclic process corresponding to the thermodynamic cyclic given in the figure? where, $1\to 2$ is adiabatic. (Graphs are schematic and are not to scale)

From p-V to V-T Diagrams

Translating a thermodynamic cycle from one set of axes to another is a classic test of your conceptual clarity. It requires you to look beyond the shape of the graph and understand the physical reality of the gas at every step. Let's break down this $p-V$ cycle and reconstruct it on a $V-T$ plane.

The Adiabatic Expansion

The problem explicitly states that the process from state 1 to state 2 is adiabatic. Looking at the $p-V$ diagram, the curve moves downwards and to the right, indicating that the volume $V$ is increasing while the pressure $p$ is decreasing. This is an adiabatic expansion.

In an adiabatic process, no heat is exchanged with the surroundings ($Q=0$). As the gas expands, it does positive work. According to the First Law of Thermodynamics, this work must come at the expense of the gas's internal energy. A drop in internal energy directly translates to a drop in temperature $T$. Mathematically, this is governed by the relation $T{V}^{\gamma -1}=\text{constant}$.

Therefore, on our new $V-T$ graph, the path from 1 to 2 must be a curve where volume increases and temperature decreases.

The Isobaric Compression

Next, we examine the path from state 2 to state 3. On the $p-V$ diagram, this is a perfectly horizontal line moving to the left. A horizontal line on a $p-V$ graph means the pressure $p$ remains constant. Since the volume is decreasing, this is an isobaric compression.

According to Charles's Law, at constant pressure, the volume of an ideal gas is directly proportional to its absolute temperature ($V\propto T$). Because the volume is decreasing, the temperature must also decrease proportionally.

On a $V-T$ graph, a direct proportionality $V=\left(\frac{nR}{p}\right)T$ is represented by a straight line that, if extended, would pass exactly through the origin.

The Isochoric Heating

Finally, the cycle closes by moving from state 3 back to state 1. On the $p-V$ diagram, this is a vertical line moving upwards. A vertical line means the volume $V$ is constant, making this an isochoric process. The pressure is increasing.

By Gay-Lussac's Law, at constant volume, pressure is directly proportional to temperature ($p\propto T$). To achieve the higher pressure of state 1, the temperature must increase.

On our $V-T$ graph, a constant volume process is simply drawn as a horizontal line.

Putting all these pieces together, our equivalent $V-T$ diagram must feature: 1. A non-linear curve for $1\to 2$ (adiabatic). 2. A straight line pointing towards the origin for $2\to 3$ (isobaric). 3. A horizontal line for $3\to 1$ (isochoric).

Comparing this synthesized mental model with the given options, Option (b) is the only graph that perfectly captures the physics of all three processes.