Animated Solution for Physics - Thermodynamics: A thermodynamic cycle xyzx is shown on a V-T diagram. The p-V diagram that best describes this cycle is (diagrams are schematic and not to scale)

From V-T to p-V Diagrams

Thermodynamics is often about telling the same physical story in different mathematical languages. A cycle plotted on a Volume-Temperature ($V-T$) diagram contains exactly the same information as when it is plotted on a Pressure-Volume ($p-V$) diagram, but the visual representation changes dramatically.

In this problem, we are given a cyclic process $xyzx$ on a $V-T$ diagram and asked to identify its equivalent $p-V$ diagram. To do this accurately, we must break the cycle down into its atomic processes and translate them one by one using the ideal gas law, $pV=nRT$.

Let's begin by analyzing the first leg of the journey, from state $x$ to state $y$. On the $V-T$ diagram, this process is represented by a straight line that, if extended backwards, passes perfectly through the origin.

Mathematically, a straight line through the origin implies a direct proportionality between the variables on the axes. Therefore, we can write:

Now, let's bring in the ideal gas equation:

For the volume $V$ to be strictly proportional to the temperature $T$, the term in the parentheses must be a constant. Since $n$ and $R$ are already constants, this means the pressure $p$ must remain constant throughout the process.

Furthermore, as we move from $x$ to $y$, the volume is increasing. Thus, process $x\to y$ is an isobaric expansion. On our new $p-V$ diagram, a constant pressure process with increasing volume is drawn as a flat, horizontal line moving to the right.

Next, we trace the path from $y$ to $z$. On the $V-T$ diagram, this line is perfectly horizontal. Because the vertical axis represents volume, a horizontal line means the volume is locked in; it is not changing. This is an isochoric process.

While the volume is constant, observe the horizontal axis: the temperature $T$ is decreasing. If you have a gas trapped in a fixed volume and you cool it down, the kinetic energy of the molecules decreases, and consequently, the pressure drops.

So, on our $p-V$ diagram, we need to draw a process where volume is constant but pressure decreases. This translates to a straight vertical line heading straight down.

Finally, we complete the cycle by moving from $z$ back to $x$. On the $V-T$ diagram, this is a vertical line. Since the horizontal axis is temperature, a vertical line means the temperature is held perfectly constant. It is an isothermal process.

As we move from $z$ to $x$, the volume is decreasing—we are compressing the gas. According to Boyle's law, when you compress an ideal gas at a constant temperature, its pressure increases inversely with volume:

On a $p-V$ diagram, this inverse relationship does not yield a straight line. Instead, it traces a hyperbolic curve moving upwards (increasing pressure) and to the left (decreasing volume), bringing us right back to our starting state $x$.

Putting all the pieces together, our translated $p-V$ diagram must consist of: 1. A horizontal line moving right ($x\to y$) 2. A vertical line moving down ($y\to z$) 3. A hyperbolic curve moving up and left ($z\to x$)

Comparing our derived sequence with the given options, we find that Option (b) is the exact match. By systematically applying the ideal gas law to the geometry of the graphs, we have successfully translated the thermodynamic story!