Animated Solution for Physics - Thermodynamics: The given diagram shows four processes, i.e. isochoric, isobaric, isothermal and adiabatic. The correct assignment of the processes, in the same order is given by

When you look at a $p-V$ (pressure-volume) diagram, you are essentially looking at the fingerprint of a thermodynamic process. Every curve tells a story of how a gas behaves as it expands or compresses. In this problem, we are presented with four distinct paths starting from the exact same initial state, and our mission is to identify which path corresponds to which fundamental thermodynamic process: isochoric, isobaric, isothermal, and adiabatic.

The easiest processes to identify on a $p-V$ diagram are the ones that form straight horizontal or vertical lines. Let's start with curve $a$. It is a perfectly horizontal line moving to the right. A horizontal line means the value on the y-axis—which is pressure $p$—remains absolutely constant while the volume $V$ increases. A process where pressure is constant is known as an isobaric process.

Next, look at curve $d$. It is a perfectly vertical line dropping straight down. A vertical line means the value on the x-axis—volume $V$—is locked at a constant value while the pressure drops. When the volume of a gas does not change, the process is called isochoric.

Isothermal vs. Adiabatic

Now comes the slightly trickier part: distinguishing between curve $b$ and curve $c$. Both curves represent an expansion process where the volume increases and the pressure decreases. One is isothermal (constant temperature) and the other is adiabatic (no heat exchange). How do we tell them apart?

The secret lies in their slopes. For an isothermal process, the ideal gas law $pV=\text{constant}$ gives us a slope of $\frac{dp}{dV}=-\frac{p}{V}$. For an adiabatic process, the governing equation is $p{V}^{\gamma }=\text{constant}$, which yields a slope of $\frac{dp}{dV}=-\gamma \frac{p}{V}$.

Since the heat capacity ratio $\gamma$ is always greater than $1$ for any ideal gas, the magnitude of the adiabatic slope is always greater than the isothermal slope at any given point. In simple visual terms, the adiabatic curve is always steeper than the isothermal curve when starting from the same point.

Looking at our diagram, curve $c$ drops much more sharply than curve $b$. Therefore, curve $c$ must be the adiabatic process, and the gentler curve $b$ is the isothermal process.

The question asks us to list the processes in a specific order: isochoric, isobaric, isothermal, and adiabatic.

Based on our analysis: - Isochoric is $d$ - Isobaric is $a$ - Isothermal is $b$ - Adiabatic is $c$

Putting it all together, the correct sequence is $d,a,b,c$, which perfectly matches option (a).