Animated Solution for Physics - Thermodynamics: Starting with the same initial conditions, an ideal gas expands from volume $V_1$ to $V_2$ in three different ways, the work done by the gas is $W_1$ if the process is purely isothermal, $W_2$ if purely isobaric and $W_3$ if purely adiabatic, then

Imagine you are holding a piston containing an ideal gas. You want to expand this gas from an initial volume $V_1$ to a final volume $V_2$. But here is the catch: you can do this in three completely different ways. You can keep the pressure constant (isobaric), keep the temperature constant (isothermal), or perfectly insulate the cylinder so no heat escapes (adiabatic).

The question is: in which process does the gas do the most work?

The p-V Diagram To answer this, we don't need to crunch massive equations. We just need to visualize the processes on a $p-V$ (pressure-volume) diagram. Why? Because the work done by a gas is simply the area under its $p-V$ curve.

Let's draw all three processes starting from the exact same initial state $A(p_1,V_1)$.

The Isobaric Process: In an isobaric expansion, the pressure remains perfectly constant. On our graph, this is a straight horizontal line moving to the right. The pressure doesn't drop at all.

2. The Isothermal Process: Here, the temperature is constant. According to Boyle's Law, $p\propto \frac{1}{V}$. As the volume increases, the pressure drops. The curve is a gentle rectangular hyperbola. Because the pressure is dropping, this curve must lie below the horizontal isobaric line.

3. The Adiabatic Process: In an adiabatic expansion, the gas does work at the expense of its own internal energy, causing it to cool down. The governing equation is $p\propto \frac{1}{V^{\gamma }}$. Since the adiabatic exponent $\gamma$ is always greater than 1, the pressure drops much faster than in the isothermal case. Therefore, the adiabatic curve is the steepest and lies below the isothermal curve.

Now, look at the areas under these three curves: - The isobaric curve is the highest, so it has the largest area ($W_2$). - The isothermal curve is in the middle, giving it a medium area ($W_1$). - The adiabatic curve drops the fastest and is the lowest, giving it the smallest area ($W_3$).

Comparing the areas, we get our beautiful, elegant result:

A Quick Trap Warning: What if the question asked about compression from $V_2$ to $V_1$? If we compress the gas, the adiabatic curve shoots up the fastest, making it the highest curve! In compression, the adiabatic work is the greatest. Always read carefully whether the gas is expanding or compressing!