Animated Solution for Physics - Thermodynamics: For an adiabatic expansion of an ideal gas, the fractional change in its pressure is equal to (where, $\gamma$ is the ratio of specific heats)

Adiabatic Expansion

Imagine a gas expanding inside a perfectly insulated cylinder. No heat enters, and no heat leaves. This is the hallmark of an adiabatic process. For an ideal gas undergoing such a process, the relationship between its pressure $p$ and volume $V$ is beautifully captured by the equation:

Here, $\gamma$ is the ratio of specific heats ($C_p/C_v$), and $C$ is a constant. Our goal is to find the fractional change in pressure, which mathematically translates to finding the expression for $\frac{dp}{p}$.

Logarithmic Differentiation

We could use the standard product rule to differentiate $p{V}^{\gamma }=C$, but there is a much more elegant way. Whenever you have variables multiplied together or raised to powers, taking the natural logarithm simplifies the landscape immensely. Let's take the natural log ($\ln$) on both sides:

Using the properties of logarithms, we can split the product into a sum and bring the exponent down as a coefficient:

This linear-looking equation is now primed and ready for differentiation.

Now, let's differentiate the entire equation. Remember that the derivative of $\ln (x)$ is $\frac{1}{x}dx$.

Differentiating $\ln (p)$ gives us $\frac{dp}{p}$.

Differentiating $\gamma \ln (V)$ gives us $\gamma \frac{dV}{V}$.

And since $\ln (C)$ is just another constant, its derivative is exactly $0$.

Putting it all together:

To isolate the fractional change in pressure, we simply rearrange the terms by moving the volume term to the right side of the equation:

And there we have it! But let's not just stop at the math; let's understand the physics. Notice the negative sign? It tells a physical story. It means that if the volume increases (an expansion where $dV$ is positive), the pressure must decrease ($dp$ becomes negative). The gas does work at the expense of its own internal energy, causing the pressure to drop.

This simple yet profound derivation is a classic example of how mathematical operations perfectly mirror physical realities.