Animated Solution for Physics - Thermodynamics: Which of the following graphs correctly represent the variation of $\beta =-\frac{dV/dp}{V}$ with $p$ for an ideal gas at constant temperature?

Have you ever wondered how a gas responds when you squeeze it? Does it fight back harder the more you compress it, or does it yield easily? This fundamental question is captured by a property called isothermal compressibility, denoted by $\beta$.

In this problem, we are asked to find the correct graphical representation of $\beta$ against pressure $p$ for an ideal gas at a constant temperature. Let's embark on this mathematical journey to decode the physical behavior of the gas!

The problem provides us with the mathematical definition of compressibility:

This expression might look intimidating, but let's break it down. The term $\frac{dV}{dp}$ represents the rate of change of volume with respect to pressure. Since increasing pressure generally decreases volume, this derivative is inherently negative. The negative sign in front of the formula ensures that our compressibility $\beta$ turns out to be a positive number. The $\frac{1}{V}$ factor makes it a fractional (or relative) change in volume.

We are also given a crucial constraint: the gas is ideal and the temperature is constant. This immediately points us to the Ideal Gas Law:

Since the temperature $T$ is constant, the entire right side of the equation ($nRT$) is a constant.

To find how $\beta$ varies with $p$, we need to evaluate the derivative $\frac{dV}{dp}$. Let's differentiate our ideal gas equation with respect to pressure $p$. Using the product rule on the left side:

Since $\frac{dp}{dp}=1$, this simplifies to:

Now, let's isolate the derivative:

This tells us exactly how the volume changes with pressure at any given moment.

Now comes the elegant part. We substitute our expression for $\frac{dV}{dp}$ back into the original definition of $\beta$:

The negative signs cancel out, and beautifully, the volume $V$ also cancels out! We are left with a remarkably simple relationship:

What does $\beta =\frac{1}{p}$ look like on a graph?

In mathematics, an equation of the form $y=\frac{k}{x}$ (where $k$ is a positive constant) represents a rectangular hyperbola in the first quadrant. As the pressure $p$ increases, the compressibility $\beta$ decreases. This makes perfect physical sense: as you compress a gas more and more, it becomes denser and harder to compress further.

Looking at our options, the graph that correctly depicts a decreasing curve that is concave upwards (a rectangular hyperbola) is option (a).

This problem is a beautiful example of how a complex-looking physical definition can collapse into a simple, elegant mathematical relationship when we apply the fundamental laws of thermodynamics!