Animated Solution for Physics - Properties of Solids and Liquids: A soap bubble, blown by a mechanical pump at the mouth of a tube, increases in volume, with time, at a constant rate. The graph that correctly depicts the time dependence of pressure inside the bubble is given by

## The Expanding Soap Bubble: A Lesson in Open Systems

Have you ever blown a soap bubble and wondered how the pressure inside changes as it grows? It’s a fascinating physics problem that tests your understanding of fluid mechanics and thermodynamics. Let's dive into a classic JEE question that often traps students with a deceptive assumption.

The pump is steadily blowing air into the bubble. The problem states a crucial condition: the volume of the bubble increases at a constant rate.

Mathematically, we can express this as: where $c$ is a positive constant.

If we integrate this with respect to time (assuming the initial volume is zero at $t=0$), we get a simple linear relationship for the volume:

We know the volume of a sphere is given by $V=\frac{4}{3}\pi r^3$. Let's equate this to our time-dependent volume to see how the radius $r$ evolves:

Solving for the radius $r$, we get:

This tells us that the radius doesn't grow linearly; it grows proportional to the cube root of time. As the bubble gets larger, it takes more and more air to increase its radius by the same amount.

Pressure Now, let's talk about pressure. A soap bubble has two liquid-air interfaces (an inner and an outer surface). The surface tension $T$ of the soap solution creates an inward squeeze, resulting in an excess pressure inside the bubble.

The total pressure $p$ inside the bubble is the sum of the atmospheric pressure $p_0$ and this excess pressure:

Now, we substitute our expression for the radius $r(t)$ into this pressure equation:

We can group all the constant terms—$4T$ and the geometric factors—into a single new constant, let's call it $K$. This simplifies our equation beautifully:

Why Boyle's Law Fails Here Before we look at the graph, we must address a massive trap. Many students (and even some textbooks!) attempt to solve this by applying Boyle's Law, assuming $pV=\text{constant}$.

This is fundamentally incorrect. Boyle's Law only applies to a closed system where the mass (or number of moles) of the gas is constant. In our scenario, a pump is actively forcing new air into the bubble. The number of moles $n$ is increasing! Therefore, the ideal gas law $pV=nRT$ tells us that $pV$ is definitely not constant. Always check if your system is open or closed before applying gas laws.

Let's look back at our derived equation:

This equation is in the classic slope-intercept form of a straight line, $y=mx+c$. - Our "y" variable is the pressure $p$. - Our "x" variable is $\frac{1}{t^{1/3}}$. - The y-intercept is the atmospheric pressure $p_0$. - The slope is the positive constant $K$.

Therefore, if we plot $p$ against $\frac{1}{t^{1/3}}$, we will get a straight line that starts at $p_0$ on the y-axis and goes upwards. This perfectly matches the graph shown in option (a).

Physics is beautiful when you follow the fundamental principles step-by-step and avoid the temptation of blindly applying formulas!