2024 A/L Physics Past Paper · Question 23 (English Medium)

Answer: (2) only (B) is true.

What the question asks

This question is about blowing more air into a soap bubble so that it grows from radius $r$ to radius $2r$.

The air is blown in slowly, so the temperature stays the same.

The question gives three statements, (A), (B) and (C), about the pressure, the surface energy and the volume of the bubble.

What we must find: which statements are true.

How we find it: we check the pressure, the surface energy and the volume one at a time, each time doubling $r$.

What we know:

The bubble starts with radius $r$ and ends with radius $2r$.

The soap film has the same surface tension $T$ all the time.

What each letter means:

$r$ ($\text{m}$): radius of the bubble.

$T$ ($\text{N m}^{-1}$): surface tension of the soap solution.

$p_1$, $p_2$ ($\text{Pa}$): pressure inside and outside the bubble.

$A$ ($\text{m}^2$): area of one surface of the bubble.

$V$ ($\text{m}^3$): volume of the bubble.

Relevant theory

If 'r' is the radius of the bubble, 'T' the surface tension of the soap solution 'p_1' the pressure inside the bubble and 'p_2' the air pressure outside it.

$p_1 - p_2 = \dfrac{4T}{r}$

The work done against surface tensional forces in increasing the area of the film by a unit

area is known as its free surface energy.

Reading the options

What is happening: the soap film has two surfaces, inside and outside. Both pull inwards, so the air inside is pushed harder than the air outside, by $4T/r$.

A small bubble is curved sharply, so it squeezes hard. A big bubble is flatter and squeezes less.

The film stores energy in every square metre of its two surfaces, $T$ per square metre.

For example, blow up a party balloon. The first puffs are the hardest. Once it is big, it gets easier.

A soap bubble is the same: the bigger it gets, the less it pushes back.

Reasoning and calculation

The pressure inside the bubble was increased.

The extra pressure inside is $p_1-p_2=4T/r$. At $2r$ it is $4T/(2r)$, half as much. The pressure inside falls, so (A) is false.

A

The surface potential energy of the bubble was increased by four times.

Surface energy is $T$ times the area of both surfaces: $E=T\times2\times4\pi r^2$. At $2r$ the area is $4\pi(2r)^2=4\times4\pi r^2$. The energy grows four times, so (B) is true.

B

The volume of the bubble was increased by four times.

Volume is $\frac{4}{3}\pi r^3$. At $2r$ it is $\frac{4}{3}\pi(2r)^3=8\times\frac{4}{3}\pi r^3$. The volume grows eight times, not four, so (C) is false.

C

Statement (A) is false: a bigger bubble has a smaller extra pressure.

Statement (B) is true: the area, and so the surface energy, grows four times.

Statement (C) is false: the volume grows eight times.

So only (B) is true.

Why the other options fail

(A) is false: the extra pressure halves when $r$ doubles. (B) is true.

(A) is false: a bigger bubble pushes on its air less, $4T/(2r)$.

(C) is false: the volume grows as $r^3$, eight times.

(A) and (C) are false. Only the surface energy grows four times.

Conclusion

The pressure inside falls, the surface energy grows four times and the volume grows eight times. Only (B) is true.

The answer is (2).

Doubling $r$: $pressure\,difference\,\times \tfrac{1}{2},\; area\,and\,surface\,energy\,\times 4,\; volume\,\times 8$.

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