Question

The excess pressure inside a soap bubble is thrice the excess pressure inside a second soap bubble. The ratio between the volume of the first and the second bubble is :

• A. 1 : 9
• B. 1 : 3
• C. 1 : 81
• D. 1 : 27

Answer 1

Answer: (D)

1 : 27

Answer 2

The excess pressure $P_{\text{excess}}$ inside a soap bubble is given by the formula:

$P_{\text{excess}} = \frac{4T}{r},$ where $T$ is the surface tension and $r$ is the radius of the bubble.

Let the radii of the two bubbles be $r_1$ and $r_2$, and their excess pressures be $P_1$ and $P_2$, respectively.

Given:

$P_1 = 3P_2.$

Using the formula for excess pressure:

$\frac{4T}{r_1} = 3 \times \frac{4T}{r_2}.$

Cancelling common terms:

$\frac{1}{r_1} = 3 \times \frac{1}{r_2} \Rightarrow r_1 = \frac{r_2}{3}.$

Since the volume of a sphere is $V = \frac{4}{3} \pi r^3$, the ratio of the volumes is:

$\frac{V_1}{V_2} = \left( \frac{r_1}{r_2} \right)^3 = \left( \frac{1}{3} \right)^3 = \frac{1}{27}.$

Thus, the ratio of the volumes is $1 : 27$.