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

The Correct Option is D

To resolve this, we must first establish the connection between the excess pressure within a soap bubble and its radius, which subsequently determines its volume.

The formula defining the excess pressure (\(\Delta P\)) inside a soap bubble is:

\(\Delta P = \frac{4T}{R}\)

Here, \(T\) denotes the surface tension of the liquid, and \(R\) represents the bubble's radius.

The problem states that the excess pressure in the first bubble is three times that in the second bubble, thus:

\(\Delta P_1 = 3 \Delta P_2\)

Substituting the excess pressure formula yields:

\(\frac{4T}{R_1} = 3 \times \frac{4T}{R_2}\)

Simplifying this equation leads to:

\(\frac{1}{R_1} = \frac{3}{R_2}\)

This implies the relationship:

\(R_2 = 3R_1\)

The volume of a sphere, or bubble, is calculated using the formula:

\(V = \frac{4}{3} \pi R^3\)

Consequently, the volumes of the bubbles are expressed as:

• Volume of the first bubble: \(V_1 = \frac{4}{3} \pi R_1^3\)
• Volume of the second bubble: \(V_2 = \frac{4}{3} \pi R_2^3 = \frac{4}{3} \pi (3R_1)^3\)

Upon substituting \(R_2 = 3R_1\) into the equation for \(V_2\), we obtain:

\(V_2 = \frac{4}{3} \pi (27R_1^3) = 27 \times \frac{4}{3} \pi R_1^3 = 27V_1\)

Therefore, the ratio of the volumes of the first to the second bubble is:

\(\frac{V_1}{V_2} = \frac{1}{27}\)

Based on these calculations, the correct answer is 1 : 27.