Question

If the excess pressures inside two soap bubbles are in the ratio \( 2:3 \), then the ratio of the volumes of the soap bubbles is: 

• A. \( 3:2 \)
• B. \( 9:4 \)
• C. \( 27:8 \)
• D. \( 81:16 \)

Hint:

For soap bubbles, excess pressure is inversely proportional to the radius. The volume ratio is found using the cube of the radius ratio.

Answer 1

The Correct Option is C

Step 1: Understanding the Relation between Pressure and Radius For a soap bubble, the excess pressure inside is given by: \[ P = \frac{4T}{R} \] where: - \( P \) is the excess pressure, - \( T \) is the surface tension, - \( R \) is the radius of the bubble. Given the ratio of excess pressures: \[ \frac{P_1}{P_2} = \frac{2}{3} \] Using the formula: \[ \frac{\frac{4T}{R_1}}{\frac{4T}{R_2}} = \frac{2}{3} \] \[ \frac{R_2}{R_1} = \frac{3}{2} \]

 Step 2: Find the Ratio of Volumes Volume of a sphere is: \[ V = \frac{4}{3} \pi R^3 \] So the ratio of volumes: \[ \frac{V_1}{V_2} = \left( \frac{R_1}{R_2} \right)^3 = \left( \frac{2}{3} \right)^3 \] \[ = \frac{8}{27} \] Thus, the ratio of volumes is: \[ \mathbf{27:8} \]