Question:
A soap bubble in vacuum has a radius of 3 cm and another soap bubble in vacuum has a radius of 4 cm. If the two bubbles coalesce under isothermal condition, then the radius of the new bubble will be:
A soap bubble in vacuum has a radius of 3 cm and another soap bubble in vacuum has a radius of 4 cm. If the two bubbles coalesce under isothermal condition, then the radius of the new bubble will be:
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When two soap bubbles coalesce, their volumes are added, and the radius of the resulting bubble is the cube root of the sum of the volumes.
Updated On: Apr 22, 2026
- 2.3 cm
- 4.5 cm
- 5 cm
- 7 cm
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The Correct Option is C
Solution and Explanation
Step 1: Volume of a Soap Bubble.
The volume \( V \) of a soap bubble is related to its radius \( r \) by the formula: \[ V = \frac{4}{3} \pi r^3 \] This volume is conserved when two soap bubbles coalesce under isothermal conditions.
Step 2: Total Volume Before Coalescence.
The total volume before coalescence is the sum of the volumes of both bubbles. If the radius of the first bubble is 3 cm and the radius of the second bubble is 4 cm, the total initial volume is: \[ V_{\text{total}} = \frac{4}{3} \pi (3^3) + \frac{4}{3} \pi (4^3) \] \[ V_{\text{total}} = \frac{4}{3} \pi (27) + \frac{4}{3} \pi (64) = \frac{4}{3} \pi (91) = 121.33 \pi \, \text{cm}^3 \]
Step 3: New Radius After Coalescence.
The volume of the new bubble is the same as the total initial volume: \[ V_{\text{new}} = \frac{4}{3} \pi r_{\text{new}}^3 \] Equating the two volumes: \[ \frac{4}{3} \pi r_{\text{new}}^3 = 121.33 \pi \] \[ r_{\text{new}}^3 = 121.33 \] \[ r_{\text{new}} = \sqrt[3]{121.33} = 5 \, \text{cm} \]
The volume \( V \) of a soap bubble is related to its radius \( r \) by the formula: \[ V = \frac{4}{3} \pi r^3 \] This volume is conserved when two soap bubbles coalesce under isothermal conditions.
Step 2: Total Volume Before Coalescence.
The total volume before coalescence is the sum of the volumes of both bubbles. If the radius of the first bubble is 3 cm and the radius of the second bubble is 4 cm, the total initial volume is: \[ V_{\text{total}} = \frac{4}{3} \pi (3^3) + \frac{4}{3} \pi (4^3) \] \[ V_{\text{total}} = \frac{4}{3} \pi (27) + \frac{4}{3} \pi (64) = \frac{4}{3} \pi (91) = 121.33 \pi \, \text{cm}^3 \]
Step 3: New Radius After Coalescence.
The volume of the new bubble is the same as the total initial volume: \[ V_{\text{new}} = \frac{4}{3} \pi r_{\text{new}}^3 \] Equating the two volumes: \[ \frac{4}{3} \pi r_{\text{new}}^3 = 121.33 \pi \] \[ r_{\text{new}}^3 = 121.33 \] \[ r_{\text{new}} = \sqrt[3]{121.33} = 5 \, \text{cm} \]
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