If the excess pressure inside a soap bubble is three times that of the other bubble, then the ratio of their volumes will be

• A. 1:3
• B. 1:10
• C. 1:17
• D. 1:27

To determine the ratio of the volumes of two soap bubbles based on their excess pressures, we can use the relationship between pressure and volume in bubbles. The excess pressure inside a soap bubble is given by the formula:

Excess Pressure Formula

The excess pressure (ΔP) inside a soap bubble is related to its radius (r) by the equation:

ΔP = 4T/r

where T is the surface tension of the soap solution.

Understanding the Relationship

If the excess pressure in one bubble is three times that of the other, we can express this as:

• ΔP1 = ΔP
• ΔP2 = 3ΔP

Volume and Radius Connection

The volume (V) of a bubble is related to its radius by the formula:

V = (4/3)πr³

From the excess pressure formula, we can derive that:

• For the first bubble: ΔP1 = 4T/r1
• For the second bubble: ΔP2 = 4T/r2

Finding the Ratio

Setting up the equations:

From ΔP1 and ΔP2, we have:

r1 = 4T/ΔP and r2 = 4T/(3ΔP)

Now, we can find the ratio of their volumes:

V1/V2 = (r1³)/(r2³)

Substituting the values of r1 and r2 gives:

V1/V2 = (4T/ΔP)³ / (4T/(3ΔP))³ = (1/(1/3)³) = 27

Final Answer

The ratio of their volumes is:

1:27

Thus, the correct answer is option D: 1:27.