Fluid Mechanics

Step 1: Find Extra Pressure Inside a Bubble
The extra (excess) pressure inside a soap bubble is p=‌

4T

r

, where T is surface tension and r is the radius.
So for bubble A : pA=‌

4T

r1

And for bubble B : pB=‌

4T

r2

Step 2: Use the Gas Law for Bubbles
The usual gas equation is pV=nRT. Here, p is pressure, V is volume, n is the number of moles, R is the gas constant, and T is temperature.
Since the temperature is constant, pV∝n. The number of moles n is related to the mass m of the gas. So, pV∝m.
Step 3: Work Out the Mass for Each Bubble
For bubble A :
The volume of a sphere is VA=‌

4

3

πr13.
So pAVA∝mA becomes:
‌

4T

r1

×‌

4

3

πr13∝mA
This simplifies to mA∝r12.
For bubble B :
Similarly, VB=‌

4

3

πr23.
So ‌

4T

r2

×‌

4

3

πr23∝mB
This gives us mB∝r22.
Step 4: Find the Ratio of the Masses
The ratio is ‌

mA

mB

=‌

r12

r22

.