Pressure inside two soap bubbles are 1.01 and 1.02 atmospheres. Ratio between their volumes is

The correct option is 8 : 1.

Step 1: Understand the formula for pressure inside a soap bubble
The pressure inside a soap bubble is greater than the atmospheric pressure ($P_0$) due to the surface tension ($T$) of the soap solution. The absolute pressure $P$ inside a bubble of radius $R$ is given by:
$P=P_0+\frac{4T}{R}$
The excess pressure ($\Delta P$) is the difference between the inside pressure and the outside atmospheric pressure:
$\Delta P=P-P_0=\frac{4T}{R}$

Step 2: Calculate the excess pressure for each soap bubble
Assuming the atmospheric pressure $P_0=1\text{atm}$:
For the first bubble with pressure $P_1=1.01\text{atm}$:
$\Delta P_1=1.01-1=0.01\text{atm}$
For the second bubble with pressure $P_2=1.02\text{atm}$:
$\Delta P_2=1.02-1=0.02\text{atm}$

Step 3: Determine the ratio of the radii
Since surface tension $T$ is constant for both soap bubbles, the excess pressure is inversely proportional to the radius of the bubble:
$\Delta P\propto \frac{1}{R}$
Therefore, the ratio of the radius of the first bubble to that of the second bubble is:
$\frac{R_1}{R_2}=\frac{\Delta P_2}{\Delta P_1}$
Substituting the excess pressures:
$\frac{R_1}{R_2}=\frac{0.02}{0.01}=\frac{2}{1}$

Step 4: Find the ratio of their volumes
The volume $V$ of a spherical soap bubble is calculated as:
$V=\frac{4}{3}\pi R^3$
This shows that the volume is directly proportional to the cube of the radius:
$V\propto R^3$
Taking the ratio of the volumes of the two bubbles:
$\frac{V_1}{V_2}={\left(\frac{R_1}{R_2}\right)}^3$
Substituting the radius ratio:
$\frac{V_1}{V_2}={\left(\frac{2}{1}\right)}^3=\frac{8}{1}$

Thus, the ratio of their volumes is 8 : 1.