Question

Pressure inside two soap bubbles are 1.01 and 1.02 atmosphere, respectively. The ratio of their volumes is:

• A. 4:1
• B. 2:1
• C. 0.8:1
• D. 8:1

Answer 1

The correct option is D

8:1

Explanation of correct answer:-

Option (D):

Step 1:- Calculation of excess pressure.

Given, Outside pressure = 1 atm
since, soap bubble has two spherical surfaces

Pressure inside first bubble = 1.01 atm

Pressure inside second bubble = 1.02 atm

Excess pressure ($△\mathrm{P}$) = Pressure inside - Atmospheric pressure

Excess pressure $△{\mathrm{P}}_1$ = 1.01 - 1 = 0.01 atm

Excess pressure $△{\mathrm{P}}_2$ = 1.02 - 1 = 0.02 atm

Step 2 :- To find the value of radius R $△\mathrm{P}=\frac{4\mathrm{T}}{\mathrm{R}}$

We know that, $△\mathrm{P}=\frac{4\mathrm{T}}{\mathrm{R}}$, where T is the surface tension and R is the radius of the bubble.

$$\begin{gathered} \therefore △\mathrm{P}\propto 1\mathrm{R} \\ \Rightarrow \mathrm{R}\propto 1△\mathrm{P} \\ \Rightarrow \frac{{\mathrm{R}}_1}{{\mathrm{R}}_2}=\frac{∆{\mathrm{P}}_1}{∆{\mathrm{P}}_2} \\ \Rightarrow \frac{{\mathrm{R}}_1}{{\mathrm{R}}_2}=\frac{0.02}{0.01} \\ \Rightarrow \frac{{\mathrm{R}}_1}{{\mathrm{R}}_2}=\frac{2}{1} \end{gathered}$$

Step 3 :- Calculation of ratio of volumes.

$$\begin{gathered} {\mathrm{R}}_1=2\mathrm{cm}\mathrm{and}{\mathrm{R}}_2=1\mathrm{cm} \\ \mathrm{we}\mathrm{also}\mathrm{know}\mathrm{that},\mathrm{Volume}\propto {\left(\mathrm{radius}\right)}^3 \\ \Rightarrow \frac{{\mathrm{V}}_1}{{\mathrm{V}}_2}={\left(\frac{{\mathrm{R}}_1}{{\mathrm{R}}_2}\right)}^3 \\ \Rightarrow \frac{{\mathrm{V}}_1}{{\mathrm{V}}_2}={\left(\frac{2}{1}\right)}^3 \\ \Rightarrow \frac{{\mathrm{V}}_1}{{\mathrm{V}}_2}=\left(\frac{8}{1}\right) \end{gathered}$$

Final answer :- The ratio of volumes is 8 : 1.