The excess pressure inside a soap bubble is thrice the excess pressure inside a second soap bubble. The ratio between the volume of the first and the second bubble is :
- 1 : 9
- 1 : 3
- 1 : 81
- 1 : 27
The Correct Option is D
Approach Solution - 1
To solve this problem, we need to understand the relationship between excess pressure inside a soap bubble and its radius, and subsequently its volume.
The formula for excess pressure (\(\Delta P\)) inside a soap bubble is given by:
\(\Delta P = \frac{4T}{R}\)
where \(T\) is the surface tension of the liquid and \(R\) is the radius of the soap bubble.
According to the question, the excess pressure inside the first bubble is thrice that inside the second bubble, so:
\(\Delta P_1 = 3 \Delta P_2\)
By substituting the formula for excess pressure, we get:
\(\frac{4T}{R_1} = 3 \times \frac{4T}{R_2}\)
After simplifying, the above equation becomes:
\(\frac{1}{R_1} = \frac{3}{R_2}\)
This implies:
\(R_2 = 3R_1\)
The volume of a sphere (or bubble) is given by:
\(V = \frac{4}{3} \pi R^3\)
Thus, the volumes of the bubbles can be represented as:
- Volume of the first bubble: \(V_1 = \frac{4}{3} \pi R_1^3\)
- Volume of the second bubble: \(V_2 = \frac{4}{3} \pi R_2^3 = \frac{4}{3} \pi (3R_1)^3\)
After substituting \(R_2 = 3R_1\) into the equation for \(V_2\), we find:
\(V_2 = \frac{4}{3} \pi (27R_1^3) = 27 \times \frac{4}{3} \pi R_1^3 = 27V_1\)
Therefore, the ratio of the volumes of the first and second bubble is:
\(\frac{V_1}{V_2} = \frac{1}{27}\)
Based on this calculation, the correct answer is 1 : 27.
Approach Solution -2
The excess pressure \( P_{\text{excess}} \) inside a soap bubble is given by the formula:
\[ P_{\text{excess}} = \frac{4T}{r}, \] where \( T \) is the surface tension and \( r \) is the radius of the bubble.
Let the radii of the two bubbles be \( r_1 \) and \( r_2 \), and their excess pressures be \( P_1 \) and \( P_2 \), respectively.
Given:
\[ P_1 = 3P_2. \]
Using the formula for excess pressure:
\[ \frac{4T}{r_1} = 3 \times \frac{4T}{r_2}. \]
Cancelling common terms:
\[ \frac{1}{r_1} = 3 \times \frac{1}{r_2} \Rightarrow r_1 = \frac{r_2}{3}. \]
Since the volume of a sphere is \( V = \frac{4}{3} \pi r^3 \), the ratio of the volumes is:
\[ \frac{V_1}{V_2} = \left( \frac{r_1}{r_2} \right)^3 = \left( \frac{1}{3} \right)^3 = \frac{1}{27}. \]
Thus, the ratio of the volumes is \( 1 : 27 \).
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