Question

A liquid drop of radius \(R = 0.1 \, \text{m}\) having surface tension \(S = \frac{0.01}{4\pi} \, \text{N/m}\) divides itself into \(n\) identical drops. In the process, the total change in the surface energy \(\Delta U = 10^{-4} \, \text{J}\). The value of \(n\) is

• A. 121
• B. 1381
• C. 8
• D. 27

Hint:

When dealing with problems involving changes in surface energy and volume, always consider the change in surface area and apply the formula for surface energy. For identical drops, the total change in surface area can be related to the number of drops formed.

Answer 1

The Correct Option is D

We are given the surface tension \(S\), the radius of the drop \(R\), and the change in surface energy \(\Delta U\), and we need to find the number of identical drops \(n\).
Step 1: Use the formula for change in surface energy. The change in surface energy when a drop of radius \(R\) divides into \(n\) identical drops can be expressed as: \[ \Delta U = S \cdot \Delta A \] where \(\Delta A\) is the change in the surface area.
Step 2: Surface area of a drop. The surface area \(A\) of a spherical drop is given by: \[ A = 4\pi r^2 \] where \(r\) is the radius of the drop.
Step 3: Initial and final surface areas. - Initial surface area of the liquid drop: \[ A_{\text{initial}} = 4\pi R^2 \] - Final surface area after division into \(n\) drops: \[ A_{\text{final}} = n \cdot 4\pi r^2 = n \cdot 4\pi \left(\frac{R}{n^{1/3}}\right)^2 = 4\pi R^2 \cdot n^{2/3} \]
Step 4: Change in surface area. The change in surface area is: \[ \Delta A = A_{\text{final}} - A_{\text{initial}} = 4\pi R^2 \left(n^{2/3} - 1\right) \]
Step 5: Calculate the change in surface energy. Now, using the relation for the change in surface energy, we have: \[ \Delta U = S \cdot \Delta A = S \cdot 4\pi R^2 \left(n^{2/3} - 1\right) \] Substitute the given values for \(S = \frac{0.01}{4\pi}\), \(R = 0.1\), and \(\Delta U = 10^{-4}\): \[ 10^{-4} = \frac{0.01}{4\pi} \cdot 4\pi (0.1)^2 \left(n^{2/3} - 1\right) \] Simplifying: \[ 10^{-4} = 0.01 \cdot 0.01 \left(n^{2/3} - 1\right) \] \[ 10^{-4} = 10^{-4} \left(n^{2/3} - 1\right) \] Thus: \[ n^{2/3} - 1 = 1 \] \[ n^{2/3} = 2 \] \[ n = 2^{3/2} = 27 \] Thus, the value of \(n\) is 27.