Question

When 'n' identical mercury drops combine to form a single big drop

• A. Surface area increases and heat is released
• B. Surface area decreases and heat is released
• C. Surface area increases and heat is absorbed
• D. Surface area decreases and heat is absorbed

Hint:

A system naturally tends to move towards a state of lower potential energy. For liquids, surface tension creates a surface energy proportional to the surface area. Combining drops reduces the total surface area, thus lowering the total surface energy. This lost energy is released, usually as heat.

Answer 1

The Correct Option is B

Step 1: Analyze the change in volume.
Let 'r' be the radius of each small drop and 'R' be the radius of the big drop.
The total volume is conserved when the drops combine.
Volume of 'n' small drops = Volume of one big drop.
\( n \times (\frac{4}{3}\pi r^3) = \frac{4}{3}\pi R^3 \).
This gives the relationship between the radii: \( n r^3 = R^3 \implies R = n^{1/3}r \).
Step 2: Analyze the change in surface area.
Total initial surface area of 'n' small drops: \( A_{initial} = n \times (4\pi r^2) \).
Final surface area of the big drop: \( A_{final} = 4\pi R^2 \).
Substitute \(R = n^{1/3}r\) into the final area equation.
\( A_{final} = 4\pi (n^{1/3}r)^2 = 4\pi n^{2/3}r^2 \).
Let's compare \(A_{initial}\) and \(A_{final}\). We can look at their ratio.
\( \frac{A_{final}}{A_{initial}} = \frac{4\pi n^{2/3}r^2}{n \times 4\pi r^2} = \frac{n^{2/3}}{n} = n^{-1/3} = \frac{1}{n^{1/3}} \).
Since n>1 for combination, \(n^{1/3}>1\), and therefore \( \frac{1}{n^{1/3}}<1 \).
This means \( A_{final}<A_{initial} \). So, the surface area decreases.
Step 3: Analyze the change in energy.
Surface energy is given by \( E = T \times A \), where T is the surface tension and A is the surface area.
Since the surface area decreases, the total surface energy of the system also decreases.
The change in energy is \( \Delta E = E_{final} - E_{initial} \), which is negative.
By conservation of energy, this loss in potential energy must be converted into another form, which is released as heat.
Therefore, heat is released.
Combining the results, the surface area decreases and heat is released.