Two water droplets merge with each other to from a larger droplet. In this process

Options

A. Energy is liberated
B. Energy is absorbed
C. Some mass is connverted into energy
D. Energy is neither liberated nor absorbed

Answer

To solve the problem of two water droplets merging to form a larger droplet, we can follow these steps:

### Step 1: Understand the Volume Conservation
When two droplets merge, the total volume before merging is equal to the volume after merging. The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
Let the radii of the two droplets be \( r_1 \) and \( r_2 \). The volume of the two droplets before merging is:
\[ V_1 + V_2 = \frac{4}{3} \pi r_1^3 + \frac{4}{3} \pi r_2^3 \]
The volume of the resulting droplet with radius \( r_3 \) is:
\[ V_3 = \frac{4}{3} \pi r_3^3 \]
Setting these equal gives:
\[ \frac{4}{3} \pi r_1^3 + \frac{4}{3} \pi r_2^3 = \frac{4}{3} \pi r_3^3 \]
This simplifies to:
\[ r_1^3 + r_2^3 = r_3^3 \]

### Step 2: Calculate Initial and Final Energy
The energy associated with the surface tension of the droplets can be expressed as:
\[ U = S \cdot A \]
where \( S \) is the surface tension and \( A \) is the surface area. The surface area \( A \) of a sphere is given by:
\[ A = 4 \pi r^2 \]
Thus, the initial energy \( U_i \) of the two droplets is:
\[ U_i = S \cdot (4 \pi r_1^2 + 4 \pi r_2^2) = 4 \pi S (r_1^2 + r_2^2) \]
The final energy \( U_f \) of the larger droplet is:
\[ U_f = S \cdot 4 \pi r_3^2 = 4 \pi S r_3^2 \]

### Step 3: Relate Initial and Final Energies
Using the relation \( r_3^3 = r_1^3 + r_2^3 \), we can express \( r_3^2 \) in terms of \( r_1 \) and \( r_2 \). However, for our purpose, we can compare the energies directly:
\[ U_i = 4 \pi S (r_1^2 + r_2^2) \]
\[ U_f = 4 \pi S r_3^2 \]
Now we need to compare \( r_1^2 + r_2^2 \) with \( r_3^2 \).

### Step 4: Compare Energies
From the identity \( r_3^3 = r_1^3 + r_2^3 \), we can derive:
\[ r_3^2 < r_1^2 + r_2^2 \]
This implies:
\[ U_f < U_i \]
Thus, the initial energy is greater than the final energy.

### Step 5: Conclusion
Since the initial energy is greater than the final energy, energy is liberated during the merging process.

### Final Answer
The correct option is: **Energy is liberated.**

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