A drop of water is broken into two droplets . The sum of which property of the two drops is equal to that of the single one ?

Options

A. surface energy
B. radius of the molecule
C. volume
D. surface tension

Answer

To solve the problem, we need to analyze the properties of a single drop of water and how they relate to two smaller droplets formed from it. The key properties to consider are surface energy, radius, and volume.

### Step-by-Step Solution:

1. **Understanding Surface Energy**:
- The surface energy (E) of a drop is given by the formula:
\[ E = T \times A \]
where \( T \) is the surface tension and \( A \) is the surface area of the drop.
- For a single large drop with radius \( R \), the surface area \( A \) is:
\[ A = 4\pi R^2 \]
- Thus, the surface energy of the large drop is:
\[ E_{\text{large}} = T \times 4\pi R^2 \]

2. **Surface Energy of Two Smaller Droplets**:
- When the large drop is broken into two smaller droplets, each with radius \( r \), the surface area of each small droplet is:
\[ A = 4\pi r^2 \]
- Therefore, the surface energy of one small droplet is:
\[ E_{\text{small}} = T \times 4\pi r^2 \]
- For two small droplets, the total surface energy is:
\[ E_{\text{total small}} = 2 \times E_{\text{small}} = 2 \times T \times 4\pi r^2 = 2T \times 4\pi r^2 \]

3. **Volume Conservation**:
- The volume of the large drop is given by:
\[ V_{\text{large}} = \frac{4}{3}\pi R^3 \]
- The volume of each small droplet is:
\[ V_{\text{small}} = \frac{4}{3}\pi r^3 \]
- Therefore, the total volume of the two small droplets is:
\[ V_{\text{total small}} = 2 \times V_{\text{small}} = 2 \times \frac{4}{3}\pi r^3 = \frac{8}{3}\pi r^3 \]
- Setting the volumes equal for conservation:
\[ \frac{4}{3}\pi R^3 = \frac{8}{3}\pi r^3 \]
- Simplifying gives:
\[ R^3 = 2r^3 \implies R = 2^{1/3} r \]

4. **Conclusion**:
- The only property that remains conserved when the large drop is broken into two smaller droplets is the volume. The total volume of the two smaller droplets is equal to the volume of the single larger drop.
- Therefore, the answer to the question is that the **sum of the volumes of the two droplets is equal to that of the single one**.

### Final Answer:
The sum of the volumes of the two droplets is equal to that of the single drop.

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