What is ratio of surface energy of 1 small drop and 1 large drop, if 1000 small drops combined to form 1 large drop

Options

A. `1 : 100`
B. `1 : 1000`
C. `1 : 10`
D. `0.1 : 1`

Explanation

To find the ratio of surface energy of one small drop to one large drop, we can follow these steps:

### Step 1: Understand the relationship between the small drops and the large drop
We know that 1000 small drops combine to form 1 large drop. Let's denote:
- \( r \) = radius of one small drop
- \( R \) = radius of the large drop

### Step 2: Use the volume conservation principle
The volume of the large drop must equal the total volume of the 1000 small drops:
\[ \text{Volume of large drop} = \text{Volume of 1000 small drops} \]
This can be expressed mathematically as:
\[ \frac{4}{3} \pi R^3 = 1000 \times \frac{4}{3} \pi r^3 \]
Cancelling out the common terms, we have:
\[ R^3 = 1000 r^3 \]

### Step 3: Solve for the radius of the large drop
Taking the cube root of both sides gives:
\[ R = 10r \]

### Step 4: Calculate the surface energy of the small drop
The surface energy \( E_1 \) of one small drop is given by:
\[ E_1 = \text{Surface Tension} \times \text{Surface Area} \]
The surface area of one small drop is:
\[ \text{Surface Area} = 4\pi r^2 \]
Thus, the surface energy of one small drop is:
\[ E_1 = T \times 4\pi r^2 \]

### Step 5: Calculate the surface energy of the large drop
The surface energy \( E_2 \) of the large drop is:
\[ E_2 = T \times \text{Surface Area of large drop} \]
The surface area of the large drop is:
\[ \text{Surface Area} = 4\pi R^2 \]
Substituting \( R = 10r \):
\[ E_2 = T \times 4\pi (10r)^2 = T \times 4\pi \times 100r^2 = 100T \times 4\pi r^2 \]

### Step 6: Find the ratio of surface energies
Now, we can find the ratio of the surface energy of the small drop to that of the large drop:
\[ \text{Ratio} = \frac{E_1}{E_2} = \frac{T \times 4\pi r^2}{100T \times 4\pi r^2} \]
Simplifying this gives:
\[ \text{Ratio} = \frac{1}{100} \]

### Conclusion
The ratio of the surface energy of one small drop to that of one large drop is:
\[ \text{Ratio} = 1 : 100 \]