A spherical drop of liquid splits into `1000` identical spherical drops.If `mu_i` is the surface energy of the original drop and `mu_f` is the total surface energy of the resulting drops,the (gnoring evaporation), `(mu_(f))/(mu_(i))=((10)/(x))`.The value of `x` is

Answer

To solve the problem, we need to analyze the surface energy of the spherical drops before and after the split.

### Step 1: Understand the volume conservation
When a spherical drop of liquid splits into 1000 identical smaller drops, the volume of the original drop must equal the total volume of the smaller drops.

Let:
- \( R \) = radius of the original drop
- \( r \) = radius of each smaller drop

The volume of the original drop is given by:
\[ V_i = \frac{4}{3} \pi R^3 \]

The volume of one smaller drop is:
\[ V_f = \frac{4}{3} \pi r^3 \]

Since there are 1000 smaller drops, the total volume of the smaller drops is:
\[ V_{total} = 1000 \times V_f = 1000 \times \frac{4}{3} \pi r^3 \]

Setting the volumes equal gives:
\[ \frac{4}{3} \pi R^3 = 1000 \times \frac{4}{3} \pi r^3 \]
Cancelling \( \frac{4}{3} \pi \) from both sides:
\[ R^3 = 1000 r^3 \]

### Step 2: Relate the radii
Taking the cube root of both sides:
\[ R = 10 r \]

### Step 3: Calculate the surface energy
The surface energy \( \mu \) of a spherical drop is given by:
\[ \mu = \sigma \times A \]
where \( \sigma \) is the surface tension and \( A \) is the surface area.

The surface area of the original drop is:
\[ A_i = 4 \pi R^2 \]
Thus, the surface energy of the original drop is:
\[ \mu_i = \sigma \times 4 \pi R^2 \]

For the smaller drops, the surface area of one drop is:
\[ A_f = 4 \pi r^2 \]
So, the total surface energy of the 1000 smaller drops is:
\[ \mu_f = 1000 \times \sigma \times 4 \pi r^2 \]

### Step 4: Find the ratio of surface energies
Now, we need to find the ratio \( \frac{\mu_f}{\mu_i} \):
\[ \frac{\mu_f}{\mu_i} = \frac{1000 \times \sigma \times 4 \pi r^2}{\sigma \times 4 \pi R^2} \]

Cancelling \( \sigma \) and \( 4 \pi \):
\[ \frac{\mu_f}{\mu_i} = \frac{1000 r^2}{R^2} \]

### Step 5: Substitute \( R \) in terms of \( r \)
From Step 2, we have \( R = 10r \), thus:
\[ R^2 = (10r)^2 = 100r^2 \]

Substituting this into the ratio:
\[ \frac{\mu_f}{\mu_i} = \frac{1000 r^2}{100 r^2} = \frac{1000}{100} = 10 \]

### Step 6: Relate to the given equation
We are given:
\[ \frac{\mu_f}{\mu_i} = \frac{10}{x} \]

Setting the two expressions equal:
\[ 10 = \frac{10}{x} \]

### Step 7: Solve for \( x \)
Cross-multiplying gives:
\[ 10x = 10 \]
Thus:
\[ x = 1 \]

### Final Answer
The value of \( x \) is \( 1 \).

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