A liquid drop has surface energy U. It breaks into n equal small droplets. If the new surface energy after division becomes 5U, then the value of n is ________.

Answer

To solve the problem, we need to analyze the situation step by step.

### Step 1: Understand the Initial Conditions
We have a large liquid drop with surface energy \( U \). When this drop breaks into \( n \) smaller droplets, we need to find the relationship between the initial and final surface energies.

### Step 2: Volume Conservation
The volume of the original drop is given by:
\[ V = \frac{4}{3} \pi R^3 \]
where \( R \) is the radius of the large drop. When it breaks into \( n \) smaller droplets, each with radius \( r \), the total volume of the smaller droplets is:
\[ n \cdot \frac{4}{3} \pi r^3 \]
Since the volume is conserved, we can set these equal:
\[ \frac{4}{3} \pi R^3 = n \cdot \frac{4}{3} \pi r^3 \]
Cancelling \( \frac{4}{3} \pi \) from both sides gives:
\[ R^3 = n r^3 \]

### Step 3: Relate the Radii
From the volume conservation, we can express \( R \) in terms of \( r \):
\[ R = n^{1/3} r \]

### Step 4: Calculate Initial Surface Energy
The surface energy \( U \) of the large drop is given by:
\[ U = \text{Surface Area} \times \text{Tension} = 4 \pi R^2 \cdot \gamma \]
where \( \gamma \) is the surface tension.

### Step 5: Calculate Final Surface Energy
The surface energy of the \( n \) smaller droplets is:
\[ U' = n \cdot \text{Surface Area of one droplet} \cdot \gamma = n \cdot 4 \pi r^2 \cdot \gamma \]

### Step 6: Set Up the Equation
According to the problem, the new surface energy after division becomes \( 5U \):
\[ n \cdot 4 \pi r^2 \cdot \gamma = 5 \cdot (4 \pi R^2 \cdot \gamma) \]
Cancelling \( 4 \pi \gamma \) from both sides gives:
\[ n r^2 = 5 R^2 \]

### Step 7: Substitute for \( R \)
Substituting \( R = n^{1/3} r \) into the equation:
\[ n r^2 = 5 (n^{1/3} r)^2 \]
This simplifies to:
\[ n r^2 = 5 n^{2/3} r^2 \]
Dividing both sides by \( r^2 \) (assuming \( r \neq 0 \)):
\[ n = 5 n^{2/3} \]

### Step 8: Solve for \( n \)
Rearranging gives:
\[ n^{1/3} = 5 \implies n = 5^3 = 125 \]

### Final Answer
Thus, the value of \( n \) is:
\[ \boxed{125} \]