If `10^(6)` tiny drops coalesce to form a big drop. The surface tension of liquid is T. Then find the `%` fractional energy loss?

Options

A. `9%`
B. `19%`
C. `99%`
D. `48%`

Answer

To find the percentage fractional energy loss when \(10^6\) tiny drops coalesce to form a big drop, we can follow these steps:

### Step 1: Understand the relationship between volume and radius
The volume of a single tiny drop and the volume of the big drop must be equal. The volume \(V\) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
For \(10^6\) tiny drops of radius \(r\), the total volume is:
\[ V_{\text{tiny}} = 10^6 \times \frac{4}{3} \pi r^3 \]
For the big drop of radius \(R\), the volume is:
\[ V_{\text{big}} = \frac{4}{3} \pi R^3 \]
Setting these equal gives:
\[ 10^6 \times \frac{4}{3} \pi r^3 = \frac{4}{3} \pi R^3 \]
Cancelling \(\frac{4}{3} \pi\) from both sides:
\[ 10^6 r^3 = R^3 \]
Taking the cube root:
\[ R = 10^2 r = 100r \]

### Step 2: Calculate the surface area before and after coalescence
The surface area \(A\) of a sphere is given by:
\[ A = 4 \pi r^2 \]
The total surface area of \(10^6\) tiny drops is:
\[ A_{\text{tiny}} = 10^6 \times 4 \pi r^2 = 4 \pi \times 10^6 r^2 \]
The surface area of the big drop is:
\[ A_{\text{big}} = 4 \pi R^2 = 4 \pi (100r)^2 = 4 \pi \times 10^4 r^2 \]

### Step 3: Calculate the energy associated with surface tension
The energy \(E\) associated with surface tension \(T\) is given by:
\[ E = \text{Surface Area} \times \text{Surface Tension} \]
Thus, the initial energy \(E_{\text{initial}}\) is:
\[ E_{\text{initial}} = A_{\text{tiny}} \times T = 4 \pi \times 10^6 r^2 \times T \]
And the final energy \(E_{\text{final}}\) is:
\[ E_{\text{final}} = A_{\text{big}} \times T = 4 \pi \times 10^4 r^2 \times T \]

### Step 4: Calculate the change in energy
The change in energy \(\Delta E\) is:
\[ \Delta E = E_{\text{initial}} - E_{\text{final}} = (4 \pi \times 10^6 r^2 T) - (4 \pi \times 10^4 r^2 T) \]
Factoring out common terms:
\[ \Delta E = 4 \pi r^2 T (10^6 - 10^4) = 4 \pi r^2 T (10^6 - 10^4) = 4 \pi r^2 T (999900) \]

### Step 5: Calculate the fractional energy loss
The fractional energy loss is given by:
\[ \text{Fractional Energy Loss} = \frac{\Delta E}{E_{\text{initial}}} \times 100 \]
Substituting the values:
\[ \text{Fractional Energy Loss} = \frac{4 \pi r^2 T (999900)}{4 \pi \times 10^6 r^2 T} \times 100 \]
Cancelling out common terms:
\[ = \frac{999900}{10^6} \times 100 = 99.99\% \]

### Final Answer
The percentage fractional energy loss is approximately \(99\%\).

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