Surface Energy Gain Calculation

This solution calculates the gain in surface energy when a large water drop breaks into smaller ones.

Initial Conditions and Goal

• Initial drop radius, $R = 1 \text{ cm} = 0.01 \text{ m}$.
• Surface tension of water, $\gamma = 0.075 \text{ N m}^{-1}$.
• Number of smaller droplets, $n = 8$.
• Value of pi, $\pi = 3.14$.
• Goal: Find the gain in surface energy ($\Delta E$) in units of $10^{-7}$ J.

Step 1: Calculate Initial Surface Area

The surface area of the initial large drop is given by:

$ A_1 = 4 \pi R^2 $

Substituting the values:

$ A_1 = 4 \times 3.14 \times (0.01 \text{ m})^2 $

$ A_1 = 4 \times 3.14 \times 0.0001 \text{ m}^2 $

$ A_1 = 0.001256 \text{ m}^2 $

Step 2: Determine Radius of Smaller Droplets

Volume is conserved when the drop breaks. Let $r$ be the radius of each smaller droplet.

$ \text{Volume of large drop} = n \times \text{Volume of small drop} $

$ \frac{4}{3} \pi R^3 = n \times \frac{4}{3} \pi r^3 $

$ R^3 = n r^3 $

$ r = \frac{R}{n^{1/3}} = \frac{0.01 \text{ m}}{8^{1/3}} = \frac{0.01 \text{ m}}{2} = 0.005 \text{ m} $

Step 3: Calculate Final Total Surface Area

The total surface area of the $n$ smaller droplets is:

$ A_2 = n \times (4 \pi r^2) $

$ A_2 = 8 \times 4 \pi (0.005 \text{ m})^2 $

$ A_2 = 32 \pi (0.000025 \text{ m}^2) $

$ A_2 = 32 \times 3.14 \times 0.000025 \text{ m}^2 $

$ A_2 = 0.002512 \text{ m}^2 $

Alternatively, note that $A_2 = n^{1/3} A_1 = 8^{1/3} A_1 = 2 A_1 = 2 \times 0.001256 = 0.002512 \text{ m}^2$.

Step 4: Calculate Gain in Surface Energy

The gain in surface energy is the difference in total surface area multiplied by the surface tension:

$ \Delta E = (A_2 - A_1) \gamma $

$ \Delta E = (0.002512 \text{ m}^2 - 0.001256 \text{ m}^2) \times 0.075 \text{ N m}^{-1} $

$ \Delta E = (0.001256 \text{ m}^2) \times 0.075 \text{ N m}^{-1} $

$ \Delta E = 0.0000942 \text{ J} $

Step 5: Express Result in Required Units

The question asks for the energy gain in units of $10^{-7}$ J.

$ \Delta E = 0.0000942 \text{ J} = 9.42 \times 10^{-5} \text{ J} $

$ \Delta E = 942 \times 10^{-7} \text{ J} $

The numerical value is 942.