A water drop of radius $1$ cm is broken into eight equal droplets. Surface tension of water is $0.075$ N $m^{-1}$. The gain in surface energy is __________$\times10^{-7}$ J.(Take $\pi = 3.14$)
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.
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