A drop of water of diameter 0.2 cm is broken up into 27,000 droplets of equal volumen. How much work will be done against surface tension in the process? (ST. = 7 x 10^-2 N/m)
Example Problem
SaitechAI Worksheet
Topic: Work done against surface tension when a single drop splits into many droplets (Equal-volume droplets)10 Questions × 2 marks = 20
Timer
Start: —
End: —
Duration: —
- Assume spherical drops. Volume conserved: \( n\frac{4}{3}\pi r^3=\frac{4}{3}\pi R^3 \Rightarrow r=\dfrac{R}{n^{1/3}}\).
- Surface-area change: \( \Delta A = 4\pi R^2\left(n^{1/3}-1\right)\).
- Work against surface tension: \( W=\sigma\,\Delta A \). Use SI units (R in m).
- Show steps in the answer box. Toggle Key to verify.
Score: 0/20
Q1. A water drop of diameter 2 mm is broken into 1000 equal droplets. Surface tension = 0.072 N m−1. Find the work done.
Marks:
Key: \(R=1\times10^{-3}\,\text{m},\ n=10^3\Rightarrow n^{1/3}=10\). \(\Delta A=4\pi R^2(10-1)=36\pi\times10^{-6}\,\text{m}^2\). \(W=\sigma\Delta A=0.072\times36\pi\times10^{-6}\approx \boxed{8.143\times10^{-6}\,\text{J}}\).
Q2. A mercury drop of radius 0.8 cm is sprayed into \(10^5\) equal droplets. Surface tension = 0.465 N m−1. Energy required?
Marks:
Key: \(R=8.0\times10^{-3}\,\text{m},\ n^{1/3}\approx46.416\). \(\Delta A=4\pi R^2(n^{1/3}-1)\). \(W\approx \boxed{1.698\times10^{-2}\,\text{J}}\).
Q3. A water drop of radius 0.2 cm splits into 27,000 equal droplets. Surface tension = 0.070 N m−1. Compute the work.
Marks:
Key: \(R=2.0\times10^{-3}\,\text{m},\ n^{1/3}=30\). \(\Delta A=4\pi R^2(29)\). \(W=\sigma\Delta A\approx \boxed{1.020\times10^{-4}\,\text{J}}\).
Q4. A soap solution (σ = 0.025 N m−1) drop of diameter 1 cm is atomized into 8000 equal droplets. Work done?
Marks:
Key: \(R=5.0\times10^{-3}\,\text{m},\ n^{1/3}=20\). \(\Delta A=4\pi R^2(19)\). \(W\approx \boxed{1.492\times10^{-4}\,\text{J}}\).
Q5. A glycerine drop of radius 0.5 cm is broken into \(10^6\) equal droplets. σ = 0.063 N m−1. Energy?
Marks:
Key: \(R=5.0\times10^{-3}\,\text{m},\ n^{1/3}=100\). \(\Delta A=4\pi R^2(99)\). \(W\approx \boxed{1.959\times10^{-3}\,\text{J}}\).
Q6. A water drop of diameter 3 mm splits into 125 equal droplets. σ = 0.072 N m−1. Work done?
Marks: