If a drop of liquid breaks into smaller droplets,it results in the lowering of the temperature of the droplets. Let a drop of radius $R$ break into $N$ small droplets,each of radius $r$. Estimate the drop in temperature.

Vedclass pdf generator app on play store
Vedclass iOS app on app store
(N/A) The volume of the liquid remains constant during the process.
Volume of large drop = $N \times$ Volume of small drop
$\frac{4}{3} \pi R^3 = N \times \frac{4}{3} \pi r^3 \implies R^3 = N r^3 \implies N = \frac{R^3}{r^3}$.
The change in surface area $\Delta A$ is given by the final surface area minus the initial surface area:
$\Delta A = N(4 \pi r^2) - 4 \pi R^2 = 4 \pi (N r^2 - R^2)$.
Since $N = R^3/r^3$,we have $\Delta A = 4 \pi (\frac{R^3}{r^3} r^2 - R^2) = 4 \pi R^2 (\frac{R}{r} - 1)$.
The energy released due to the increase in surface area is $E = S \Delta A = 4 \pi S R^2 (\frac{R}{r} - 1)$,where $S$ is the surface tension.
This energy is absorbed from the internal energy of the liquid,causing a temperature drop $\Delta \theta$.
$E = m C \Delta \theta$,where $m = \rho V = \rho (\frac{4}{3} \pi R^3)$ and $C$ is the specific heat capacity.
Equating the energy: $4 \pi S R^2 (\frac{R}{r} - 1) = \rho (\frac{4}{3} \pi R^3) C \Delta \theta$.
Solving for $\Delta \theta$: $\Delta \theta = \frac{3 S}{\rho C R} (\frac{R}{r} - 1) = \frac{3 S}{\rho C} (\frac{1}{r} - \frac{1}{R})$.

Explore More

Similar Questions

The temperature at which the surface tension of water is zero is:

On mixing salt in water,the surface tension of water will:

$A$ square wire frame of each side $L$ is dipped in soap solution. On taking out, a membrane is formed. If the surface tension of the solution is $T$, the force acting on the frame will be (in $T L$)

$A$ thread is tied slightly loose to a wire frame as shown in the figure. The frame is dipped into a soap solution and taken out,such that the frame is completely covered with a soap film. When the portion $A$ is punctured with a pin,the thread:

Give two practical illustrations of surface tension.

Vedclass Products

For Students

Vedclass Test Series

Mock tests in real JEE/NEET style with performance analysis. 5-day free trial.

Start Free Trial
For Teachers

Exam Paper Generator

Generate Set A/B/C/D exam papers from 7.5L+ questions in 2 minutes. 3 chapters free.

Try Free
For Institutes

Online Exam Module

Live online exams with unlimited students, 360° analytics & white-label branding.

See Demo