Concept: The energy released when multiple small drops coalesce into one large drop is equal to the decrease in the total surface energy of the system.
Formula/Principle:1. Volume of a sphere:
V=‌πr32. Surface Area of a sphere:
A=4Ï€r23. Surface Energy:
U=A×S4. Energy Released:
∆U=Uinitial−UfinalSolution/Analysis:1.
Volume Conservation: Since the eight small drops coalesce into one large drop, the total volume remains constant. Let
rbe the radius of the small drops and
Rbe the radius of the final large drop.
8×(‌πr3)=‌πR32.
Finding the Final Radius (R): Canceling the common terms
‌πfrom both sides:
8r3=R3R=‌3√8r3=2r3.
Calculating Initial Surface Energy ( Uinitial ): The initial surface energy is the sum of the surface energies of the eight individual drops.
Uinitial=8×(Surface Area of one drop)×SUinitial=8×(4πr2)×S=32πr2S4.
Calculating Final Surface Energy ( Ufinal ): The final surface energy is the surface energy of the single large drop of radius
R.
Ufinal=(Surface Area of large drop)×SUfinal=4πR2SSubstituting
R=2r:
Ufinal=4Ï€(2r)2S=4Ï€(4r2)S=16Ï€r2S5.
Calculating Energy Released ( ∆U ): The energy released is the difference between the initial and final surface energies.
Energy Released =
Uinitial−Ufinal Energy Released =
32πr2S−16πr2S Energy Released =
16Ï€r2S