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Mechanical Properties Of Fluids

Question
CBSEENPH11020579

A certain number of spherical drops of a liquid of radius r coalesce to form a single drop of radius R and volume V. If T is the surface tension of the liquid, then

  • Energy = 4VT open parentheses 1 over straight r minus 1 over straight R close parentheses is released.

  • Energy = 3VTopen parentheses 1 over straight r plus 1 over R close parentheses is absorbed

  • Energy = 3 VT open parentheses 1 over straight r minus 1 over straight R close parentheses is released

  • Energy is neither absorbed nor released.

Solution

C.

Energy = 3 VT open parentheses 1 over straight r minus 1 over straight R close parentheses is released

If the surface area changes, it will change the surface energy also.
When the surface area decreases, it means energy is released ad vice-versa.
Change in surface energy is increment straight A space straight x space straight T   ... (i)
Let, there be n number of drops initially.
So, increment straight A space equals space 4 πR squared space minus space straight n space left parenthesis 4 πr squared right parenthesis       ... (ii)
Volume is constant.
So, straight n space 4 over 3 πr cubed space equals space 4 over 3 πr cubed space equals space straight V         ... (iii)
From equations (ii) and (iii), we have
increment straight A space equals space 3 over straight R fraction numerator 4 straight pi over denominator 3 end fraction xR cubed space minus space 3 over straight r open parentheses straight n fraction numerator 4 straight pi over denominator 3 straight r cubed end fraction close parentheses
space space space space space space space space equals space space 3 over straight R xV space minus space 3 over straight r straight V
increment straight A space space equals space 3 straight V space open parentheses 1 over straight R minus 1 over straight r close parentheses
space space space space space space space space equals space left parenthesis negative right parenthesis ve space value
Since, R > r, incrementA is negative.
That is, the surface area is decreased.
Hence, energy must be released.
Energy released = increment straight A space straight x space straight T space equals space minus 3 VT open parentheses 1 over straight r minus 1 over straight R close parentheses
In the above expression, (-)ve sign shows that amount of energy is released.