A) W
B) 2W
C) \[\sqrt{2}W\]
D) \[{{4}^{1/3}}W\]
Correct Answer: B
Solution :
From the definition of surface tension (T), the surface tension of a liquid is equal to the work (W) required to increase the surface area of the liquid film by unity at constant temperature, \[\therefore \] \[W=T\times \Delta A\] Since, surface area of a sphere is \[4\pi {{R}^{2}}\] and there are two free surfaces, we have \[W=T\times \Delta A\] ?(i) and volume of sphere \[=\frac{4}{3}\,\pi {{R}^{3}}\] i.e., \[V=\frac{4}{3}\,\pi {{R}^{3}}\] \[\Rightarrow \] \[R={{\left( \frac{3V}{4\pi } \right)}^{1/3}}\] ?(ii) From Eqs. (i) and (ii), we get \[W=T\times 8\pi \times {{\left( \frac{3V}{4\pi } \right)}^{2/3}}\] \[\Rightarrow \] \[W\propto \,{{V}^{2/3}}\] \[\therefore \] \[{{W}_{1}}\propto \,{{V}_{1}}^{2/3}\] and \[{{W}_{2}}\propto \,{{V}_{2}}^{2/3}\] \[\therefore \] \[\frac{{{W}_{2}}}{{{W}_{1}}}=\,{{\left( \frac{2V}{V} \right)}^{2/3}}\] \[\Rightarrow \] \[{{W}_{2}}={{2}^{2/3}}{{W}_{1}}={{4}^{1/3}}W\]
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