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Question: Which of the following represents correctly the variation of surface tension \(T\) with temperature ...

Which of the following represents correctly the variation of surface tension TT with temperature θ\theta ?
A. T∝θc−θT \propto {\theta _c} - \theta
B. T∝θc−θ−1T \propto {\theta _c} - {\theta ^{ - 1}}
C. T∝θc−θ0T \propto {\theta _c} - {\theta ^0}
D. T∝θc−θ−2T \propto {\theta _c} - {\theta ^{ - 2}}

Explanation

Solution

Hint: The relation between the surface tension of the liquid and the temperature of the liquid is given by Eotvos rule. By Eotvos rule, it is clear that molar surface tension is proportional to the difference between the critical temperature and the current temperature of the liquid.

Step by step solution:
(i) Surface tension:
The tension on the surface of the liquid due to the cohesive force of attraction between the liquid molecules placed below the surface of the liquid and it leads to minimize surface area.

(ii) Impact of temperature on surface tension:
When there is an increase in the temperature of the liquid, it leads to the decrease in the cohesive force of the liquid molecule because while the temperature tends to increase the molecular vibration leads to increase. Hence, the surface tension of the liquid will automatically drop down.

(iii) Eotvos rule:
The Eotvos rule states that the molar surface tension is directly proportional to the algebraic difference between the critical temperature of the liquid and the current temperature of the liquid.
Tmol=k(θc−θ)  .............................(1){T_{mol}} = k\left( {{\theta _c} - \theta } \right)\;.............................\left( 1 \right)
Since, the molar surface tension is Tmol=T×V(23)  ...........................................(2){T_{mol}} = T \times {V^{\left( {\dfrac{2}{3}} \right)}}\;...........................................\left( 2 \right)
Where, Tmol{T_{mol}} is the molar surface tension of the liquid, TT is the surface tension of the liquid and VV is the molar volume of the liquid.

By using equation (2) in equation (1), we get
T×V(23)=k(θc−θ)   T=k(θc−θ)V(23)  .................................................(3)  T \times {V^{\left( {\dfrac{2}{3}} \right)}} = k\left( {{\theta _c} - \theta } \right)\; \\\ T = \dfrac{{k\left( {{\theta _c} - \theta } \right)}}{{{V^{\left( {\dfrac{2}{3}} \right)}}}}\;.................................................\left( 3 \right) \\\
From the equation (3), it is clear that T∝(θc−θ)T \propto \left( {{\theta _c} - \theta } \right)
Hence, the option (A) is correct.

Note: The surface tension of the liquid is always proportional to the temperature of the liquid. An impact in the temperature of the liquid leads to a change in the surface tension of the liquid. At critical temperature, the surface tension of the liquid is zero.