Three liquids have same surface tension and have densities ρ 1 , ρ 2 and ρ 3 ( ρ 1 ρ 2 ρ 3 ) . In three identical capillaries rise of liquid is same. The corresponding angles of contac

Question

Three liquids have same surface tension and have densities ρ 1 ​ , ρ 2 ​ and ρ 3 ​ ( ρ 1 ​ > ρ 2 ​ > ρ 3 ​ ) . In three identical capillaries rise of liquid is same. The corresponding angles of contact θ 1 ​ , θ 2 ​ and θ 3 ​ are related as

TestHub · Accepted Answer

The height of capillary rise is given by the formula:

$h = \frac{2 T cos θ}{r ρ g}$

Rearranging the formula for $cos θ$ , we get:

$cos θ = \frac{h r g}{2 T} ρ$

It is given that the surface tension $T$ , capillary radius $r$ , and capillary rise $h$ are the same for all three liquids. Therefore, the term $\frac{h r g}{2 T}$ is a constant.

This implies that $cos θ \propto ρ$ .

Given the relation between the densities is $ρ_{1} > ρ_{2} > ρ_{3}$ , it follows that:

$cos θ_{1} > cos θ_{2} > cos θ_{3}$

For capillary rise, the angle of contact $θ$ is acute ( $0 \leq θ < 9 0^{\circ}$ ). In this interval, the cosine function is strictly decreasing, meaning a larger value of $cos θ$ corresponds to a smaller angle $θ$ .

Thus, the relation between the angles of contact is:

$θ_{1} < θ_{2} < θ_{3}$

Answer: $θ_{1} < θ_{2} < θ_{3}$

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