Question

What should be the radius of a capillary tube if water has to rise to a height of 6 cm in it, if surface tension of water is 7.2 x 10$^{ - 2}$ Nm$^{ - 1}$:
A. $0.24mm$
B. $2.4mm$
C. $2cm$
D. $1.2cm$

Answer 1

This problem can be solved using rise of water due to capillarity. We can use the formula for height up to which liquid rises in a tube due to capillary action, $h = \dfrac{{2\sigma \cos \theta }}{{\rho gr}}$to get the radius of the capillary tube.

Formula used:
$h = \dfrac{{2\sigma \cos \theta }}{{\rho gr}}$
where$\rho$ density of the liquid, $\sigma$ is surface tension of the liquid, θ is the angle made by the liquid meniscus with the capillary's surface and r is the inner radius of the capillary.

Complete step by step answer:
Taking density of water$\rho = 1000kg/{m^3}$. Given, the surface tension of water,
$\sigma = 7.2 \times {10^{ - 2}}N{m^{ - 1}}$.
Let the angle made by the water meniscus with the capillary's surface be zero $\left( {\theta = 0} \right)$ and r is the inner radius of the capillary. The formula for the capillary rise of water in a tube is, $h = \dfrac{{2\sigma \cos \theta }}{{\rho gr}}$ .

Therefore the formula for the radius of the capillary is, $r = \dfrac{{2\sigma }}{{\rho gh}}$.
$\left( {\because \cos 0^\circ = 1} \right)$
Substituting the values given in the question we get,
$r = \dfrac{{2 \times 7.2 \times {{10}^{ - 2}}}}{{1000 \times 10 \times 6 \times {{10}^{ - 2}}}} \\\ \Rightarrow r = 2.4 \times {10^{ - 4}}m \\\ \therefore r = 0.24mm$.

Therefore the radius of the capillary is 0.24mm.

Note: Student should take care about the units of the different quantities used in the equation of height up to which liquid rises to get error free solution.The formula for capillary rise of a liquid in a tube can be derived by balancing the weight of the liquid column $(\pi {r^2}h\rho g)$ with the upward force due to surface tension $\left( {2\pi r\sigma \cos \theta } \right)$.