Capillary Rise Physics Explained

The height ($h$) a liquid rises in a capillary tube is determined by several factors, including surface tension ($T$), liquid density ($\rho$), tube radius ($r$), the contact angle ($\theta$) between the liquid and the tube, and acceleration due to gravity ($g$). The governing equation is:

$h = \frac{2T \cos \theta}{\rho g r}$

Assuming constant contact angle ($\theta$) and gravity ($g$), the height is directly proportional to surface tension and inversely proportional to density and radius:

$h \propto \frac{T}{\rho r}$

Analyzing Liquid Heights and Surface Tensions

The problem describes capillary rise experiments in two tubes with inner radii $r_1$ and $r_2$, where $r_1 > r_2$. It is given that the densities of the two liquids are the same ($\rho_1 = \rho_2$).

We need to find the relationship between the measured heights ($h_1, h_2$) and surface tensions ($T_1, T_2$). The specific relation being considered is $h_1 > h_2$ and $T_1 = T_2$.

Relation Between Height and Surface Tension

If the surface tensions are equal ($T_1 = T_2$) and the densities are equal ($\rho_1 = \rho_2$), the capillary rise height formula simplifies to:

$h \propto \frac{\cos \theta}{r}$

For the condition $h_1 > h_2$ to be observed when $r_1 > r_2$, it must be that the term $\frac{\cos \theta}{r}$ is greater for the first scenario:

$\frac{\cos \theta_1}{r_1} > \frac{\cos \theta_2}{r_2}$

This shows that the observed relationship $h_1 > h_2$ is consistent with equal surface tensions ($T_1 = T_2$) under the given conditions of equal densities and differing radii, possibly indicating differences in the contact angles ($\theta_1$ and $\theta_2$).