Question

The ratio of surface tensions of mercury and water is given to be $7.5$ while the ratio of their densities is $13.6$ . Their contact angles, with glass, are close to ${135}^{\circ }$ and $0^{\circ }$ , respectively. It is observed that mercury gets depressed by an amount $h$ in a capillary tube of radius $r_1$ , while water rises by the same amount $h$ in a capillary tube of radius $r_2$ . The ratio, $\frac{r_1}{r_2}$ , is then close to

• A. $\frac{2}{3}$
• B. $\frac{4}{5}$
• C. $\frac{3}{5}$
• D. $\frac{2}{5}$

Answer 1

The correct option is D $\frac{2}{5}$

Given ratio of surface tensions of mercury and water, $\frac{T_{Hg}}{T_W}=7.5$

Ratio of densities, $\frac{{\rho }_{Hg}}{{\rho }_W}=13.6$

Contact angle for mercury,
${\theta }_{Hg}={135}^{\circ }$

Contact angle for water, ${\theta }_W=0^{\circ }$

Mercury get depressed by same height of water rises in capillary tube

$|h_{Hg}|=|h_{\text{water}}|$

$\frac{2T_{Hg}\cos {\theta }_{Hg}}{{\rho }_{Hg}{R}_{Hg}g}=\frac{2T_W\cos {\theta }_W}{{\rho }_W{R}_{W}g}$

$\frac{R_{Hg}}{R_W}=\frac{\rho W}{\rho Hg}\frac{T_{Hg}}{T_W}\frac{\cos {\theta }_{Hg}}{\cos {\theta }_W}$

$\frac{R_{Hg}}{R_W}=\frac{1}{13.6}\times 7.5\times \frac{1}{\sqrt{2}}$

$\frac{R_{Hg}}{R_W}=0.4=\frac{2}{5}$