Question

Three liquids of densities $p_1,p_2$ and $p_3$ (with $p_1>p_2,p_3$) having the same value of surface tension $T$, rise to the same height in three identical capillaries The angles of contact ${\theta }_1,{\theta }_2$ and ${\theta }_3$ obey:-

• A. $\pi >{\theta }_1>{\theta }_2>{\theta }_3>\frac{\pi }{2}$
• B. $\frac{\pi }{2}>{\theta }_1>{\theta }_2>{\theta }_3\ge 0$
• C. $0\le {\theta }_1<{\theta }_2<{\theta }_3<\frac{\pi }{2}$
• D. $\frac{\pi }{2}<{\theta }_1<{\theta }_2<{\theta }_3<\pi$

Answer 1

The correct option is C $0\le {\theta }_1<{\theta }_2<{\theta }_3<\frac{\pi }{2}$

Rise of a liquid in a capillary tube is given by ,

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

or $\cos \theta =hr\rho g/2T$

where $\theta =$ angle of contact ,

$r=$ radius of capillary tube ,

$T=$ surface tension ,

$\rho =$ density of liquid ,

now given that $h$ , $T$ and $r$ are constants for all three liquids ,

and ${\rho }_1>{\rho }_2>{\rho }_3$ ,

therefore $\cos {\theta }_1>\cos {\theta }_2>\cos {\theta }_3$

or ${\theta }_1<{\theta }_2<{\theta }_3$

now as the liquid is rising in all three capillaries therefore angles of contact will be acute ,

$0\le {\theta }_1<{\theta }_2<{\theta }_3<\frac{\pi }{2}$