Question

In Jaeger’s method for determination of surface tension of water, the capillary was dipped $4cm$ in water. The radius of the capillary was $0.1cm$ and the difference in water levels of the manometer was $5.5cm$ .Taking $g=10m{{s}^{-1}}$ determines surface tension of water.

Answer 1

Hint: The surface tension leads a liquid to acquire minimum surface area. Jaeger’s method is an experiment to calculate the surface tension using pressure. According to this method, the surface tension depends on the pressure difference and the radius of the capillary used in the experiment.

Formulas used:
$\sigma =\dfrac{\Delta P\times {{R}_{cap}}}{2}$
$P=h\rho g$

Complete step-by-step solution:
Surface tension is the force that allows the liquids to acquire minimum surface area. Surface tension comes into play because the force of attraction between liquid molecules is stronger than the force of attraction between air and liquid molecules.
One of the methods used to determine surface tension is Jaeger’s method which was used to determine the surface tension of the mercury.
In this experiment a capillary of known radius is dipped into a liquid of which surface tension is to be calculated. A bubble is blown into the capillary. Then,
$\sigma =\dfrac{\Delta P\times {{R}_{cap}}}{2}$ - (1)
Here, $\sigma $ is the surface tension of the liquid
$\Delta P$ is the maximum pressure drop
${{R}_{cap}}$is the radius of the capillary
We know that,
$P=h\rho g$
Here, $P$ is the pressure
$h$ is height inside the liquid column
$g$ is acceleration due to gravity
Let $\Delta P={{P}_{\max }}-{{P}_{0}}$
$\begin{align}
  & {{P}_{\max }}=5.5\times {{10}^{-2}}\times 10 \\
 & {{P}_{0}}=4\times {{10}^{-2}}\times 10 \\
 & \Rightarrow \Delta P=5.5\times {{10}^{-2}}\times 10-4\times {{10}^{-2}}\times 10 \\
 & \therefore \Delta P=0.15P \\
\end{align}$
Substituting values in eq (1), we get,
$\begin{align}
  & \sigma =\dfrac{0.15\times 0.1\times {{10}^{-3}}}{2} \\
 & \therefore \sigma =7.5\times {{10}^{-6}}N{{m}^{-1}} \\
\end{align}$

Therefore, the surface tension of water is equal to $7.5\times {{10}^{-6}}N{{m}^{-1}}$.

Note:
A manometer is a bent tube that is used to calculate the pressure in a column of liquid. The pressure increases as we go deeper. For a water bubble to be stable, the pressure inside must be higher than the pressure outside. Surface tension decreases as surface area increases.