Determination of the surface tension of a liquid by Jaeger’s method

Let $T$ be the surface tension of the liquid, $\rho_1$ its density, $\rho_2$ the density of kerosene oil used in the manometer, $r$ the radius of the capillary tube in the apparatus, $h_2$ the maximum height difference between the liquid levels in the manometer, $h_1$ the depth of the lower end of the capillary tube from the liquid level in the beaker, and $p_0$ the atmospheric pressure.

Then,

$$\text{Pressure inside the bubble } (p_1) = p_0 + h_2 \rho_2 g$$ $$\text{Pressure outside the bubble } (p_2) = p_0 + h_1 \rho_1 g$$ $$\text{Excess pressure in the bubble} = p_1 - p_2$$ $$= \left( h_2 \rho_2 - h_1 \rho_1 \right) g = \frac{2T}{r}$$

Therefore,

$$T = \frac{r}{2} \left( h_2 \rho_2 - h_1 \rho_1 \right) g$$

Method

Set up the Jaeger’s apparatus as shown in Figure 42.1. Introduce a sufficient amount of kerosene oil into the manometer. Fix the capillary tube vertically on a stand. Place the beaker containing the liquid on the adjustable bench or any other suitable stand and adjust it so that the lower end of the capillary tube is immersed in the liquid as shown in Figure 42.1.

Attach a bent pin or a pointer to the capillary tube so that its tip touches the liquid level in the beaker. Open the tap $T_1$ so that water flows into the larger flask. Adjust the setup until the increasing pressure in the flask forces air out of the capillary tube into the liquid in the beaker in the form of an air bubble.

To find the maximum height difference in the manometer, first observe the highest position of the liquid level in limb A of the manometer. Focus the travelling microscope on the lowest position of the meniscus and record the reading $h_1$ on the vertical scale of the microscope.

Next, observe the lowest position of the meniscus in limb B and, after focusing the microscope on the meniscus, record the reading $h_2$ on the vertical scale.

Enter these readings in Table 42.1.

Remove the liquid beaker and observe the tip of the pin or pointer attached to the capillary tube through the microscope. Focus the tip on the horizontal cross wire of the microscope and record the reading $h_3$ on the vertical scale.

Similarly, focus the microscope on the lower end of the capillary tube and record the reading $h_4$ on the vertical scale.

Record these readings in Table 41.2.

$$h_1 = h_3 - h_4$$ $$h_2 = h_1 - h_2$$ $$\text{Diameter of the capillary tube} = \left[ \frac{(y_2 - y_1) + (x_2 - x_1)}{2} \right]$$

Substituting the values of $\rho_1$, $\rho_2$, $h_1$, and $h_2$, and the radius

$$r = \frac{d}{2}$$

into the expression given in the theory, calculate the value of the surface tension $T$.