Answer:
The ratio of surface tensions of mercury and water is given to be X, while the ratio of their densities is Y. The contact angles of mercury and water with glass are close to θ1 and θ2 respectively. It is observed that mercury gets depressed by an amount Δh1 in a capillary tube of radius r1, while water rises by the same amount Δh2 in a capillary tube of radius r2.
To find the ratio between X and Y, we can use the relationship between capillary rise (Δh) and the surface tension (T) of the liquid.
For water:
Δh2 = (2T2 * cos(θ2))/ (ρ2 * g * r2)
For mercury:
Δh1 = (2T1 * cos(θ1))/ (ρ1 * g * r1)
Since Δh1 = Δh2 (both liquids rise/depress by the same amount), we can equate the two equations:
(2T1 * cos(θ1))/ (ρ1 * g * r1) = (2T2 * cos(θ2))/ (ρ2 * g * r2)
Rearranging the equation, we can find the ratio between X and Y:
(X * cos(θ1) * ρ2 * r2) / (Y * cos(θ2) * ρ1 * r1) = 1
Therefore, the ratio between X and Y is (Y * cos(θ2) * ρ1 * r1) / (X * cos(θ1) * ρ2 * r2).
Step-by-step explanation: