Three liquids of densities `rho_1`,`rho_2` and `rho_3` (with `rho_1gtrho_2gtrho_2)` 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

Options

A. `(pi)/(2)lttheta_1lttheta_2ltthetaltpi`
B. `pigttheta_1gttheta_2gttheta_3gt(pi)/(2)`
C. `(pi)/(2)gttheta_1gttheta_2gttheta_3ge0`
D. `0letheta_1lttheta_2lttheta_3lt(pi)/(2)`

Explanation

To solve the problem, we need to analyze the relationship between the angles of contact (θ₁, θ₂, θ₃) of three liquids with different densities (ρ₁, ρ₂, ρ₃) that rise to the same height in identical capillaries.

### Step-by-Step Solution:

1. **Understanding Capillary Rise**: The height of liquid rise in a capillary tube is given by the formula:
\[ h = \frac{2T \cos \theta}{\rho g r} \]
where:
- \( h \) = height of the liquid column,
- \( T \) = surface tension,
- \( \theta \) = angle of contact,
- \( \rho \) = density of the liquid,
- \( g \) = acceleration due to gravity,
- \( r \) = radius of the capillary tube.

2. **Identifying Constants**: Since all three liquids rise to the same height \( h \) in identical capillaries, we can rearrange the formula to express \( \cos \theta \):
\[ \cos \theta = \frac{\rho g r h}{2T} \]
Here, \( \frac{g r h}{2T} \) is a constant for all three liquids.

3. **Analyzing Density and Cosine Relationship**: From the rearranged equation, we see that:
\[ \cos \theta \propto \rho \]
This means that as the density increases, the value of \( \cos \theta \) also increases.

4. **Relating Angles of Contact**: Since \( \cos \theta \) is directly proportional to the density:
- For the highest density liquid (ρ₁), \( \cos \theta_1 \) will be the largest, which means \( \theta_1 \) will be the smallest.
- For the medium density liquid (ρ₂), \( \cos \theta_2 \) will be smaller than \( \cos \theta_1 \), meaning \( \theta_2 \) will be larger than \( \theta_1 \).
- For the lowest density liquid (ρ₃), \( \cos \theta_3 \) will be the smallest, which means \( \theta_3 \) will be the largest.

5. **Conclusion**: Therefore, we can conclude the order of angles of contact:
\[ \theta_1 < \theta_2 < \theta_3 \]
This indicates that the angle of contact increases as the density of the liquid decreases.

### Final Answer:
The angles of contact obey the relation:
\[ \theta_1 < \theta_2 < \theta_3 \]