A capillary tube of radius r is immersed in water and water rises in to a height h. The mass of water in the capillary tube is 5g. Another capillary tube of radius 2 r is immersed in water. The mass of water that will rise in this tube is

Options

A. 2.5 g
B. 5.0 g
C. 10 g
D. 20 g

Explanation

To solve the problem, we need to understand how the mass of water that rises in a capillary tube is related to the radius of the tube.

### Step-by-Step Solution:

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

2. **Mass of Water in the Tube**: The mass of water that rises in the tube can be expressed as:
\[ m = \rho V \]
where \( V \) is the volume of water in the tube. The volume of water in a cylindrical tube is given by:
\[ V = \pi r^2 h \]
Therefore, the mass can be rewritten as:
\[ m = \rho \pi r^2 h \]

3. **Relation of Mass to Radius**: From the above equation, we can see that the mass \( m \) is directly proportional to the square of the radius \( r \):
\[ m \propto r^2 \]

4. **Given Data**: For the first capillary tube with radius \( r \), the mass of water that rises is given as 5 g. Therefore, we can write:
\[ m_1 = 5 \text{ g} \quad \text{for} \quad r \]

5. **New Capillary Tube**: For the second capillary tube with radius \( 2r \), we need to find the new mass \( m_2 \):
\[ m_2 \propto (2r)^2 = 4r^2 \]

6. **Calculate the New Mass**: Since \( m_1 \propto r^2 \), we can relate the two masses:
\[ \frac{m_2}{m_1} = \frac{4r^2}{r^2} = 4 \]
Thus:
\[ m_2 = 4 \times m_1 = 4 \times 5 \text{ g} = 20 \text{ g} \]

7. **Final Answer**: The mass of water that will rise in the second capillary tube of radius \( 2r \) is 20 g.

### Summary:
The mass of water that will rise in the second capillary tube is **20 g**.