When a long capillary tube of radius 0.015 cm is dipped in a liquid, the liquid rises to a height of 15 cm within it. If the contact angle between the liquid and glass to close to `0^@`, the surface tension of the liquid, in milliNewton `m^(-1)`, is `[rho_("liquid") = 900 kgm^(-3), g = 10 ms^(-2)]` (give anwer is closet integer)

Explanation

To solve the problem of finding the surface tension of the liquid in the capillary tube, we can use the formula for capillary rise, which is given by:

\[ h = \frac{2S \cos \theta}{\rho g r} \]

Where:
- \( h \) is the height of the liquid rise in the capillary (in meters),
- \( S \) is the surface tension of the liquid (in Newton per meter),
- \( \theta \) is the contact angle (in degrees),
- \( \rho \) is the density of the liquid (in kg/m³),
- \( g \) is the acceleration due to gravity (in m/s²),
- \( r \) is the radius of the capillary tube (in meters).

### Step 1: Convert given values to SI units
- Radius of the capillary tube:
\[ r = 0.015 \, \text{cm} = 0.015 \times 10^{-2} \, \text{m} = 0.00015 \, \text{m} \]
- Height of liquid rise:
\[ h = 15 \, \text{cm} = 15 \times 10^{-2} \, \text{m} = 0.15 \, \text{m} \]
- Density of the liquid:
\[ \rho = 900 \, \text{kg/m}^3 \]
- Acceleration due to gravity:
\[ g = 10 \, \text{m/s}^2 \]

### Step 2: Substitute values into the capillary rise formula
Since the contact angle \( \theta \) is close to \( 0^\circ \), we have:
\[ \cos \theta \approx 1 \]

Substituting the values into the formula:
\[ 0.15 = \frac{2S \cdot 1}{900 \cdot 10 \cdot 0.00015} \]

### Step 3: Rearranging the equation to solve for surface tension \( S \)
Rearranging gives:
\[ S = \frac{0.15 \cdot 900 \cdot 10 \cdot 0.00015}{2} \]

### Step 4: Calculate the surface tension
Calculating the right-hand side:
\[ S = \frac{0.15 \cdot 900 \cdot 10 \cdot 0.00015}{2} \]
Calculating step-by-step:
1. \( 0.15 \cdot 900 = 135 \)
2. \( 135 \cdot 10 = 1350 \)
3. \( 1350 \cdot 0.00015 = 0.2025 \)
4. Finally, divide by 2:
\[ S = \frac{0.2025}{2} = 0.10125 \, \text{N/m} \]

### Step 5: Convert to milliNewton per meter
To convert to milliNewton per meter:
\[ S = 0.10125 \, \text{N/m} = 101.25 \, \text{mN/m} \]

### Step 6: Round to the closest integer
The closest integer to \( 101.25 \) is \( 101 \).

### Final Answer
The surface tension of the liquid is approximately **101 mN/m**.