A barometer is constructed with its tube having radius 1.0 mm. Assume that the surface of mercury in the tube is spherical in shape. If the atmospheric pressure is equal to 76 cm of mercury, what will be the height raised in the barometer tube? The contact angle of mercury with glass`135^@` and surface tension of mercury `=0.466Nm^-1`. Density of mercury`=13600kgm^-3`.

Explanation

To solve the problem of determining the height of mercury raised in a barometer tube, we will follow these steps:

### Step 1: Understand the Given Values
We have the following values provided in the problem:
- Radius of the tube, \( r = 1.0 \, \text{mm} = 1.0 \times 10^{-3} \, \text{m} \)
- Atmospheric pressure equivalent to mercury height, \( h_0 = 76 \, \text{cm} = 0.76 \, \text{m} \)
- Contact angle of mercury with glass, \( \theta = 135^\circ \)
- Surface tension of mercury, \( \sigma = 0.466 \, \text{N/m} \)
- Density of mercury, \( \rho = 13600 \, \text{kg/m}^3 \)
- Acceleration due to gravity, \( g = 9.8 \, \text{m/s}^2 \)

### Step 2: Calculate the Capillary Depression
The formula for capillary rise (or depression) is given by:

\[ h_c = \frac{2\sigma \cos \theta}{\rho r g} \]

Where:
- \( h_c \) is the height of capillary rise (or depression),
- \( \sigma \) is the surface tension,
- \( \theta \) is the contact angle,
- \( \rho \) is the density of the liquid,
- \( r \) is the radius of the tube,
- \( g \) is the acceleration due to gravity.

### Step 3: Calculate \( \cos(135^\circ) \)
We need to calculate \( \cos(135^\circ) \):

\[ \cos(135^\circ) = -\frac{1}{\sqrt{2}} \approx -0.7071 \]

### Step 4: Substitute the Values into the Formula
Now we can substitute the values into the formula for \( h_c \):

\[ h_c = \frac{2 \times 0.466 \times (-0.7071)}{13600 \times (1.0 \times 10^{-3}) \times 9.8} \]

### Step 5: Calculate \( h_c \)
Calculating the numerator:

\[ 2 \times 0.466 \times (-0.7071) \approx -0.659 \]

Calculating the denominator:

\[ 13600 \times (1.0 \times 10^{-3}) \times 9.8 \approx 133.28 \]

Now, substituting these values into the equation:

\[ h_c = \frac{-0.659}{133.28} \approx -0.00494 \, \text{m} \approx -0.494 \, \text{cm} \]

### Step 6: Calculate the Height of Mercury in the Barometer Tube
The height of mercury in the barometer tube can be calculated as:

\[ h = h_0 + h_c \]

Substituting the values:

\[ h = 76 \, \text{cm} - 0.494 \, \text{cm} = 75.506 \, \text{cm} \approx 75.5 \, \text{cm} \]

### Final Answer
The height of mercury raised in the barometer tube is approximately **75.5 cm**.

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