Mercury in a capillary tube is depressed by 1.16cm. The diameter of the capillary tube is 1mm. Calculate the angle of contact of mercury with the glass if the surface tension of mercury is `54 xx 10Nm^(-1)` and its density is `13.6 xx 10^(3) kg m^(-3)`.

Explanation

To solve the problem step by step, we will use the formula for the angle of contact (θ) in a capillary tube, which relates the height of the liquid column (h), the radius of the tube (R), the density of the liquid (ρ), the acceleration due to gravity (g), and the surface tension (σ).

### Step 1: Convert the given measurements to SI units.
- Height (h) = 1.16 cm = 1.16 × 10^(-2) m
- Diameter of the capillary tube = 1 mm = 1 × 10^(-3) m
- Radius (R) = Diameter / 2 = (1 × 10^(-3) m) / 2 = 0.5 × 10^(-3) m
- Surface tension (σ) = 54 × 10^(-2) N/m
- Density (ρ) = 13.6 × 10^(3) kg/m^3
- Acceleration due to gravity (g) = 9.8 m/s²

### Step 2: Use the formula for cos(θ).
The formula for cos(θ) in terms of the given parameters is:
\[ \cos(θ) = \frac{R \cdot ρ \cdot g \cdot h}{2 \cdot σ} \]

### Step 3: Substitute the values into the formula.
Substituting the values we have:
\[ \cos(θ) = \frac{(0.5 \times 10^{-3}) \cdot (13.6 \times 10^{3}) \cdot (9.8) \cdot (1.16 \times 10^{-2})}{2 \cdot (54 \times 10^{-2})} \]

### Step 4: Calculate the numerator and denominator.
Calculating the numerator:
\[ Numerator = (0.5 \times 10^{-3}) \cdot (13.6 \times 10^{3}) \cdot (9.8) \cdot (1.16 \times 10^{-2}) \]
Calculating step by step:
1. \(0.5 \times 13.6 = 6.8\)
2. \(6.8 \times 9.8 = 66.64\)
3. \(66.64 \times 1.16 \times 10^{-5} = 7.72 \times 10^{-5}\)

Now calculating the denominator:
\[ Denominator = 2 \cdot (54 \times 10^{-2}) = 1.08 \]

### Step 5: Calculate cos(θ).
Now we can calculate cos(θ):
\[ \cos(θ) = \frac{7.72 \times 10^{-5}}{1.08} \]
Calculating this gives:
\[ \cos(θ) = 0.0000714 \]

### Step 6: Find the angle θ.
To find θ, we take the inverse cosine:
\[ θ = \cos^{-1}(0.0000714) \]
Calculating this gives:
\[ θ \approx 135° \text{ (since the value is negative, we take the angle in the second quadrant)} \]

### Final Answer:
The angle of contact of mercury with the glass is approximately **135 degrees**.

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