Mercury has an angle of contact equal to `140^(@)` with soda lime galss. A narrow tube of radius `1.00mm` made of this glass is dipped in a through containing mercury. By what amount does the mercury dip down in the tube relative to the mercury surface outside? Surface tension of mercury at the temperature of the experiment is `0.465 Nm^(-1)`. Density of mercury = `13.6xx10^(3) kg m^(-3)`.

Options

A. 5.34 mm
B. 2.35 mm
C. 6.25 mm
D. 1.44 mm

Explanation

To solve the problem of how much mercury dips down in a tube relative to the mercury surface outside, we can follow these steps:

### Step-by-Step Solution

1. **Identify the Given Data:**
- Angle of contact (θ) = 140°
- Radius of the tube (r) = 1.00 mm = 1.00 × 10^(-3) m
- Surface tension of mercury (S) = 0.465 N/m
- Density of mercury (ρ) = 13.6 × 10^(3) kg/m³
- Acceleration due to gravity (g) = 9.8 m/s²

2. **Convert the Angle of Contact:**
- Since we need to use the cosine of the angle in our calculations, we convert the angle from degrees to radians if necessary. However, for cosine calculations, we can directly use degrees in most calculators.
- cos(140°) = -0.7660 (approximately)

3. **Use the Formula for Capillary Rise (or Dip):**
- The formula that relates surface tension, height of the liquid column (h), density, gravitational acceleration, and radius is:
\[ h = \frac{2S \cos \theta}{\rho g r} \]

4. **Substitute the Values into the Formula:**
- Plugging in the values:
\[ h = \frac{2 \times 0.465 \times \cos(140°)}{(13.6 \times 10^{3}) \times (9.8) \times (1 \times 10^{-3})} \]
- Calculate cos(140°):
\[ h = \frac{2 \times 0.465 \times (-0.7660)}{(13.6 \times 10^{3}) \times (9.8) \times (1 \times 10^{-3})} \]

5. **Calculate the Denominator:**
- Calculate the denominator:
\[ (13.6 \times 10^{3}) \times (9.8) \times (1 \times 10^{-3}) = 133.28 \, \text{(approximately)} \]

6. **Calculate the Numerator:**
- Calculate the numerator:
\[ 2 \times 0.465 \times (-0.7660) = -0.7125 \, \text{(approximately)} \]

7. **Final Calculation:**
- Now, substituting back:
\[ h = \frac{-0.7125}{133.28} \approx -0.00534 \, \text{m} \]

8. **Interpret the Result:**
- The negative sign indicates that the mercury level dips down relative to the mercury surface outside. Therefore, the amount by which the mercury dips down is approximately 0.00534 m, or 5.34 mm.

### Final Answer:
The mercury dips down by approximately **5.34 mm**.