To determine how much mercury dips down in a narrow tube when it is dipped into a trough of mercury, we can use the concept of capillarity. The angle of contact, surface tension, and the dimensions of the tube all play crucial roles in this phenomenon. Let’s break it down step by step.
Understanding Capillarity
Capillarity refers to the ability of a liquid to flow in narrow spaces without the assistance of external forces. This is influenced by the surface tension of the liquid and the angle of contact with the surface of the tube. In this case, we have mercury, which has a contact angle of 140 degrees with soda lime glass. Since this angle is greater than 90 degrees, mercury will not rise in the tube but will instead dip down.
Key Parameters
- Radius of the tube (r): 1 mm = 0.001 m
- Surface tension of mercury (γ): 0.465 N/m
- Density of mercury (ρ): 13.6 x 103 kg/m3
- Contact angle (θ): 140 degrees
Calculating the Height of the Mercury Column
The height to which the mercury will dip can be calculated using the formula derived from the balance of forces acting on the liquid column:
h = (2 * γ * cos(θ)) / (ρ * g * r)
Where:
- h: height of the liquid column (in meters)
- g: acceleration due to gravity (approximately 9.81 m/s2)
Substituting Values
First, we need to convert the contact angle from degrees to radians because most calculations in physics use radians. The conversion is:
θ (in radians) = 140 degrees × (π / 180) = 2.443 radians
Now, we can calculate cos(θ):
cos(140 degrees) = cos(2.443) ≈ -0.766
Now, substituting the values into the formula:
h = (2 * 0.465 N/m * (-0.766)) / (13.6 x 103 kg/m3 * 9.81 m/s2 * 0.001 m)
Calculating the numerator:
Numerator = 2 * 0.465 * (-0.766) ≈ -0.712 N/m
Calculating the denominator:
Denominator = 13.6 x 103 * 9.81 * 0.001 ≈ 133.416 kg/(m·s2)
Now, substituting these values into the height equation:
h ≈ (-0.712) / (133.416) ≈ -0.00533 m
Final Result
The negative sign indicates that the mercury dips down in the tube. Therefore, the height by which the mercury dips down relative to the liquid surface outside is approximately:
h ≈ 5.33 mm
This means that when the narrow tube is dipped into the mercury, the mercury level inside the tube will be about 5.33 mm lower than the mercury level outside the tube. This phenomenon is a fascinating demonstration of the principles of fluid mechanics and surface tension in action!
