A capillary tube of inner diameter 0.5 mm is dipped ina liquid of specific gravity 13.6, having surface tension 545 dyn/cm and angle of contact 130. Find the depression or elevation in the tube

To determine the depression or elevation of a liquid in a capillary tube, we can use the capillary rise (or fall) formula. This formula relates the height of the liquid column in the tube to the properties of the liquid and the tube itself. The general equation is given by:

The Capillary Rise Formula

The height (h) of the liquid column can be calculated using the formula:

h = (2 * γ * cos(θ)) / (ρ * g * r)

• γ = surface tension of the liquid (in dyn/cm)
• θ = angle of contact (in degrees)
• ρ = density of the liquid (in g/cm³)
• g = acceleration due to gravity (approximately 980 cm/s²)
• r = radius of the capillary tube (in cm)

To apply this formula, we need to convert the given values into appropriate units and understand the concept behind each parameter.

Given Values

• Inner diameter of the capillary tube = 0.5 mm = 0.05 cm
• Specific gravity of the liquid = 13.6
• Surface tension (γ) = 545 dyn/cm
• Angle of contact (θ) = 130 degrees

Calculating Density

The density (ρ) of the liquid can be derived from its specific gravity:

Density (ρ) = Specific Gravity × Density of Water

Using the density of water as approximately 1 g/cm³:

ρ = 13.6 × 1 g/cm³ = 13.6 g/cm³

Calculating the Radius

The radius (r) of the capillary tube is half of the diameter:

r = 0.5 mm / 2 = 0.25 mm = 0.025 cm

Applying the Values in the Formula

Now, we can substitute the values into the capillary rise equation. First, we need to convert the angle of contact from degrees to radians for the cosine function:

θ = 130 degrees = 130 × (π/180) radians

Calculating cos(130 degrees):

cos(130°) ≈ -0.8192

Now substituting the values into the formula:

h = (2 * 545 * (-0.8192)) / (13.6 * 980 * 0.025)

Final Calculation

Calculating the numerator:

Numerator = 2 * 545 * (-0.8192) ≈ -892.7

Calculating the denominator:

Denominator = 13.6 * 980 * 0.025 ≈ 332.0

Now, substituting these values:

h = -892.7 / 332.0 ≈ -2.69 cm

Interpreting the Result

The negative sign indicates that the liquid will be depressed in the capillary tube due to the high angle of contact (130 degrees) and the properties of the liquid. Thus, the liquid will be depressed by approximately 2.69 cm in the tube.

In summary, this exercise illustrates how various physical properties of a liquid and the geometry of a capillary tube interact to influence the behavior of liquids in small spaces. Understanding these principles is essential in fields ranging from fluid dynamics to biological systems where capillary action plays a vital role.