Question

In to a vessel containing pure water a clean glass tube of radius \(3.6 \times 10^{-4}\,\text{m}\) is held vertically with \(12\,\text{cm}\) of the tube above the water level. Now the capillary tube is moved down in to the water so that only \(2\,\text{cm}\) of its length is above the water surface. Angle of contact \(\Theta\) at this position is (given surface tension of water \(= 7.2 \times 10^{-2}\,\text{N m}^{-1}\) and \(g = 10\,\text{m s}^{-2}\))

• A. \(\Theta = 45^\circ\)
• B. \(\Theta = 30^\circ\)
• C. \(\Theta = 15^\circ\)
• D. \(\Theta = 60^\circ\)

Hint:

When the available tube length above water is less than natural capillary rise, use \(h = h_0\cos\Theta\) to find the changed angle of contact.

Answer 1

The Correct Option is D

Step 1: Use capillary rise formula.
The height of liquid rise in a capillary tube is given by:
\[ h = \frac{2T\cos\Theta}{r\rho g} \]
where \(T\) is surface tension, \(r\) is radius of capillary tube, \(\rho\) is density of water, and \(\Theta\) is angle of contact.

Step 2: Find maximum capillary rise for clean glass and water.
For pure water in clean glass, initially \(\Theta = 0^\circ\), so:
\[ \cos 0^\circ = 1 \]
Thus,
\[ h_0 = \frac{2T}{r\rho g} \]

Step 3: Substitute given values.
\[ T = 7.2 \times 10^{-2}\,\text{N m}^{-1} \]
\[ r = 3.6 \times 10^{-4}\,\text{m} \]
\[ \rho = 1000\,\text{kg m}^{-3} \]
\[ g = 10\,\text{m s}^{-2} \]
\[ h_0 = \frac{2 \times 7.2 \times 10^{-2}}{3.6 \times 10^{-4} \times 1000 \times 10} \]

Step 4: Simplify maximum rise.
\[ h_0 = \frac{14.4 \times 10^{-2}}{3.6} \]
\[ h_0 = 4 \times 10^{-2}\,\text{m} \]
\[ h_0 = 4\,\text{cm} \]

Step 5: Use actual available height of tube above water.
When only \(2\,\text{cm}\) of tube is above water surface, water can rise only up to:
\[ h = 2\,\text{cm} \]
So, the capillary rise becomes less than natural rise due to change in meniscus condition.

Step 6: Relate actual rise with maximum rise.
\[ h = h_0 \cos\Theta \]
\[ 2 = 4\cos\Theta \]
\[ \cos\Theta = \frac{1}{2} \]

Step 7: Find angle of contact.
\[ \Theta = 60^\circ \]
\[ \boxed{60^\circ} \]