Angle of Contact and Capillarity

Transcript

0:00Introduction

Hello everyone! Okay, a small puzzle to start. Dip a very thin glass tube into water. The water climbs up the tube, on its own, against gravity. Now dip the same tube into mercury. The mercury inside goes down, below the level outside! Same tube, opposite results. Why does that happen? By the end of this video, you will know. The same idea lets a cotton wick carry oil up in a lamp. Today: angle of contact and capillarity. Questions on this come up often in JEE, so let's begin.

0:38Angle of contact: does the liquid wet the glass?

So, look at the figure. Inside each tube, the liquid surface is curved where it meets the glass wall. So it makes an angle with the wall. We call it the angle of contact, theta, the Greek letter on screen. And remember, it is measured inside the liquid. Okay? Its two arms are the solid wall, and the tangent to the liquid surface, drawn right where the liquid touches the wall. With me? Now think. Water on clean glass. Is the angle of contact small, or large? Small! Very good! For water on clean glass, theta is almost zero; the wavy equals sign means approximately. The rule: when theta is less than ninety degrees, the liquid wets the solid. The small circle means degrees, and the pointed sign means less than. Clear? Now mercury on glass. Here theta is about one hundred and forty degrees: greater than ninety, the sign pointing the other way. So mercury does not wet glass. In the figure, water rises in the left tube, and mercury is pushed down in the right one. Different angles of contact, opposite results. Right?

1:49Jurin's law: how high does it rise?

Now, how high does the liquid rise? Take a capillary, a very thin tube, of radius small r. The rise is h equals two capital T cos theta, over r rho g. Capital T is the surface tension, rho, the Greek letter, is the density of the liquid, and g is the acceleration due to gravity. This is Jurin's law. Write it down. Got it? For water, theta is less than ninety degrees, so cos theta is greater than zero. Then h is positive, and the liquid rises. Okay? Now mercury, with theta about one hundred and forty degrees. What is the sign of h? Negative! Yes, exactly! The cosine of an angle between ninety and one hundred and eighty degrees is negative, so h is negative. And a negative h means depression: the mercury in the tube stands below the level outside, just like the right side of the figure. So that is our puzzle solved: the sign of cos theta decides rise or fall. With me? Quick question. Same liquid, but now the radius of the tube is halved. What happens to h? It doubles! Correct! Small r sits at the bottom of the formula, so h times r is constant for a given liquid. Narrower tube, higher rise.

3:10Short, tilted, or in a lift: what changes?

Clear? Now, a famous trap. Suppose the tube sticks out above the water by less than the height h the water wants to reach. Does the water spill over the top, like a fountain that never stops? No, no, careful! No overflow. The water rises to the top of the tube, and it stops there. Okay? What changes is the meniscus, the curved surface of the water at the top. It flattens. Its radius, capital R, grows, so that the height times capital R stays constant: a smaller height, a bigger radius. So, no endless fountain! Right? Next, tilt the tube. The vertical height stays the same h. So along the slanted tube, the column of liquid gets longer. Got it? Now put the whole experiment in a lift, accelerating upward with acceleration small a. Does h grow, or shrink? It shrinks! Very good! Inside the lift, g becomes g plus a; the arrow means becomes. And g sits at the bottom of the formula, so a bigger g gives a smaller h. With me? And in free fall, the effective g becomes zero. Then the liquid rises and fills the whole tube. Remember these four cases; they come up often in questions. Clear?

4:31Example 1: water rise in a glass capillary of radius 0.25 mm

Now, Example one. Find the rise of water in a glass capillary of radius zero point two five millimetre. The surface tension is zero point zero seven newton per metre, theta is zero, and g is nine point eight metres per second squared. Pause the video, and try it first. Done? Okay, let's check. Radius zero point two five, straight into the formula, right? Oh no, careful! That is in millimetres. S I units first: zero point two five millimetre is zero point two five times ten to the power minus three metre. Okay? Theta is zero, so cos theta is one; the double arrow means so. And for water, rho is one thousand kilogram per cubic metre; m to the power minus three means per cubic metre. Right? Now substitute. On top: two times zero point zero seven times one. At the bottom: zero point two five times ten to the power minus three, times one thousand, times nine point eight. Clear? The top is zero point one four, and the bottom is two point four five. So h is about zero point zero five seven metre: five point seven centimetres. Almost six centimetres, in such a thin tube! Got it?

5:53Example 2: mercury depression in a glass tube of radius 1 mm

Now, Example two: the mercury half of our puzzle. Find the depression of mercury in a tube of radius one millimetre. The surface tension is zero point four six five newton per metre, theta is one hundred and forty degrees, rho is thirteen thousand six hundred kilogram per cubic metre, and g is nine point eight metres per second squared. Pause the video and try it. Hint: watch the sign of cos theta. Done? Okay, let's check. One millimetre is ten to the power minus three metre. And cos one hundred and forty degrees is about minus zero point seven six six. Negative, remember! With me? Substitute. On top: two, times zero point four six five, times minus zero point seven six six, in brackets. At the bottom: ten to the power minus three, times thirteen thousand six hundred, times nine point eight. Okay? The top is minus zero point seven one two, and the bottom is one hundred and thirty three point two eight. So h is about minus five point three times ten to the power minus three metre: minus five point three millimetres. Right? Hmm, a negative height. Did we make a mistake somewhere? No mistake at all! Negative h means depression. The mercury in the tube stands five point three millimetres below the level outside, just like the right side of the figure. The minus sign gives the direction; the depression itself is five point three millimetres. Puzzle solved! Clear?

7:26Revision: wetting, Jurin's law, short tube, lift, free fall

So, quick revision. Angle of contact below ninety degrees: the liquid wets the glass and rises. Above ninety degrees: it is depressed. Jurin's law: h equals two capital T cos theta, over r rho g, and h times r is constant for a given liquid. Short tube: no overflow. And a tilted tube keeps the same vertical height. In a lift going up, h decreases; in free fall, the liquid fills the tube. Now go and try the practice problems on this page. All the best!