Surface Tension by Capillary Rise

Transcript

0:00Introduction

Hello everyone, here is a small puzzle before we start. Take a capillary, a very thin glass tube, in which water would climb six centimetres all by itself. Now suppose only four centimetres of that tube stand above the water. So does the water overflow from the top, like a little fountain? We will settle it soon. This rise is also how we measure surface tension, an experiment in your JEE syllabus, so let's begin.

0:29The experiment in one picture

Look at this figure first. Here a glass tube stands in a beaker, and the water inside has climbed above the level outside, all by itself. At the top, two times small r, with the double arrow, is the diameter of the bore, the hole through the tube. So small r is the radius of the bore, okay? The long arrow marked small h is the rise. It goes from the water level in the beaker up to the dashed line, at the bottom of the curved surface, which we call the meniscus. The writing on the right of the figure is what we build next. Clear?

1:03The capillary rise formula

Small h equals two times capital T times cos theta, over small r times rho times small g. Capital T is the surface tension, and theta, the Greek letter, is the angle of contact. Rho, the Greek letter like a small p, is the density of the liquid, and small g is the acceleration due to gravity. Right? We want the surface tension, so turn this formula around in your head. What does capital T equal? Yes, exactly, capital T equals small r times rho times small g times small h, over two times cos theta. So the two lengths we must measure are small r and small h. With me? Now a condition to remember. For pure water and clean glass, theta is approximately zero, that is the wavy equals sign, and cos theta is one. So the cos theta drops out, and capital T is simply small r times rho times small g times small h, over two. Got it?

2:06Measuring h and r

Now the measurements. Small h is the height of the bottom of the meniscus above the free surface, the level of the water in the beaker. We measure it with a travelling microscope. Okay? But the microscope cannot focus on the water surface itself. So how do we locate its level? We use a pointer, a needle fixed beside the tube, lowered until its tip just touches the surface. Then the beaker is removed, and the microscope is focused on the tip. Nice trick, isn't it? The radius comes from the same microscope. We read two perpendicular diameters of the bore and take their mean, and half of that is small r, right? Now be careful here. For a bore one point zero millimetres across, small r is zero point five millimetres, so do not put in the diameter.

2:56One small correction

Clear? Now picture the meniscus once more. Small h stops at the bottom of the curve. So what has it left out? Yes, exactly, it leaves out the liquid in the meniscus itself, okay? So more exactly, capital T equals small r times rho times small g, over two times cos theta, times the bracket, small h plus small r over three. That small r over three is the correction for the meniscus, and our simple form neglects it. Right?

3:27What changes the rise?

Now read this formula again. For a given liquid, suppose we make the tube narrower. Does the water rise more, or less? Yes, exactly, it rises more, because small r sits in the bottom of the fraction. Small h times small r is constant for a given liquid, so a narrower tube gives a greater rise. Okay? Now suppose we add a little detergent to the water. Will the rise go up, or go down? Very good if you said down, because detergents lower the surface tension of water, so the rise is smaller. Hotter water does the same, because surface tension falls as the temperature rises. Got it? Also, the tube and the water must be clean, because grease changes both the angle of contact and the surface tension. Clear?

4:16Back to our puzzle: a tube that is too short

Now back to our puzzle. Water would rise six centimetres, but the tube projects only four centimetres above the surface. So does the water overflow? No, it does not overflow. The water rises to the top, and the meniscus becomes flatter. With me? Flatter means that its radius of curvature increases, here to one point five times the tube radius. Okay? The pressure difference then just supports a four centimetre column. So there is no fountain, and our puzzle is solved. Nice, isn't it?

4:51Which reading matters most?

Now one question on errors. Small r is zero point three zero, plus or minus zero point zero one, millimetres. Small h is four point eight zero, plus or minus zero point zero two, centimetres. Which reading spoils capital T more? It is the radius that does the damage. The per cent error in capital T comes from adding two fractions: zero point zero one over zero point three zero, plus zero point zero two over four point eight zero. As a per cent, that is about three point eight. Most of it comes from the first fraction, so the radius dominates.

5:32Example 1: surface tension from a 5.8 cm rise of water

Clear? Now the easier example first. Water rises to a height of five point eight centimetres in a capillary of radius zero point two five millimetres. Find the surface tension of water, with small g as nine point eight metres per second squared. Pause the video and try it first. Done? Okay, let's check. Can we put zero point two five and five point eight straight into the formula? No, we cannot, because one is in millimetres and the other in centimetres. In metres, small r is zero point two five times ten to the power minus three, and small h is zero point zero five eight. Rho for water is one thousand kilograms per cubic metre. Right? For water in a clean tube, we take cos theta as one, so capital T is small r times rho times small g times small h, over two. Now put in the numbers on screen, with nine point eight for small g. Okay? The top comes to zero point one four two one, and half of that gives the surface tension, zero point zero seven one newtons per metre. If you got that, very good indeed. With me?

6:51Example 2: T with and without the meniscus correction

Now the harder one. In a wide capillary of radius one point five millimetres, the water rises nine point six millimetres. Find the surface tension in two ways: part A, neglecting the correction for the liquid in the meniscus, and part B, including it. Pause the video and try it first. Done? Okay, let's check. We take cos theta as one again, so both parts share the factor small r times rho times small g, over two. Let us find that first. With small r as one point five times ten to the power minus three metres, rho as one thousand and small g as nine point eight, this gives seven point three five newtons per square metre, okay? For part A, we neglect the correction and multiply by small h alone, in metres. Seven point three five times nine point six times ten to the power minus three gives zero point zero seven one newtons per metre. Right? Now part B, and here is the trap. Do we add the whole radius, one point five millimetres, to the nine point six? No, we add only small r over three, and one point five over three gives zero point five millimetres. So seven point three five times the bracket, nine point six plus zero point five, times ten to the power minus three, gives zero point zero seven four newtons per metre. See, in a wide tube like this, the correction really shows. Got it?

8:28Quick revision: T formula, h + r/3, narrow and short tubes

So, quick revision. Capital T equals small r times rho times small g times small h, over two times cos theta, and for pure water and clean glass, cos theta is one. Small h goes to the bottom of the meniscus, and more exactly we use small h plus small r over three. Small h times small r is constant, so a narrower tube gives a greater rise, and detergent or heat lowers the surface tension. And in a tube too short, there is no overflow, the meniscus just becomes flatter. You now know this experiment well, so go and try the practice problems on this page.