Excess Pressure Inside Drops and Bubbles

Transcript

0:00Introduction

Hello everyone! Okay, a small puzzle to start. Take two soap bubbles, one small and one big, and join them with a thin tube. Which way does the air flow? Most people say the big bubble pushes air into the small one, until they become equal. Sounds sensible, right? Hold that thought; the answer may surprise you. Questions on drops and bubbles come up often in JEE, so let's begin.

0:28Curved surface: more pressure on the concave side

So, recall: the surface of a liquid behaves like a stretched membrane. Capital T is the surface tension, in newton per metre, written N m to the power minus one. Okay? Now bend that membrane into a curve, like the skin of a drop. The stretched skin pulls inwards, towards the centre of the curve. So tell me: which side has the higher pressure, the inside of the curve, or the outside? The inside, yes! The pressure on the inside has to push back against that inward pull. The inside of a curve is called the concave side. So a curved surface always has higher pressure on its concave side. Right? How much higher? For one surface of radius capital R, the excess pressure, delta P, equals two T over R, okay? The little triangle is the Greek letter capital delta, and delta P means pressure inside minus pressure outside. Write this down. Everything today comes from this line. Now look where capital R sits: in the denominator. So a smaller radius means a larger excess pressure. The double arrow on screen means implies. Smaller drop, higher pressure inside. Remember this; it will solve our puzzle. Clear?

1:48Drop, soap bubble, air bubble: count the surfaces

Alright, now let's apply it. First, a liquid drop, like a raindrop. It has just one surface, with air all around it. So delta P equals two T over R, okay? Next, a soap bubble. Think carefully here. The soap bubble is a thin film of soapy water, with air on both sides of the film. So how many surfaces does it have? Two, exactly! One outer surface and one inner surface, and each one adds two T over R to the pressure. So, for a soap bubble, four T over R is the excess pressure. With me? Now an air bubble inside water, like the bubbles rising in a glass of soda. It is also a bubble. So, four T over R again, right? No, no, careful! Inside water there is no film. The air simply meets the water at one surface. So an air bubble inside a liquid has one surface: two T over R, just like a drop. Classic trap: count the surfaces, not the word bubble. Okay?

2:54Air bubble at depth h

Now, what if the air bubble sits deep inside the liquid? First, P naught, that is P with the subscript zero: the pressure at the open surface of the liquid. If the surface is open to the air, that is atmospheric pressure. Right? Now the bubble is at depth small h below the surface. What is the pressure in the liquid just outside the bubble? Yes! P naught, plus rho g h. Rho, the Greek letter, is the density of the liquid, and g is the acceleration due to gravity. The liquid above the bubble adds rho g h. Clear? And the air inside is two T over R higher than that. One surface, remember. So the pressure inside equals P naught, plus rho g h, plus two T over R, okay? Three terms: the atmosphere, the liquid above, and surface tension. Don't drop any. Got it?

3:50Smaller bubble, higher pressure

Now, back to our puzzle. Two soap bubbles, a small one and a big one, each with excess pressure four T over R inside. Smaller radius, larger four T over R, okay? So the small bubble has the higher pressure inside. With me? Now join them with a tube. Air always flows from higher pressure to lower pressure. So which way does it flow? From the small bubble into the big one! Yes, exactly. The small bubble shrinks, and the big one grows. So the sensible answer from the start was wrong. Mystery solved! Right? Now, when two bubbles touch, they share a common film. Let the radii be r one and r two, small r with subscripts one and two; the less than sign says r one is the smaller. The common film has radius r equals r one times r two, over r two minus r one. Clear? From one example on this page: radii two and three centimetres. Two times three, over three minus two, gives six centimetres. And the film bulges into the larger bubble: the smaller bubble has the higher pressure, on the concave side. Got it?

4:59Example 1: excess pressure in a water drop and a soap bubble

Now, example one. Find the excess pressure inside a water drop of radius one millimetre, with T equal to zero point zero seven newton per metre, and inside a soap bubble of radius two centimetres, with T equal to zero point zero three newton per metre. Pause the video, and try it first. Done? Okay, let's check. The drop first. Change the radius to metres: one millimetre is ten to the power minus three metres. And the drop has one surface. Right? So delta P equals two times zero point zero seven, over ten to the power minus three. That is one hundred and forty pascals, P a for short. Okay? Now the soap bubble, radius two centimetres. Which one do we use here, two T over R, or four T over R? Four T over R, yes! Two surfaces. And two centimetres is zero point zero two metres. Got it? So delta P equals four times zero point zero three, over zero point zero two: six pascals. Look, one hundred and forty pascals for the tiny drop, only six for the much bigger bubble. The small radius does most of that work. With me?

6:16Example 2: pressure inside an air bubble 20 cm under water

Now a harder one, example two. Find the pressure inside an air bubble of radius zero point five millimetres, twenty centimetres below the surface of water. P naught is ten to the power five pascals, T is zero point zero seven newton per metre, and g is ten metres per second squared. Pause the video, and try it. Done? Okay, let's check. Units first: twenty centimetres is zero point two metres, and zero point five millimetres is five times ten to the power minus four metres. Okay? Now, this is a bubble. So which excess pressure do we add, two T over R, or four T over R? Two T over R! It is air inside water, so just one surface. No soap film here. So P equals P naught, plus rho g h, plus two T over R, right? Now rho g h. For water, rho is one thousand kilograms per cubic metre. One thousand times ten times zero point two: two thousand pascals. Clear? Next, two T over R: two times zero point zero seven, over five times ten to the power minus four. That is two hundred and eighty pascals. Got it? Now add all three. Ten to the power five, plus two thousand, plus two hundred and eighty: one point zero two two eight times ten to the power five pascals. Forgot P naught? Then your answer is far too small. Always add all three terms. With me?

7:50Quick revision: 2T over R, 4T over R, bubble at depth h

So, quick revision. Curved surfaces have higher pressure on the concave side: two T over R for each surface. Drops and air bubbles in a liquid have one surface, two T over R; soap bubbles have two, four T over R, okay? At depth h, add P naught and rho g h as well. And the smaller bubble has the higher pressure. Now try the practice problems on this page. All the best!