Math Detours · Grade 7 · Connections · 16

Squeeze the air

Block the end of a syringe and push the plunger. Halve the volume of the trapped air and its pressure doubles. Gases squeeze in a precise way.

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Tonight's idea

Fill a syringe with air, block the end with your finger and push the plunger. The air squeezes into less room, and pushes back harder the further you go. Water in the same syringe hardly squeezes at all. Gases are compressible.

Pressure is push on each unit of area, measured in kilopascals (kPa). The air around you already presses with about 100 kPa. Pressure counted from zero, where zero means no gas at all, is called absolute pressure.

Boyle's law: for a fixed amount of gas at a fixed temperature, absolute pressure \(\times\) volume stays the same.

So halve the volume and the pressure doubles. Three conditions matter: no gas leaks in or out, the temperature ends where it started, and the pressure is absolute. The law is a model, and a very good one for air at everyday pressures. How gases squeeze as they flow, around a fast aircraft for example, is a whole branch of math(s).

Watch one: a blocked syringe

A blocked syringe holds 20 mL of air at 100 kPa, the same as the air outside. You push the plunger in slowly, so the air stays at room temperature.

air 20 mL of air100 kPa air 10 mL of air200 kPa

The constant. Pressure \(\times\) volume starts at \(100 \times 20 = 2000\). With no leaks and no change of temperature, it stays 2000.

Squeeze to 10 mL. The pressure is \(2000 \div 10 = 200\) kPa. Half the volume, twice the pressure.

Squeeze to 16 mL instead. The pressure is \(2000 \div 16 = 125\) kPa. The volume is \(\tfrac45\) of what it was, and the pressure is \(\tfrac54\) of what it was. Flip the fraction. Check: \(125 \times 16 = 2000\).

Your turn

  1. Use the example's syringe, moved slowly. Find the pressure when the air is (a) squeezed to 5 mL, (b) stretched to 40 mL by pulling the plunger out. (c) What volume makes the pressure 250 kPa? (d) What does the law give at 1 mL? Could any pressure squeeze the air to 0 mL?
  2. A diver takes a floppy balloon holding 12 L of air under the sea. The surface pressure is about 100 kPa, and each 10 m of sea water adds about 100 kPa. Its air stays at one temperature. (a) Find the absolute pressure at 10 m and at 30 m. (b) Find the balloon's volume at each. (c) At what depth is its volume 4 L? (d) How much does it shrink in each 10 m step down to 30 m? Where does it change fastest?
  3. A tyre gauge reads only the extra pressure above the outside air, here 100 kPa. (a) A gauge reads 200 kPa. What is the tyre's absolute pressure? (b) What would a gauge read on the example's syringe at 20 mL, and at 10 mL? Does reading \(\times\) volume stay the same? (c) Air is let out until the gauge falls from 200 kPa to 100 kPa. Has the absolute pressure halved? (d) The tyre holds 30 L and keeps its size. What volume would all its first air fill at the new pressure? What fraction of that air is still in the tyre?
  4. A bike pump's barrel holds a 30 cm column of air at 100 kPa. The barrel has one width, so volume goes with length. The tyre's valve opens only when the pump's air reaches 300 kPa absolute. (a) Pushed slowly, how long is the column when the valve opens? (b) Pump fast and the barrel gets warm. Which of the law's conditions fails? (c) Hot gas pushes harder than cool gas in the same space. You push fast to 15 cm. Is the pressure 200 kPa, more, or less? (d) Hold it at 15 cm and wait. Nothing leaks. What happens to the pressure?

Stretch: why divers breathe out on the way up

A scuba diver breathes air supplied at the pressure of the water around them. A diver at 30 m has 6 L of air in their lungs, a round figure. Use exercise 2's pressures. (a) If the diver rose to the surface holding their breath, what volume would that air need? (b) By what number does the volume multiply from 30 m to 20 m, from 20 m to 10 m, and from 10 m to the surface? (c) Which part is most dangerous for someone holding their breath? Why do divers breathe out as they rise?

Answers

  1. Pressure \(\times\) volume is 2000 throughout. (a) \(2000 \div 5 = 400\) kPa. A quarter of the volume, four times the pressure. (b) \(2000 \div 40 = 50\) kPa. The law works for stretching the air too. This is less than the air outside, so the plunger is pushed back in when you let go. (c) \(2000 \div 250 = 8\) mL. (d) \(2000 \div 1 = 2000\) kPa. No pressure reaches 0 mL: any pressure times 0 is 0, never 2000. In the model, the pressure grows without limit as the volume shrinks. A real syringe would leak or break long before.
  2. (a) \(100 + 100 = 200\) kPa at 10 m, and \(100 + 300 = 400\) kPa at 30 m. (b) Pressure \(\times\) volume is \(100 \times 12 = 1200\). At 10 m the volume is \(1200 \div 200 = 6\) L, and at 30 m it is \(1200 \div 400 = 3\) L. (c) \(1200 \div 4 = 300\) kPa, which is 200 kPa more than at the surface, so 20 m. (d) The volumes at 0, 10, 20 and 30 m are 12, 6, 4 and 3 L. It loses 6 L, then 2 L, then 1 L. It changes fastest near the surface. The same step in depth always adds 100 kPa, but that is a doubling at the top and a much smaller change deeper down. This is a model: a real balloon's skin squeezes a little too, and deep water is often colder.
  3. (a) \(200 + 100 = 300\) kPa. (b) It would read 0 kPa at 20 mL and \(200 - 100 = 100\) kPa at 10 mL. The products are \(0 \times 20 = 0\) and \(100 \times 10 = 1000\). They are not equal, so the law fails with gauge readings. It needs absolute pressure. (c) No. The absolute pressure falls from 300 kPa to 200 kPa, which is \(\tfrac23\) of what it was. (d) Follow all the first air, once its temperature has settled. \(300 \times 30 = 9000\), and \(9000 \div 200 = 45\) L. Only 30 L of that is inside the tyre, so \(\tfrac{30}{45} = \tfrac23\) of the air is left. The gauge reading halved, but only a third of the air came out.
  4. (a) Pressure \(\times\) length is \(100 \times 30 = 3000\), and \(3000 \div 300 = 10\) cm. You push two thirds of the way before any air goes in. (b) The fixed temperature. Squeezing a gas fast heats it, and the heat has no time to leak away. (c) More than 200 kPa. Slowly, 15 cm would give \(3000 \div 15 = 200\) kPa. The fast-squeezed air is hotter, so it pushes harder. A calculation beyond this sheet gives about 260 kPa if no heat escaped at all, so a real pump lands somewhere between 200 and 260 kPa. (d) The air cools back to the temperature of the room, and the pressure falls to 200 kPa. Then Boyle's law holds again. The law compares the start and the end, as long as both are at the same temperature.

Stretch. (a) At 30 m the pressure is 400 kPa, so pressure \(\times\) volume is \(400 \times 6 = 2400\). At the surface the air would need \(2400 \div 100 = 24\) L, four times what the lungs held. (b) The pressures at 30, 20, 10 and 0 m are 400, 300, 200 and 100 kPa, so the volumes are 6, 8, 12 and 24 L. The multipliers are \(\tfrac43\), then \(\tfrac32\), then 2. (c) The last 10 m. There the air doubles, the biggest change of the whole climb, as in exercise 2. Lungs cannot stretch to hold several times their fill, and the trapped air could tear them. Breathing out lets the extra air escape, so the air left in the lungs always fits. That is why scuba divers are taught never to hold their breath on the way up. Someone who dives on one breath taken at the surface is in a different position: their air is squeezed going down, and on the way up it only returns to the size it started.

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