9702 · 15.2Equation of state

Pressure, temperature and moles

0:00Take a sealed plastic bottle of air on a cold morning and leave it in a hot car. By the afternoon, the bottle is tight and hard. Nobody added any air. The same molecules are in there, but the pressure has gone up, just because the temperature did. Now squeeze a syringe with your finger over the end. The air gets pushed into a smaller space, and you can feel it pushing back harder.

0:22Pressure, volume and temperature are clearly tied together. Change one, and the others respond. What's remarkable is that, for most gases in everyday conditions, the rule linking them is almost exactly the same, whether the gas is air, helium, nitrogen or carbon dioxide. In this video we'll build that rule up from experiments, see why it only works in kelvin, write it in its two standard forms, pV equals nRT and pV equals NkT, and then use it on real problems.

0:53Three quick reminders before we start. First, pressure is force per unit area, measured in pascals. One pascal is one newton per square metre. Normal atmospheric pressure is about one times ten to the five pascals. Second, thermodynamic temperature is measured in kelvin. To convert from degrees Celsius, you add 273.15. So twenty degrees Celsius is 293.15 kelvin. Zero kelvin is absolute zero, the lowest temperature there can be. Third, from the last video: amount of substance is measured in moles.

1:31One mole contains the Avogadro constant's worth of particles, N A, which is about 6.02 times ten to the twenty-three. That's everything we need.

Gas laws and absolute zero

1:41Long before anyone could see a molecule, scientists measured how gases behave. They trapped a fixed mass of gas, held one quantity constant, and watched how the other two changed. Three simple patterns came out. The first is Boyle's law. Keep the temperature constant and squeeze the gas. If you halve the volume of the gas, its pressure doubles. If you reduce the volume to a third, its pressure triples.

2:08So pressure is inversely proportional to volume, which means p times V stays constant. It makes sense if you think about molecules. Squash the same molecules into half the space, and each bit of the wall gets hit twice as often. Twice the collisions per second means twice the pressure. The second pattern: keep the pressure constant, for example with a gas trapped under a freely sliding piston, and heat it.

2:34The gas expands. And if you plot volume against temperature, you get a straight line. The third: keep the volume constant, in a rigid sealed flask, and heat it. The pressure rises, and again, pressure against temperature is a straight line. Now here's the key discovery. Extend those straight lines backwards, to lower and lower temperatures. For every gas, they hit zero pressure, or zero volume, at the same temperature: minus 273.15 degrees Celsius.

3:05That's absolute zero. So if you measure temperature from absolute zero, in other words in kelvin, the straight lines go right through the origin. That means volume is directly proportional to thermodynamic temperature at constant pressure, and pressure is directly proportional to thermodynamic temperature at constant volume. Notice the word "proportional" only works in kelvin. Heat a gas from ten degrees Celsius to twenty degrees Celsius at constant volume. The Celsius number has doubled, but the pressure has not.

3:36In kelvin, that's 283.15 to 293.15, a rise of only about three and a half percent, and that's how much the pressure goes up. Celsius has its zero in the wrong place for this. Kelvin has its zero where the gas actually runs out of pressure. We now have three results for a fixed mass of gas. At constant temperature, p V is constant. At constant pressure, V is proportional to T.

4:05At constant volume, p is proportional to T. There's one single statement that contains all three: p V is proportional to T. Check it. If T is fixed, p V is fixed: that's Boyle's law. If p is fixed, V must be proportional to T. If V is fixed, p must be proportional to T. All three drop straight out. This means that for any fixed mass of gas, p V divided by T is a constant.

4:36That gives a very useful form for "before and after" questions. If a gas starts with pressure p one, volume V one and temperature T one, and ends with p two, V two and T two, then p one V one over T one equals p two V two over T two. Here's the definition the syllabus uses. A gas that obeys p V proportional to T, where T is the thermodynamic temperature, is called an ideal gas.

5:06No real gas obeys it perfectly. But many come very close. Air in a room, helium in a balloon, or the gas in a car tyre all behave almost exactly like an ideal gas. Real gases are closest to ideal at low pressure and high temperature. At low pressure, the molecules are far apart, so the space taken up by the molecules themselves is tiny compared with the container, and the forces between them hardly matter.

5:32At high temperature, the molecules move so fast that those weak attractions have little effect. Real gases stray from ideal behaviour when they're squeezed to very high pressures or cooled until they're close to turning into a liquid. We'll look at the molecular picture of an ideal gas in detail in the next video, on kinetic theory. For now, the definition is the relationship: p V proportional to T.

The ideal gas equations

5:56So far, we've said p V over T is constant for a fixed mass of gas. But what does that constant depend on? Imagine two identical containers, side by side, at the same pressure and the same temperature, each holding one mole of gas. Now remove the wall between them. Nothing changes: same pressure, same temperature, but now twice the volume and two moles. So doubling the amount of gas doubles p V over T.

6:24The constant is proportional to the number of moles, n. We write p V over T equals n times a constant, R, and rearrange to get the equation of state for an ideal gas: p V equals n R T. p is the pressure in pascals. V is the volume in cubic metres. n is the amount of substance in moles. T is the thermodynamic temperature in kelvin. And R is the molar gas constant, 8.31 joules per mole per kelvin.

6:58The striking thing is that R is the same for every ideal gas. One mole of helium and one mole of oxygen, at the same pressure and temperature, take up the same volume. At zero degrees Celsius and normal atmospheric pressure, that's about 0.022 cubic metres, around twenty-two litres, whatever the gas. What matters is how many particles there are, not what they are. And one quick unit check: pressure times volume is pascals times cubic metres, which is newtons per square metre times cubic metres, which is newton metres, or joules.

7:33That's why R is measured in joules per mole per kelvin. Sometimes we don't want to count moles. We want to count individual molecules. Let's call the number of molecules capital N. The number of moles is the number of molecules divided by the Avogadro constant. So n equals N over N A. Substitute that into p V equals n R T, and you get p V equals N times R over N A, times T.

8:01That combination, R divided by N A, is itself a constant. It's called the Boltzmann constant, k. k equals R divided by N A. When you put the numbers in, 8.31 divided by 6.02 times ten to the twenty-three gives 1.38 times ten to the minus twenty-three joules per kelvin. So the equation of state can also be written as p V equals N k T. Think of it this way.

8:31R is the gas constant per mole. k is the gas constant per molecule. Since one mole is 6.02 times ten to the twenty-three molecules, k is R shared out between that many molecules, which is why it's so tiny. The two equations say exactly the same thing. Use p V equals n R T when you're working in moles. Use p V equals N k T when you're working with numbers of molecules.

8:59The pairing matters: small n goes with R, capital N goes with k. You'll need k again in the next video, when we connect temperature to the kinetic energy of individual molecules.

Worked examples with gas equations

9:11Let's use this. A bicycle pump contains 120 cubic centimetres of air at a pressure of 1.0 times ten to the five pascals and a temperature of seventeen degrees Celsius. The outlet is sealed, and the piston is pushed in quickly, so the volume falls to 40 cubic centimetres and the temperature of the air rises to forty-seven degrees Celsius. Find the new pressure. Step one: is it the same mass of gas?

9:38Yes, the outlet is sealed, so no air escapes. That means we can use p one V one over T one equals p two V two over T two. Step two: convert the temperatures to kelvin. This step is not optional. T one is seventeen plus 273.15, which is 290.15 kelvin. T two is forty-seven plus 273.15, which is 320.15 kelvin. Step three: rearrange for p two. p two equals p one, times V one over V two, times T two over T one.

10:19Step four: put the numbers in. V one over V two is 120 over 40, which is 3. Because this is a ratio of two volumes, the units cancel, so we can leave them in cubic centimetres here. T two over T one is 320.15 over 290.15, which is about 1.1. So p two equals 1.0 times ten to the five, times 3, times 1.1, which is 3.3 times ten to the five pascals.

10:51Does that make sense? Squashing the air to a third of its volume alone would triple the pressure. Warming it up pushes it a little higher still. So a bit more than three times the starting pressure is exactly what we'd expect. Now watch what happens if you forget to convert to kelvin and use forty-seven over seventeen instead. You'd get about 8.3 times ten to the five pascals, which is more than two and a half times too big.

11:17The temperature ratio in Celsius is meaningless, because Celsius doesn't start from absolute zero. Here's a different kind of question. A sealed flask has a volume of 250 cubic centimetres. It contains air at a pressure of 1.01 times ten to the five pascals and a temperature of twenty degrees Celsius. How many molecules of air are in the flask? We're asked for a number of molecules, so p V equals N k T is the natural choice.

11:48The first step is to convert everything to SI units. This time we're not taking a ratio, so the volume must be in cubic metres. One cubic centimetre is ten to the minus six cubic metres, so 250 cubic centimetres is 2.5 times ten to the minus four cubic metres. And the temperature is twenty plus 273.15, which is 293.15 kelvin. Step two: rearrange. N equals p V divided by k T.

12:20Step three: calculate. p V is 1.01 times ten to the five, times 2.5 times ten to the minus four, which is 25.25 joules. k T is 1.38 times ten to the minus twenty-three, times 293.15, which is about 4.05 times ten to the minus twenty-one joules. Divide one by the other, and N is about 6.2 times ten to the twenty-one molecules. That's six thousand two hundred billion billion molecules, in a flask you can hold in one hand.

12:59You could also do this in two steps, using moles. p V over R T gives n, about 0.0104 moles. Multiply by the Avogadro constant, 6.02 times ten to the twenty-three, and you get the same answer, 6.2 times ten to the twenty-one. Both routes agree, because k is just R divided by N A.

Common mistakes and gas release

13:24Let's go through the mistakes that cost the most marks. Mistake one: using degrees Celsius. Every form of the ideal gas equation needs thermodynamic temperature, in kelvin. The equation says p V is proportional to T, and that proportionality only exists when T is measured from absolute zero. Put in a Celsius temperature and the answer is simply wrong, as we saw in the first example. Worse, at zero degrees Celsius you'd be dividing by zero.

13:54Mistake two: forgetting SI units in p V equals n R T or p V equals N k T. R and k are in joules, so pressure must be in pascals and volume in cubic metres. A pressure in kilopascals has to be multiplied by a thousand. A volume in cubic centimetres has to be multiplied by ten to the minus six, and a volume in litres has to be multiplied by ten to the minus three.

14:20The only time you can leave volumes or pressures in other units is in a ratio, like p one V one over T one equals p two V two over T two, where the same unit appears on both sides and cancels. Temperature is never allowed to stay in Celsius, even in a ratio. Mistake three: mixing up small n and capital N. Small n is the number of moles, and goes with R.

14:46Capital N is the number of molecules, and goes with k. If you use R with a number of molecules, your answer will be out by a factor equal to the Avogadro constant, about six times ten to the twenty-three. Mistake four: using the before-and-after equation when gas has escaped or been added. p one V one over T one equals p two V two over T two is only true for a fixed amount of gas.

15:14If a tyre leaks, or a gas cylinder has been partly used, n has changed, and you need p V equals n R T for each state separately. Mistake five: thinking the gas equation depends on which gas it is. For an ideal gas, it doesn't, because the constants R and k are exactly the same for every ideal gas. One mole of any ideal gas at the same pressure and temperature takes up the same volume.

15:40Here's a typical exam-style question. "A gas cylinder of volume 0.020 cubic metres contains an ideal gas at a pressure of 3.0 times ten to the six pascals and a temperature of fifteen degrees Celsius. Some gas is released, and the pressure falls to 1.2 times ten to the six pascals. The temperature stays at fifteen degrees Celsius. Calculate the number of moles of gas that were released from the cylinder."

16:09First, spot the trap. Gas has been released, so the mass isn't fixed. You can't use p one V one over T one equals p two V two over T two here. Instead, use p V equals n R T for each state. Convert the temperature: fifteen plus 273.15 is 288.15 kelvin. The volume is already in cubic metres, and the pressures are in pascals. Before: n one equals p V over R T, which is 3.0 times ten to the six, times 0.020, divided by 8.31 times 288.15.

16:51That's about 25.1 moles. After: n two is 1.2 times ten to the six, times 0.020, divided by the same 8.31 times 288.15, which is about 10.0 moles. So the amount released is about 15 moles. Notice a shortcut. The volume and temperature didn't change, so the amount of gas is proportional to the pressure. The pressure fell by 1.8 times ten to the six pascals, and 1.8 times ten to the six, times 0.020, divided by R T gives the same 15 moles directly.

17:30Examiners like this question because it tests whether you understand the conditions behind the equations, not just whether you can substitute numbers. Ask yourself every time: is the amount of gas fixed? Are my temperatures in kelvin? Are my units SI?

Equation of state recap

17:46Let's pull it all together. Experiments on a fixed mass of gas show three things. At constant temperature, p V is constant. At constant pressure, volume is proportional to thermodynamic temperature. At constant volume, pressure is proportional to thermodynamic temperature. All three combine into one statement: p V is proportional to T. A gas that obeys p V proportional to T, with T in kelvin, is called an ideal gas.

18:16Real gases come closest to this at low pressure and high temperature. Including the amount of gas gives the equation of state: p V equals n R T, where n is the number of moles and R is the molar gas constant, 8.31 joules per mole per kelvin. Counting molecules instead gives p V equals N k T, where N is the number of molecules and k is the Boltzmann constant, 1.38 times ten to the minus twenty-three joules per kelvin.

18:46The two are linked by k equals R divided by the Avogadro constant. Always use kelvin, always use SI units, and check whether the amount of gas is fixed before reaching for p one V one over T one equals p two V two over T two. In the next video, we'll look inside the gas, and see how molecules bouncing off the walls produce this pressure in the first place.