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Where the ideal model fails

The two assumptions behind the gas relationship, the conditions under which each gives way, and why a bottle of liquid butane refutes the model outright.

Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.

1. What you will learn

By the end of this lesson you will be able to state the two assumptions the ideal gas model rests on, say which of them fails under a given set of conditions, and say which way that failure moves the measured volume — attraction making a gas smaller than predicted, the molecules' own size making it larger. You will also be able to rank gases by how well the model fits them from their boiling points alone, and to say why an ideal gas could never become a liquid.

2. What you already have

You can use the relationship between pressure, volume and absolute temperature, and you can turn a gas volume into an amount with the molar volume. You also know, from the unit on intermolecular forces, that molecules pull on one another and that how strongly they do it decides what a substance boils at. This lesson puts those two things together and asks what the first one was quietly assuming.

3. Words for this lesson

4. Where does the relationship stop being true?

The last two lessons used $pV/T = $ constant and it worked every time. A relationship that always works is either a law of nature or one that has only been tried in easy places.

One mole of each gas, predicted volume against measured. A ratio of $1.00$ means the model was exactly right.

Where is the model good? Name the rows where it would not have cost you a mark.

Then the part worth the lesson: the errors do not all go the same way. Nitrogen at 500 atm comes out bigger than predicted; ammonia at 100 atm comes out much smaller. One number cannot be wrong in two directions for one reason, so two things are going wrong.

Propose a cause for each direction. The model treats molecules as points that take up no room and pull on each other not at all, and you know that ammonia pulls on itself far harder than helium does.

5. Two assumptions, both false, both usually harmless

The ideal gas model assumes the molecules take up no room and pull on each other not at all. Both assumptions are wrong, and both are nearly harmless when a gas is warm and spread out. Squeeze it or cool it and the two errors stop cancelling: the volume the molecules themselves occupy stops being negligible, and the attractions that eventually make it a liquid start to show.

Those are the two assumptions, and each accounts for one direction in the table you have just read.

• Warm and spread out — the first two rows — the molecules are about ten times further apart than they are wide and far too fast for an attraction to take hold. Both errors are tiny and partly cancel.
• Squeezed hard — nitrogen and helium at high pressure — the molecules are nearly touching, so the room they occupy is a real fraction of the container and the gas comes out larger than predicted.
• Cooled, or strongly attracting — ammonia and carbon dioxide — they are held long enough to strike the walls less hard, and the gas comes out smaller than predicted. A little further and the attractions win outright: it condenses.

So the single most useful sentence in this lesson is the one the two directions forced: attraction makes a gas smaller than predicted; the size of the molecules makes it larger. Which you see depends on which is winning.

6. The method, step by step, and how to check it

Locate the conditions. How far is the gas above its boiling point, and how far above ordinary pressure? Those two distances decide everything.

Check for the easy case. Well above its boiling point and near ordinary pressure, the model is good to about one percent. Use $pV/T$ and the molar volume with confidence.

If squeezed hard, expect the gas to come out larger than predicted, because the molecules' own volume is no longer negligible.

If cold or strongly attracting, expect it to come out smaller, because the molecules hold one another back from the walls.

If a ratio is given, multiply the ideal prediction by measured over predicted to get the real value, or divide a real measurement by the ideal molar volume to see how far off the model is.

Check the work. Does the direction of your correction match the cause you named? A gas below 1 must be attraction-dominated; above 1, size-dominated. And does the size of the departure fit the conditions — a few percent at a hundred atmospheres for nitrogen, tens of percent for a gas near its boiling point? A huge departure at ordinary conditions means a slip, not a strange gas.

7. Why each step is allowed

Using boiling point as a gauge is allowed because boiling is the moment molecules have enough energy to escape one another. The temperature it takes measures exactly the attraction the model ignores.

Predicting larger for a squeezed gas is allowed because the space the molecules themselves fill cannot be compressed. The container's volume is no longer all available, so a given amount of gas needs more of it than the model allows.

Predicting smaller for an attracting gas is allowed because a molecule about to hit a wall is tugged back by its neighbors. It strikes less hard, the pressure falls short of the model, and at a fixed pressure the gas shrinks to compensate.

Multiplying by measured over predicted is allowed because that ratio is defined as exactly the correction factor: it says what the real volume is as a fraction of the ideal one.

Trusting the model at room conditions is allowed because both corrections are tiny there and partly cancel, which is why the last two lessons worked.

8. Boiling point is the reading you already have

You do not need a new measurement to say how far from ideal a gas is: its boiling point already says it. A substance boils when its molecules have enough energy of motion to break free of one another, so the temperature it boils at is a measure of how strongly they hold on — which is precisely the assumption the model gets wrong. Reading a few off:

Read the last column as the third: there is almost nothing in helium for the model to get wrong, and a great deal in steam. So the ranking question in this lesson is not a new skill — it is the intermolecular-forces ranking with a different question written on the front.

A bond and an intermolecular force are not the same size of thing. Boiling water pulls whole molecules apart from each other and leaves every $\mathrm{O-H}$ bond exactly where it was, which is why water boils at a hundred degrees and does not decompose there. A substance with strong bonds inside its molecules can still boil low, and the question which forces act between the molecules is a different question from what holds each molecule together.

9. What the model cannot explain at all

Being a few percent out and being unable to describe something at all are different failures. These are the second kind, and they are the evidence that this is a model.

An ideal gas can never become a liquid. Condensing means molecules staying near one another, and there is nothing in the model to make them stay. Squeeze an ideal gas as hard as you like and it just gets denser, forever. Every tank of liquid propane, every carbon dioxide extinguisher, every tank of liquid ammonia on a farm is a standing refutation.

An ideal gas cannot be cooled by letting it expand. Real gases cool on expanding through a valve, because the molecules do work pulling away from one another. That is how a refrigerator works and how air is liquefied; in the model there is no attraction to pull away from, so nothing cools.

An ideal gas has no surface tension and no critical temperature. Both come of molecules feeling each other.

None of this makes the model bad. It makes it a model: excellent for the job it was built for, and silent about everything that depends on the simplification.

10. In the world: the propane tank under a gas grill

The 20-pound propane tank that sits under millions of American backyard grills is a working demonstration of where the ideal model fails. Shake a full one and you can hear liquid slosh. At ordinary temperature, propane squeezed to about 9 atmospheres condenses, and that is exactly what the tank relies on: a liquid holds far more propane than the same steel bottle could hold as a gas.

The ideal model cannot even describe this. An ideal gas never condenses, so it would predict that 20 pounds of propane — about 206 moles — at room temperature would need roughly $206 \times 24 = 4900$ liters at ordinary pressure, or about 550 liters at 9 atmospheres. The real tank holds it in under 20 liters of liquid, because propane molecules attract one another strongly enough to cling together once pushed close.

The same failure explains the tank's safety rules. The liquid sits at the bottom with vapor above it, and the pressure in the tank is set by how readily the liquid evaporates at that temperature, not by how much propane is left. That is why the gauge reads nearly the same pressure until the tank is almost empty, and why a tank must never be filled more than about 80 percent: the liquid expands when warm, and a completely full tank has no vapor space to absorb the expansion. Every one of those rules comes from attraction between molecules — the very thing the ideal model leaves out.

11. In the world: scuba tanks

A standard aluminum scuba tank is filled to about 200 atmospheres. Air at that pressure is about five percent larger than the ideal model predicts, because the molecules' own size is starting to matter, so a tank holds a little less air than a naive calculation suggests.

12. Where this goes wrong

*Thinking a real gas is always smaller than the model predicts, or that high pressure is the condition the model was built for.* Neither. It depends which failure is winning — attraction makes a gas smaller, the molecules' own size makes it larger — and at very high pressure the second wins for every gas. The model is at its best when a gas is thin, warm and spread out.

Treating a departure from ideal behavior as a change of state. A gas that disagrees with the relationship by five percent is still entirely a gas. Condensing is a different and much later event.

Thinking heavier gases are less ideal. Mass is not the point; attraction is. Ammonia, at $17$ g/mol, is much further from ideal than chlorine at $71$ g/mol, because ammonia hydrogen bonds and chlorine does not.

Thinking the molecules stop moving when a gas is compressed or condensed. Temperature, and only temperature, sets how fast molecules move. A liquid's molecules move just as fast as the gas they came from at the same temperature; they simply no longer escape.

Throwing the model away because it fails somewhere. Every model fails somewhere. Knowing where — and in which direction — is what lets you keep using it everywhere else.

13. Reading the investigation's table

1. Read nitrogen at ordinary pressure.

   $1.000$

   Warm and spread out: the model is exact.

2. Read nitrogen at 500 atm.

   $1.620 > 1$

   Larger than predicted.

3. Name the cause.

   $\text{the molecules' own size}$

   They are nearly touching.

4. Read ammonia at 100 atm.

   $0.610 < 1$

   Smaller than predicted.

5. Name the cause.

   $\text{attraction: hydrogen bonding}$

   Two directions, two causes.

14. Two gases in the same tank

1. Set up the comparison.

   $\text{hydrogen and ammonia, same } p, V, T$

   The ideal model predicts equal amounts.

2. Gauge hydrogen's attraction.

   $\text{boils at } 20 \text{ K}$

   Almost none.

3. Name hydrogen's failure.

   $\text{size: ratio above } 1$

   The tank holds less than predicted.

4. Gauge ammonia's attraction.

   $\text{hydrogen bonds; boils at } -33 \ {}^{\circ}\mathrm{C}$

   Strong.

5. Name ammonia's failure.

   $\text{attraction: ratio below } 1$

   The tank holds more than predicted.

6. Push ammonia further.

   $\text{high enough pressure: liquid}$

   The attractions win outright.

15. Correcting a prediction with a ratio

1. Read the conditions.

   $1 \text{ mol CO}_2, \text{ room temperature}, 100 \text{ atm}$

   Squeezed hard.

2. Make the ideal prediction.

   $24 \div 100 = 0.24 \text{ L}$

   A hundredfold pressure.

3. Read the measured ratio.

   $0.730$

   Below one.

4. Correct the prediction.

   $0.24 \times 0.730 = 0.175 \text{ L}$

   The real volume.

5. Name the cause.

   $\text{attraction wins over size}$

   Carbon dioxide attracts fairly strongly.

6. Compare with nitrogen.

   $1.070 \text{ at the same pressure}$

   Weak attraction, so size wins there.

7. Draw the conclusion.

   $\text{same conditions, opposite departures}$

   The gas's own forces decide.

16. Your turn: steam at $110$ °C and ordinary pressure occupies a little *less* volume than the ideal relationship predicts. Which assumption has failed, and why that one?

1. Read the direction.

   $\text{smaller than predicted}$

   Start from the direction of the error.

2. Name the failed assumption.

   $\text{no attraction between molecules}$

   Only attraction makes a gas smaller.

3. Your turn: work this step out. Its working is at the end of the packet.

   Name the cause.

17. Guided practice

Consider a gas at room temperature and ordinary pressure. Which of the ideal gas model's two assumptions gives way first under those conditions?

• The molecules take up room of their own, so part of the container cannot be squeezed into and the gas comes out larger than predicted.
• The molecules pull on one another, so they strike the walls less hard than the model expects and the gas comes out smaller than predicted.
• Neither: under these conditions both assumptions are near enough true, and the model describes the gas well.
• The molecules have stopped moving, so the gas can no longer be compressed at all.

18. Guided practice

Complete the worked solution: one mole of carbon dioxide at room temperature is squeezed to one hundred times ordinary pressure. The ideal model starts from twenty-four liters per mole at ordinary pressure. Measured over predicted comes out at seventy-three hundredths. Find the predicted volume, the measured volume, and how many percent below prediction the gas is.

1. Predict the ideal volume.

   $\text{twenty-four} \div \text{one hundred} =$ _p

   A hundredfold pressure, a hundredth of the volume.

2. Find the measured volume.

   $(\text{prediction}) \times \text{seventy-three hundredths} =$ _m

   The real gas is smaller.

3. Find the shortfall percent.

   $\text{one hundred} - \text{seventy-three} =$ _s

   Attraction pulls it below prediction.

19. Guided practice

Put these three gases in order, starting with the one the ideal model describes best at room temperature and pressure and ending with the one it describes worst.

Number the steps in order (write the number in the box):

• propane
• helium
• methane

20. Practice

A cylinder fitted with a piston holds one mole of gas: $24$ L at $100$ kPa and $300$ K. Get it to $600$ K while ending with the volume back at $24$ L. One button breaks a rule the apparatus cannot break; leave it alone.

This task has no paper form; do it on a device.

21. Practice

A natural gas pipeline in Texas runs at high pressure. At that pressure the ideal model predicts one mole of methane fills $0.25$ L, but measured over predicted is $0.8$, so each mole really fills $0.2$ L. A segment holds $40$ L of gas. How many moles of methane does it really hold?

The answer: _a mol.

22. Somewhere new

A carbon dioxide fire extinguisher is a steel bottle holding *liquid* carbon dioxide under pressure at ordinary room temperature; shaking a full one, you can hear the liquid move. What does the existence of that liquid say about the ideal gas model for carbon dioxide under those conditions?

• The model has failed outright: an ideal gas can never become a liquid, because condensing needs exactly the attractions the model says are absent.
• Nothing at all: the ideal relationship still describes the contents, because they are still carbon dioxide.
• The model still holds, because the high pressure inside the bottle is exactly the condition it was written for.
• The model fails because the molecules in the liquid have stopped moving.

23. Lesson test

Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.

24. Test question

For each set of conditions, say whether the ideal gas model describes the gas well, and which of its two assumptions gives way first. The two assumptions are that the molecules take up no room of their own and that they do not pull on one another.

25. What you can do now

You can say what the ideal gas model assumes and where each assumption gives way. Say which way a real gas's volume moves when its molecules attract one another, and why helium is the closest thing there is to an ideal gas. That completes the unit on gases; what follows is energy, and the reason a reaction warms or cools the flask it happens in.

Working for the steps left to you

16. Your turn: steam at $110$ °C and ordinary pressure occupies a little *less* volume than the ideal relationship predicts. Which assumption has failed, and why that one?, step 3