Molar volume · printable packet

Twenty-four liters per mole, the same for every gas, and what that lets you do with a balanced equation.

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 convert between a volume of gas at room temperature and pressure and an amount in moles, using the molar volume of 24 liters per mole, and to take a gas volume through a balanced equation in either direction — balancing it first, because the coefficients are the ratio the calculation runs through. You will be able to say why one number serves every gas, and why that means the molar volume tells you nothing about which gas it is.

2. What you already have

You can turn a mass into an amount with a molar mass, and you can take an amount through a balanced equation with its coefficients. This lesson adds one number that turns a volume of gas into an amount, and once you have it, a gas goes through an equation in exactly the way a weighed solid already does.

3. Words for this lesson

4. One number turns a gas volume into an amount

Term	What it means
Molar volume	The volume one mole of a gas occupies under stated conditions.
Room conditions	About 293 K and 100 kPa; the molar volume there is 24 L/mol.
Avogadro's law	Equal volumes of gases at the same conditions hold equal numbers of molecules.
Gas density	A gas's mass per liter, its molar mass over 24.
STP	Standard temperature and pressure, 273 K and 100 kPa; about 22.7 L/mol.

with $V$ in liters and $n$ in moles, at room temperature and pressure. Twenty-four liters is about the volume of a large bucket, and it is what one mole of any gas fills.

That last word is the one to stop at. One mole of hydrogen weighs $2$ g and one mole of chlorine weighs $71$ g, and they occupy the same twenty-four liters. A balloon of hydrogen and an identical balloon of chlorine hold the same number of molecules and differ in mass by a factor of thirty-five.

Why that is not as strange as it sounds. The molecules of a gas are a very long way apart compared with their own size — roughly ten times their own width in every direction, so the gas is more than 99.9% empty space. What sets the volume is how many molecules there are and how hard they are bouncing, and at the same temperature and pressure those are the same for every gas.

$$\text{mass} \xrightarrow{\ \div M\ } \textbf{moles} \xleftarrow{\ \div 24\ } \text{volume of gas}$$

Another way: picture

Picture twenty-four one-liter cartons stacked in a crate — about the size of a small bucket. That crate holds one mole of a gas, and it holds one mole whichever gas you pour into it. Put hydrogen in and the crate weighs 2 g more than when it was empty; put chlorine in and it weighs 71 g more. Same crate, same count, different weight.

Another way: steps

To take a gas volume through a reaction:

Divide the volume in liters by 24 to get the amount in moles.
Use the balanced equation's coefficients — the coefficient of what you want over the coefficient of what you have.
Come back out: multiply by 24 for another gas volume, or by a molar mass for a mass.

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

Balance the equation, if there is one. The coefficients are the ratio every later step uses.

Get to moles. A gas volume at room conditions: divide by 24. A mass of anything: divide by its molar mass. A solution: concentration times liters.

Apply the ratio. Coefficient of what you want over coefficient of what you have.

Come back out. A gas: multiply moles by 24 for liters. Anything else: multiply by its molar mass for grams.

Check the work. First, size: a volume less than 24 L must be less than a mole, and more than 24 L more than a mole. Second, where the molar volume was used: only for substances that are gases at room conditions — never for a solid, a liquid or a dissolved substance. Third, if every substance in the step is a gas, the volume ratio should equal the coefficient ratio directly; if your volumes do not stand in that ratio, something was inverted.

6. Why each step is allowed

Dividing a gas volume by 24 is allowed because, at room conditions, every gas fills 24 liters per mole — Avogadro's observation, confirmed by the gas law relationship from the last lesson. It is the same kind of conversion as dividing a mass by a molar mass, with a rate that happens to be the same for every gas.

Using one number for all gases is allowed because the volume of a gas is set almost entirely by the empty space between molecules, and that space depends only on the count of molecules, the temperature and the pressure.

Applying the coefficients only to moles is required, as in every earlier lesson, because coefficients count particles.

Using volumes directly in the coefficient ratio, when every substance is a gas, is allowed because each volume is the same constant times its moles. The constant cancels from any ratio of two gas volumes.

Refusing to use 24 for a solid is required because a solid has no empty space between its particles to speak of, so its volume depends on the size of the particles and differs from substance to substance.

7. Which number belongs to which step

The two errors in this lesson both come from using the right number in the wrong place, so it is worth naming the jobs.

Number	What it converts	Does it depend on the substance?
the molar mass, $M$	grams and moles	yes — it is a sum over the formula
the molar volume, $24$ L/mol	liters of gas and moles	no — the same for every gas
the coefficients	moles of one substance and moles of another	yes — they come from the balanced equation

Read the middle row again. The molar volume does not depend on the gas, so an item that gives you the formula of the gas has told you something you do not need for that step. That is not a trap; it is information you need for a different step, and knowing which is which is most of the skill.

A useful consequence: gas volumes react in the ratio of the coefficients directly. In $\mathrm{N_2 + 3H_2 \rightarrow 2NH_3}$, one liter of nitrogen needs three liters of hydrogen and makes two liters of ammonia, with no arithmetic at all — because every volume goes through the same division by 24 and it cancels. That shortcut works only where every substance involved is a gas.

8. Density, which is taught here and not graded

The density of a gas is its mass divided by its volume, and the molar volume gives it in one line: one mole weighs $M$ grams and occupies $24$ liters, so

Ethane, $\mathrm{C_2H_6}$, has a molar mass of $30$, so its density is $30 \div 24 = 1.25$ g/L. Propene, $\mathrm{C_3H_6}$, has a molar mass of $42$ and a density of $1.75$ g/L. Ozone, $\mathrm{O_3}$, has a molar mass of $48$ and a density of exactly $2$ g/L.

Air is a mixture and behaves like a single gas of molar mass about $29$, so its density is about $1.2$ g/L. That one number explains a great deal of ordinary life: helium ($M = 4$) and hydrogen ($M = 2$) are far lighter than air and a balloon of either rises, while carbon dioxide ($M = 44$) and propane ($M = 44$) are heavier than air and both pool near the floor. A propane leak in a basement is dangerous for precisely this reason, and a carbon dioxide extinguisher works for it too.

No item in this course grades a gas density, and the reason is the course's rule about terminating answers: $M \div 24$ does not come out exactly for most gases — methane's density is $0.666\ldots$ g/L — and a tolerance that quietly demands three figures marks a learner wrong for a reason nobody told them. So the relationship is taught, shown and used in reasoning, and the graded questions stay on the arithmetic that comes out exact.

9. In the world: the carbon dioxide in a craft brewery

Fermentation turns sugar into alcohol and carbon dioxide, $\mathrm{C_6H_{12}O_6 \rightarrow 2C_2H_5OH + 2CO_2}$, and for most of brewing history the gas simply vented to the air. A growing number of American craft breweries now capture it, clean it, and use it to carbonate their own beer instead of buying carbon dioxide by the truckload.

The molar volume is how a brewer sizes the equipment. A 30-barrel batch might ferment about 360 kg of sugars. In moles that is $360{,}000 \div 180 = 2000$ mol of glucose, which by the equation gives $4000$ mol of carbon dioxide. At room conditions that is $4000 \times 24 = 96{,}000$ liters — nearly 100 cubic meters of gas, enough to fill a large living room, from a single batch.

The same number tells the brewer how much is needed back. Carbonating that beer takes only about a tenth of what fermentation released, so a capture system can in principle make the brewery self-sufficient with gas to spare. Breweries in Wisconsin and Colorado that installed capture systems during the carbon dioxide shortages of recent years did the arithmetic exactly this way. The density matters too: carbon dioxide is heavier than air, so it collects in low spots of a cellar, which is why brewery cellars carry gas monitors near the floor.

10. In the world: a helium party balloon

A standard 11-inch latex balloon holds about 14 liters, so about $14 \div 24 \approx 0.6$ mol of helium, weighing about 2.3 g. The same balloon of air would weigh about 17 g, and the 15 g difference is the lift that holds the string up.

11. Where this goes wrong

Multiplying by 24 when the question wanted a division. An answer 576 times too large or too small is always this. Sanity-check it: is the volume bigger or smaller than 24 liters, and therefore is the amount bigger or smaller than one mole?

Using a molar volume for a solid or a liquid. There is no such thing. The whole reason one number works for every gas is that a gas is mostly empty space; a solid is not, and a mole of iron and a mole of lead occupy quite different volumes. The molar volume applies to the gases in an equation and to nothing else in it.

Forgetting the conditions. Twenty-four liters per mole is a room-temperature figure. The same mole of gas at $0$ °C fills about $22.7$ L, and in a furnace it fills far more. Every question in this course says at room temperature and pressure, and it says it because the number would otherwise be undefined.

Treating mass as amount. Mass is not amount. Two grams of hydrogen and two grams of lead are the same mass and nothing like the same number of atoms, and a reaction counts atoms. Every question in this unit that looks like it is about mass is about moles with mass at each end.

Using the coefficient shortcut with a solid. Volumes stand in the coefficient ratio only when every substance compared is a gas. With a solid in the step, go through moles.

12. Hydrogen from magnesium

13. Gas volumes straight from the coefficients

14. A weighed solid to a gas volume, with balancing

15. Your turn: what volume of carbon dioxide, at room conditions, comes from heating $25$ g of calcium carbonate in $\mathrm{CaCO_3 \rightarrow CaO + CO_2}$?

16. Guided practice

A sealed flask holds $19.2$ L of chlorine at room temperature and pressure. How many moles of it are in the flask?

17. Guided practice

Complete the worked solution: hydrogen peroxide decomposes as $\mathrm{2H_2O_2 \rightarrow 2H_2O + O_2}$. A sample of one hundred two grams, at thirty-four grams per mole, breaks down fully. Find the moles of peroxide, the moles of oxygen, and the oxygen's volume at room conditions.

18. Guided practice

In limestone roasted in a lime kiln, the equation is $\mathrm{CaCO_3 \rightarrow CaO + CO_2}$. What volume of carbon dioxide, measured at room temperature and pressure, does $280$ g of calcium carbonate give? Answer in liters.

19. Practice

A volume of gas cannot be worked out from an equation that is not balanced: the coefficients *are* the ratio the whole calculation runs through, and wrong coefficients give a wrong volume with nothing in the arithmetic to show for it. Balance the equation for an old bottle of hydrogen peroxide going off on a shelf, and the calculation that follows becomes answerable.

20. Practice

A craft brewery in Milwaukee captures the carbon dioxide its yeast gives off: $\mathrm{C_6H_{12}O_6 \rightarrow 2C_2H_5OH + 2CO_2}$. A batch ferments $180$ g of glucose (180 g/mol). What volume of carbon dioxide, at room conditions, can it capture, in liters?

21. Somewhere new

A cylinder bound for a hospital cylinder is labeled as holding $2688$ L of oxygen, measured at room temperature and pressure. The trucking company is paid by weight and needs to know what the gas alone weighs. What mass of oxygen is in the cylinder? Answer in kilograms.

22. Lesson test

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

23. Test question

Three identical flasks of $21.6$ L each are filled, one with ammonia, one with nitrogen and one with hydrogen chloride, all at room temperature and pressure. For each flask, give the amount of gas in it and the mass of that gas.

24. What you can do now

You can turn a volume of gas into an amount and back, and run a gas through a balanced equation. Say what volume 0.5 mol of any gas occupies at room conditions, and why the answer did not depend on which gas. Next: what the model behind that number assumes, and the conditions under which it stops being true.

Working for the steps left to you

	molar mass, in g/mol	amount of gas, in mol	mass of gas, in g
ammonia	17
nitrogen	28
hydrogen chloride	36.5

15. Your turn: what volume of carbon dioxide, at room conditions, comes from heating $25$ g of calcium carbonate in $\mathrm{CaCO_3 \rightarrow CaO + CO_2}$?, step 3