Yield and atom economy · printable packet

What was collected against what the equation allowed, and what fraction of the atoms were ever going to be the product.

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 work out a percentage yield from a theoretical and an actual mass, and to use a known yield to predict what a batch will produce. You will be able to work out the atom economy of a reaction from its balanced equation, and to say clearly which of the two numbers careful work can change and which is fixed by the choice of reaction — and why a yield above 100 percent is a signal that something is wrong rather than a good result.

2. What you already have

You can work out the mass of product a reaction allows, from one reagent or from a mixture with a limiting one. Everything so far has assumed that mass actually appears in the flask. It never quite does, and this lesson is about the gap.

3. Words for this lesson

4. Two percentages, two different questions

$$\text{percentage yield} = \frac{\text{mass collected}}{\text{theoretical mass}} \times 100.$$

It is below 100 for reasons that have nothing to do with the chemistry being wrong: product left on the glassware, lost in a filtration, dissolved in the washings, or turned into something else by a side reaction. A reaction that does not go all the way to completion also shows up here.

Term	What it means
Theoretical yield	The mass of product the balanced equation allows.
Actual yield	The mass of product actually collected.
Percentage yield	The actual yield as a percentage of the theoretical yield.
Atom economy	The wanted product's mass in the equation as a percentage of all products' mass.
By-product	Something the reaction makes that nobody wanted.

Atom economy compares the wanted product with everything the reaction makes:

$$\text{atom economy} = \frac{\text{mass of the wanted product in the equation}}{\text{mass of all the products in the equation}} \times 100.$$

This one is worked out from the balanced equation and nothing else. It does not depend on how much you made, how carefully you worked, or whether the reaction went at all. It is a property of the route.

So: yield is about the day; atom economy is about the reaction. Careful work raises the first and cannot touch the second. Changing the second means finding a different reaction.

Another way: picture

Think of a bone-in roast. The yield is how much of it reaches the plate rather than being left on the cutting board — that is about the cook. The atom economy is how much of what you bought was meat rather than bone — that is about the cut, and no amount of skill changes it.

Another way: steps

For percentage yield:

Work out the theoretical mass from the equation and the limiting reagent.
Divide the mass collected by it.
Multiply by 100.

For atom economy:

Write the balanced equation.
Work out the mass of the wanted product in it, coefficient included.
Divide by the mass of everything the equation produces, and multiply by 100.

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

For a percentage yield. First find the theoretical yield, exactly as in the last two lessons: limiting reagent, mole ratio, molar mass of the product. Then take the mass actually collected, divide by the theoretical mass, and multiply by 100.

For a predicted mass. If the yield is already known from past runs, run it backwards: theoretical mass times the yield, divided by 100, is what to expect in the flask.

For an atom economy. Write the balanced equation. Multiply the wanted product's molar mass by its coefficient. Add up the coefficient-times-molar-mass of every product. Divide the first by the second and multiply by 100.

Check the work. A percentage yield must be below 100; a result above it means a wet product, an inverted division, or contamination. An atom economy must also be at most 100, and it equals 100 only when the wanted product is the only product. And a quick conservation check is free: the total mass of products in the equation must equal the total mass of reactants. If it does not, the equation was not balanced, and the atom economy built on it means nothing.

6. Why each step is allowed

Dividing collected by theoretical is the definition of a yield: a share of what was possible. The theoretical mass is the right denominator because it is the most the atoms could ever make.

Running a known yield backwards is allowed because a well-understood process loses about the same fraction each time, so the past percentage predicts the next batch.

Using the balanced equation alone for atom economy is allowed because it asks a question about the reaction, not about the run. The coefficients fix how much of each product forms per run, so the share of mass that is wanted product is fixed too.

Including the coefficients is required because they say how many of each product form. Two moles of iron weigh twice what one does.

Using the products as the denominator is equivalent to using the reactants, because mass is conserved: everything that goes in comes out as one product or another.

7. A reaction that is good at one and bad at the other

As a preparation it is easy and the yield is high — the gas comes off cleanly and there is little to lose. Call it 95%.

The atom economy is another matter. The equation produces $136 + 2 = 138$ of mass, and the hydrogen is 2 of it. That is $2 \div 138 \times 100 = 1.4\%$. Ninety-eight percent of the mass consumed leaves as zinc chloride, which nobody asked for.

Neither number is wrong and neither is the whole story. This is a perfectly good way to make a test tube of hydrogen and a terrible way to make a ton of it, and the two percentages are exactly what says so.

8. Why a yield above 100 is a signal, not a triumph

The theoretical yield is a ceiling set by conservation of atoms. Collecting more product than the equation allows would mean atoms appeared from nowhere, and they do not.

So a percentage yield above 100 always means something else, and it is usually one of three things: the product was weighed before it was dry, the division was done the wrong way up, or the product is contaminated with something that was not separated out.

It is worth saying because it makes the number self-checking. Any yield over 100 is a result to investigate rather than to write down — and impossible is a useful thing for a calculation to be, because it is the only kind of wrong answer that announces itself. A yield just under 100 deserves a second look too: very few real preparations lose nothing at all, so 99.8% usually means a damp product rather than flawless work.

9. Green chemistry and the numbers that drive it

In 1998 two American chemists, Paul Anastas and John Warner, set out twelve principles of green chemistry, and atom economy is the second of them: design reactions so that as much as possible of what goes in ends up in the product. The Environmental Protection Agency has given annual awards for green chemistry since 1996, and many of the winning entries are, at heart, a better atom economy.

The reason is economic as much as environmental. Every kilogram of by-product was bought as a reactant, has to be separated from the product, and then has to be sold, recycled or disposed of. A route with a high atom economy buys less, separates less and throws away less.

Yield matters too, and the two multiply. A route with a 90% atom economy run at an 80% yield turns only 72% of its inputs into saleable product. Improving either number helps, but only a new route can lift the atom economy. That is why a process chemist, handed a poor atom economy, goes back to the drawing board rather than to the plant floor: the waste is written into the equation itself.

10. In the world: the ibuprofen in a medicine cabinet

Ibuprofen, sold as Advil and Motrin, is one of the most widely used drugs in the United States, and the way it is made is a textbook case in green chemistry. The original route, developed in the 1960s, took six steps and had an atom economy of only about 40%: for every kilogram of ibuprofen, about a kilogram and a half of other products had to be separated out and dealt with.

In the 1990s a new three-step route was developed and commercialized in Texas, and it won one of the first Presidential Green Chemistry Challenge Awards from the EPA in 1997. Its atom economy is about 77%, and close to 99% if the acetic acid by-product is recovered and sold.

The arithmetic is the whole argument. A plant making 3,000 metric tons of ibuprofen a year on the old route produces about $3000 \times 60 / 40 = 4500$ tons of by-product. On the new route, about $3000 \times 23 / 77 \approx 900$ tons. That difference — 3,600 tons a year — is waste that never has to be handled, from reactants that never have to be bought. Percentage yield matters on top of that: the plant still tunes its handling to bring the yield up. But no amount of careful handling could have lifted the old route's 40% atom economy; only a new reaction could.

11. In the world: a school chemistry lab

A student preparing aspirin in a high school lab who collects 2.6 g from a theoretical 3.4 g has a 76% yield, a respectable result for a recrystallization, where some product always stays dissolved in the solvent.

12. Where this goes wrong

Treating yield as a property of the reaction. A percentage yield below 100 is not a failure of the chemistry. It is what was lost in the flask, on the filter paper and to a side reaction, and it says nothing about how wastefully the equation itself uses its atoms. That second question is atom economy, and a reaction can be perfect at one and poor at the other.

Dividing the wrong way up. Theoretical over actual gives a number above 100, which cannot be a yield. The collected mass goes on top.

Reading a low yield as bad chemistry. A 60% yield on a difficult separation may be excellent work. The number measures the whole procedure and not the reaction, and comparing yields is only fair between people doing the same preparation.

Working atom economy out from what was collected. It comes from the balanced equation and never from a measurement. Two chemists running the same reaction with wildly different yields have exactly the same atom economy.

Forgetting the coefficient in the atom economy. The mass of the wanted product in the equation is its molar mass times its coefficient. Using the molar mass alone understates it whenever the coefficient is not 1.

13. A percentage yield

14. An atom economy

15. Both numbers for one preparation

16. Your turn: the Haber process, $\mathrm{N_2 + 3H_2 \rightarrow 2NH_3}$, wanting the ammonia. What is its atom economy, and what would a yield of 80% mean?

17. Guided practice

The process $\mathrm{2KClO_3 \rightarrow 2KCl + 3O_2}$ is wanted for its oxygen. It ran at a yield of $80.0\%$ on the day, and its atom economy is $39.2\%$. A manager asks which number would change if the operators became more careful. Which is it?

18. Guided practice

Complete the worked solution: a blast furnace runs $\mathrm{Fe_2O_3 + 3CO \rightarrow 2Fe + 3CO_2}$, wanting the iron. Iron is fifty-six grams per mole and carbon dioxide forty-four. Find the iron's mass in the equation, the carbon dioxide's, and the atom economy to one decimal place.

19. Guided practice

In the reaction $\mathrm{CaCO_3 \rightarrow CaO + CO_2}$, the balanced equation allows $56$ g of calcium oxide and $42$ g was collected. What is the percentage yield?

20. Practice

A batch of zinc chloride is expected to give $136$ g, and the process is known to run at a yield of $85.0\%$. Work out what will actually be collected, and how much will be lost.

21. Practice

Ibuprofen made in the United States today mostly comes from a three-step route with an atom economy of about $77\%$, replacing an older six-step route at about $40\%$. A batch on the newer route produces $700$ kg of products in total. How many kilograms of that are ibuprofen, assuming a perfect yield?

22. Somewhere new

A company is deciding whether to build a plant around $\mathrm{CaCO_3 \rightarrow CaO + CO_2}$ for its calcium oxide, and wants to know what fraction of the mass it produces is the product it can sell. Work out the atom economy, to one decimal place.

23. Lesson test

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

24. Test question

A preparation runs $\mathrm{N_2 + 3H_2 \rightarrow 2NH_3}$ to make ammonia. The equation allows $34$ g, and $27.2$ g was collected after filtering and drying. Complete the table.

25. What you can do now

You can calculate a percentage yield and an atom economy, and say what each one measures. Explain why making hydrogen from zinc and acid can have a high yield and a dreadful atom economy at the same time. Next: reactions in solution, where the amount is set by a concentration and a volume rather than by a balance.

Working for the steps left to you

	theoretical mass, in g	mass collected, in g	percentage yield
$\mathrm{NH_3}$	34	27.2

16. Your turn: the Haber process, $\mathrm{N_2 + 3H_2 \rightarrow 2NH_3}$, wanting the ammonia. What is its atom economy, and what would a yield of 80% mean?, step 3

A fifth lost or unreacted, a question about the plant.