Concentration and dilution · printable packet

Amount per liter as a third route to moles, what a dilution leaves unchanged, and the arithmetic of a titration.

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1. What you will learn

By the end of this lesson you will be able to work out the amount of solute in a solution from its concentration and volume, and the mass to weigh out to make a solution up to a stated concentration. You will be able to handle a dilution by reasoning that the amount of solute is unchanged, converting milliliters to liters wherever the calculation needs it, and to work an unknown concentration out from a titration result using the equation's mole ratio.

2. What you already have

You can get an amount in moles from a mass, and take an amount through a balanced equation. This lesson adds the third route to an amount — the one you use when the substance is dissolved and cannot be weighed — and then the course's toolkit is complete.

3. Words for this lesson

4. Amount per volume

Term	What it means
Solute	The substance that is dissolved.
Solvent	The liquid it is dissolved in, usually water.
Concentration	The amount of solute per liter of solution, in mol/L, also called molarity.
Standard solution	A solution whose concentration is known accurately, made from a weighed mass.
Titration	Measuring an unknown concentration by reacting it with a known one.

with $c$ in moles per liter, $n$ in moles and $V$ in liters. American textbooks call this concentration molarity and write it with a capital M: a 0.1 M solution is 0.1 mol/L.

That gives a third way into the same place. A weighed solid gives an amount through its molar mass; a count of particles gives one through the Avogadro constant; a solution gives one through its concentration and volume. All three arrive at moles, and from moles the balanced equation takes over exactly as before.

To make up a solution of a stated concentration and volume: $n = c \times V$ for the amount, then $m = n \times M$ for the mass to weigh. Two multiplications, and both of them are things you can already do.

Volumes come in milliliters and the formula wants liters. A buret reads in milliliters, a volumetric flask is marked in milliliters, and every one of those numbers has to be divided by a thousand before it is multiplied by a concentration. It is the commonest arithmetic error in this lesson, and it is out by exactly a factor of a thousand, which is at least easy to recognize.

Another way: picture

Think of lemonade made from concentrate. The concentration is how strong it tastes, which does not depend on how big the glass is; the amount of concentrate in the glass does. Two glasses of the same strength, one twice the size, hold twice the concentrate.

Another way: steps

To make 250 mL of 0.1 mol/L sodium hydroxide:

Volume in liters: $250 \div 1000 = 0.25$ L.
Amount: $0.1 \times 0.25 = 0.025$ mol.
Molar mass of $\mathrm{NaOH}$ is 40, so mass $= 0.025 \times 40 = 1.0$ g.
Dissolve the gram, then make the solution up to the mark — not add 250 mL of water to it.

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

Convert every volume to liters before it meets a concentration. Divide milliliters by 1000.

To find a concentration: amount in moles divided by volume in liters. If you were given a mass, turn it into moles first.

For a dilution: the amount before equals the amount after, so $c_1 V_1 = c_2 V_2$; here the volumes only need to share a unit.

For a titration: moles from the buret, through the mole ratio, divided by the flask's volume.

Check the work. Three checks catch nearly everything. Is the concentration after a dilution smaller than before? It must be. Is a mass to weigh out sensible for a lab — usually a few grams, not a few thousand or a few thousandths? An answer out by a thousand means a volume stayed in milliliters. And do the units cancel? $\text{mol/L} \times \text{L} = \text{mol}$, and $\text{mol} \div \text{L} = \text{mol/L}$; if they do not, the operation is wrong.

6. Why each step is allowed

Multiplying concentration by volume is allowed because a concentration is a rate: moles per liter. Moles per liter times liters is moles, exactly as miles per hour times hours is miles.

Converting milliliters to liters is required because the concentration is per liter. Multiplying a per-liter rate by a number of milliliters gives an answer a thousand times too big.

Setting the amounts equal in a dilution is allowed because only solvent was added. The solute particles in the flask are the same particles, now spread through more liquid.

Using any matching unit for the two dilution volumes is allowed because the volumes appear as a ratio, and a ratio of two volumes in the same unit has no unit at all.

Using the mole ratio in a titration is required because the reaction consumes the two reagents in the equation's proportion, not one for one in general. Only for a one-to-one equation do the moles match directly.

7. Dilution, and why the formula is obvious

Adding water to a solution changes the volume and does not change the solute. Nothing was taken out of the flask and nothing but water was put in, so the number of moles of solute is exactly what it was.

That is the whole derivation, and it is worth doing once rather than memorizing the result, because the memorized version is easy to apply upside down and the reasoning is not. If the volume went up, the concentration went down, and by the same factor.

The volumes may be in any unit as long as both are in the same unit — milliliters on both sides is fine here, because the factor of a thousand cancels. That is the one place in this lesson where the conversion can be skipped, and it is worth knowing precisely so that it is not skipped anywhere else.

8. Other ways a concentration is written

Moles per liter is the chemist's unit because it plugs straight into a balanced equation. Outside the lab you will meet others, and each one converts to it with arithmetic you already have.

Grams per liter is a mass concentration: divide by the molar mass to get mol/L. Percent by mass, as on a saline bag or a bottle of hydrogen peroxide, is grams of solute per 100 g of solution; for a dilute water solution, 100 g is very close to 100 mL, so 3% hydrogen peroxide is about 30 g per liter.

Parts per million and parts per billion are for very dilute solutions. In water, 1 ppm is about 1 mg per liter. The Environmental Protection Agency sets drinking-water limits this way: the action level for lead is 15 parts per billion, which is 0.015 mg per liter, or about $7 \times 10^{-8}$ mol/L.

None of these is wrong; they suit different readers. A water utility reports ppb because the numbers are easy to compare with the limit. A chemist converts to mol/L when the question is how much reagent will react.

9. What a titration is doing

A titration answers a question a balance cannot: how concentrated is this solution?

A known volume of the unknown goes in a flask. A solution of known concentration — a standard solution — is run in from a buret until the reaction is exactly complete, which an indicator shows. Then:

Every one of those steps is something this course has already done. What the apparatus adds is the ability to measure an amount by reacting it, and that is why titration arithmetic belongs at the end of this unit rather than in a lesson of its own: it is the same calculation with a buret reading on the front.

10. In the world: the IV bag at a hospital bedside

Normal saline is the most common fluid given in American hospitals, with hundreds of millions of bags used every year. It is 0.9% sodium chloride by mass, which means 9 g of salt in each liter. That figure is chosen so the solution has about the same concentration of dissolved particles as blood, and this lesson's arithmetic is what connects the two.

In moles: $9 \div 58.5 = 0.154$ mol of sodium chloride per liter. Each formula unit dissolves into two ions, so the bag carries about 0.308 mol of dissolved particles per liter — close to the roughly 0.29 mol per liter in blood plasma. Close enough that red blood cells neither swell nor shrink when it drips into a vein.

Get the concentration wrong and the consequences are physical. Pure water would flood the cells and burst them; a solution several times too concentrated would draw water out and shrivel them. That is why hospital pharmacies treat dilution calculations with great care. When a nurse dilutes a concentrated drug into a saline bag, $c_1 V_1 = c_2 V_2$ is the safety check, and the commonest error is the one this lesson warns about most: a volume left in milliliters where liters were needed, giving a dose off by a factor of a thousand. Hospitals use double checks and smart pumps precisely because that slip is so easy to make.

11. In the world: testing pool water

A pool test kit is a small titration. Drops of a reagent of known concentration go into a measured water sample until the color changes, and the number of drops, times a fixed factor, gives the chlorine or alkalinity in parts per million.

12. Where this goes wrong

Milliliters used as liters. Every answer out by a thousand comes from here. A buret reads milliliters; the formula wants liters.

Thinking dilution changes the amount of solute. It changes the concentration precisely because it does not change the amount. Holding that straight makes every dilution question a one-line calculation.

Adding the stated volume of water instead of making up to the mark. Dissolving 1 g of solid in 250 mL of water gives rather more than 250 mL of solution, and so a concentration below the one asked for. The solid is dissolved first and the flask is then filled to the line.

Forgetting the mole ratio in a titration. Sulfuric acid neutralizes twice its own amount of sodium hydroxide, and a titration calculation that assumes one to one is out by a factor of two with nothing to show for it.

Confusing grams per liter with moles per liter. A label reading 9 g/L is a mass concentration. Dividing by the molar mass turns it into mol/L; the two numbers are only equal for a substance with a molar mass of 1.

13. Making up a standard solution

14. A dilution

15. A titration with a two-to-one ratio

16. Your turn: 20 mL of 0.5 mol/L sulfuric acid is diluted to 100 mL. What is the new concentration?

17. Guided practice

$24$ mL of potassium manganate(VII) solution is diluted to $96$ mL with water. What is the same after the dilution as it was before?

18. Guided practice

Complete the worked solution: a technician makes two hundred fifty milliliters of sodium hydroxide at one tenth of a mole per liter. Sodium hydroxide is forty grams per mole. Find the volume in liters, the moles needed, and the grams to weigh out.

19. Guided practice

$44$ mL of hydrogen chloride solution at $2$ mol/L is poured into a flask and made up to $176$ mL with water. What is the concentration now, in mol/L?

20. Practice

A technician has to make $1.25$ L of sodium hydroxide, $\mathrm{NaOH}$, at a concentration of $0.5$ mol/L. What mass has to be weighed out? Answer in grams.

21. Practice

Hospitals across the country use normal saline for IV drips. A $5$ L bag holds $45$ g of sodium chloride, molar mass $58.5$ g/mol. What is its concentration in mol/L, to three decimal places?

22. Somewhere new

A buret delivers $40$ mL of hydrochloric acid at $0.4$ mol/L to neutralize $25$ mL of sodium hydroxide solution of unknown concentration. The reaction is $\mathrm{HCl + NaOH \rightarrow NaCl + H_2O}$. What is the concentration of the base, in mol/L?

23. Lesson test

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

24. Test question

Three solutions of potassium hydroxide, $\mathrm{KOH}$, molar mass $56$ g/mol, are to be made up at a concentration of $0.1$ mol/L. For each volume, work out how many moles of solute it contains and what mass has to be weighed out.

25. What you can do now

You can move between concentration, volume, amount and mass, and you know what a dilution leaves alone. Say what mass of sodium hydroxide you would weigh out for 250 mL of a 0.1 mol/L solution, and what happens to the concentration if you then double the volume with water. That is the course: a formula is a count, an equation conserves that count, and the mole is what turns the count into something you can weigh, measure or pour.

Working for the steps left to you

	volume, in L	amount of solute, in mol	mass to weigh out, in g
Flask A	0.75
Flask B	0.75
Flask C	0.25

16. Your turn: 20 mL of 0.5 mol/L sulfuric acid is diluted to 100 mL. What is the new concentration?, step 3

$0.5 \div 5 = 0.1 \text{ mol/L}$