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molarity = moles of solute or moles of solute = molarity × volume (in L) litres of solution
TOPIC 10.
CHEMICAL CALCULATIONS IV - solution stoichiometry.
Calculations involving solutions. Frequently reactions occur between species which are present in solution. One type of chemical analysis called VOLUMETRIC ANALYSIS makes use of the fact that volumes are easier and faster to measure than mass. Volumetric analysis is done using specialised glassware in a process called TITRATION. In order to deduce the amount of a dissolved species (called the SOLUTE) which is present in a given volume of a solution, it is necessary to know the CONCENTRATION of the solution. Concentration is most commonly expressed as how much solute is present per unit volume (e.g. per mL or L) of the solution. The concentration of a solution is therefore independent of the volum e taken and to calculate the amount of solute in any given volume of solution, the concentration must be multiplied by that volume. There are many ways of expressing concentrations, for example % m/v means "the mass of solute in 100 mL of solution". In chemical calculations, by far the most commonly used concentration unit is the number of moles of solute present per litre of solution, and this is termed the MOLARITY of the solution, abbreviated as M.
Thus a 1 molar solution (written as 1 M) of a compound would contain 1 mole of that compound dissolved in solvent so that the total volume of solution is 1 litre, while a 2 molar solution (2 M) would have 2 moles of the compound per litre of solution and a 10 M solution contains 10 moles of compound per litre of solution. As an example, a 1 M solution of sodium chloride, NaCl (1 M), would contain 1 mole of NaCl (= 22.99 + 35.45 g) dissolved in enough water so that the final volume of the solution was 1 litre.
Note: The molarity applies to the solute form ula. Thus, while the amount of sodium chloride, NaCl, dissolved is 1 mole, there are actually 1 mole of Na+^ ions and 1 mole of Cl –^ ions present in that solution. Similarly, a 1 M solution of barium chloride, BaCl 2 , contains 1 mole of the solute per litre of solution which would provide 1 mole of Ba2+^ ions and 2 mole of Cl –^ ions per litre of solution.
The following examples show how molarity, volume and moles are related. If two of these quantities are known, then the third can be deduced.
Example 1. What mass of sodium chloride is present in 500 mL of N aCl (2.00 M) solution?
molarity = moles or moles = molarity × litres litres
i.e. moles of NaCl = 2.00 × 0.500 mol = 1.00 mol
and mass of NaCl = moles × molar mass = 1.00 × (22.99 + 35.45) g = 58.4 g
Example 2. Calculate the concentration in moles per litre of a solution containing 45.2 g of magnesium chloride, MgCl 2 , in a total volume of 800 mL.
The first step is to calculate the molar mass (gram formula weight) of MgCl 2.
Molar mass = (24.31 + 2 × 35.45) = 95.21 g mol–
i.e. 95.21 g of MgCl 2 is exactly 1 mole.
Then calculate how many moles are in 45.2 g of MgCl 2.
Moles of MgCl 2 in 45.2 g = 45.2 (^) = 0.475 mol
Finally, from its definition, calculate the molarity.
Concentration of MgCl 2 = moles / litres = 0.475 = 0.594 M
[Again, note that the concentration always applies to the solute specified, MgCl 2 here, not its component ions if the solute is ionic. Thus in this example, while the concentration of MgCl 2 dissolved is 0.594 M, the solution actually contains Mg2+ ions at a concentration = 0.594 M and Cl–^ ions at a concentration = 2 × 0.594 M = 1.19 M, because there are 1 Mg2+^ and 2 Cl–^ ions in each formula unit of the compound.]
Example 3. What volume of 0.450 M sodium carbonate solution contains 10.0 g of the solute?
Firstly, it is necessary to convert the mass of sodium carbonate to moles.
Moles of Na 2 CO 3 = 10.0 = 0.0943 mol (2 × 22.99 + 12.01 + 3 × 16.00)
As molarity = moles of solute volume of solution in litres
then the volume containing a specified amount of solute is given by the expression
volume in litres = moles of solute = 0.0943 = 0.210 L or 210 mL molarity 0.
The next set of examples shows how stoichiometric calculations can be carried out when solutions are involved.
Example 4. A solution of sodium hydroxide of unknown concentration is titrated against a STANDARD sulfuric acid solution (i.e. one of known concentration). The volume of 0.104 M sulfuric acid needed for complete reaction with 25.00 mL of the sodium hydroxide solution was 20.05 mL. Calculate the concentration of the sodium hydroxide solution.
Test your understanding of this section. Define the terms (i) solute (ii) concentration. What does the concentration unit % v/v mean? What is a 1 molar solution? A 2 molar solution of a solute is prepared. What would be the concentration of that solute in (i) 10 mL and (ii) 100 mL of the solution? Distinguish between moles and molarity. What are the abbreviations used for (i) moles (ii) molarity? In a 1 molar solution of calcium chloride, which ions would be present and at what concentrations? If 500 mL of a 1.00 M solution of a given solute is diluted to a total volume of 1.00 L, what would be the concentration of the final solution?
Alternatively the final step can be completed using the expression
V (litres) = n (moles) / M (molarity)
Substituting, V = 0.1887 / 0.115 = 1.64 L
[In Example 5, note that the known compound, Na 2 CO 3 , was not in solution, so in the calculation the number of moles is simply = mass ÷ molar mass.]
Example 6. A solution of sodium chloride (0.0823 M) is added to 21.40 mL of a solution of silver nitrate (0.962 M) and a precipitate of silver chloride results.
Calculate the minimum volume of the sodium chloride solution required for complete reaction.
NaCl + AgNO 3 v AgCl + NaNO 3
1 mol 1 mol
ˆ equation factor = 1
Moles of AgNO 3 = volume in litres × molarity
= 0.02140 × 0.962 mol = 0.02059 mol
From the equation factor, 1 mole of NaCl requires 1 mole of AgNO 3
ˆ moles of NaCl = 0.02059 mol
1000 mL NaCl solution contains 0.0823 mole NaCl
ˆ 1000 / 0.0823 mL of NaCl solution contains 1 mole NaCl
and (^1000) × 0.02059 mL of NaCl solution contains 0.02059 mole NaCl
= 250 mL
Objectives of this Topic.
When you have completed this Topic, including the tutorial questions, you should have achieved the following goals:
- Understand the terms solute, volumetric analysis; concentration; titration; standard solution.
- Understand the concept of molarity of a solution and be able to use it to
calculate a solution concentration or the amount of solute in a given volume of a solution.
- Be able to use the concept of concentration in stoichiometric calculations.
SUMMARY
Reactions between species in solution (solutes) can be used as a convenient basis for one form of chemical analysis, called volumetric analysis, because volumes are easier and quicker to measure than mass. The amount of solute per unit volume of
solution is called its concentration and therefore concentration is independent of the volume of solution taken. The amount of solute in a given volume of a solution is
the product of the volume taken and the concentration of the solute in the solution. The concentration of a solution can be expressed in a number of ways, but the most
common in chemical analysis is as the number of moles of solute per litre of solution, called the molarity of the solution. Thus the molarity of a given solute in
a solution is the number of moles of that solute divided by the number of litres of the solution. Conversely, the number of moles of the solute in a given volume is its molarity multiplied by the volume of solution.
Using the concept of concentration, stoichiometric calculations can be done by basically the same procedure as used earlier except that the amount of each reacting
solute is obtained from its molarity and volume reacting.
- Hydrochloric acid (23.95 mL) reacts completely with sodium carbonate (0.217g). Calculate the concentration of the hydrochloric acid.
- What volume of sulfuric acid (0.171 M) would be required to react completely with 0.217 g of sodium carbonate?
- Calculate the molarity of a sodium chloride solution, 25.00 mL of which requires 21.40 mL of silver nitrate (0.0962 M) to reach an end point. [Note that this is not an acid/base reaction but a precipitation reaction - see Topic 6.]
- A solution of potassium permanganate containing 79.0 g of solute dissolved in water to give a total volume of 1.00 L is prepared.
(a) Calculate the molarity of the solution.
(b) The solution is then diluted to a final volume of 4.00 L. Calculate the molarity of the new solution.
(c) Calculate the number of MnO 4 –^ ions present in 1.00 mL of the final solution.
- One for the road: The alcohol c ontent o f various a lcoholic beverages is quoted on their labels as % v/v which means the number of mL of pure ethanol per 100 mL of the beverage. The following table lists the ethanol concentration for a range of common alcoholic beverages. Using the relationship density = mass ÷ volume, convert each to % m/v and thence calculate the molar concentration of ethanol for each drink. The density of ethanol at 25^0 C = 0.785 g mL!^1.
Beverage Ethanol concentration Molarity of ethanol as % v/v as % m/v
light beer 3.
full strength beer 5.
wine 13.
overproof rum 75.
vodka 38.
whisky 40.
ANSWERS TO TUTORIAL TOPIC 10
- 0.108 mol 2. 8.77 g
- 0.330 M 4. 3.45 g
- 0.65 L 6. 3.7 mL
- 0.600 M 8. 0.400 M
- 5.94 g 10. 93.8 mL
- 0.333 M 12. 6.53 mL
- 0.171 M 14. 12.0 mL
- 0.0823 M
- (a) 0.500 M (b) 0.125 M (c) 7.53 × 10^19
- Beverage Ethanol concentration Molarity of ethanol as % v/v as % m/v light beer 3.50 2.75 0.
full strength beer 5.00 3.93 0.
wine 13.5 10.6 2.
overproof rum 75.0 58.9 12.
vodka 38.0 29.8 6.
whisky 40.0 31.4 6.
WORKED SOLUTIONS
- Amount of solute = volume of solution × concentration If amount is expressed in moles and concentration in moles/litre (mol L–1^ or M), then moles of solute = molarity × volume in litres = 0.864 × 0.125 mol = 0.108 mol
- First the number of moles of sodium chloride must be calculated from the volume and concentration of the solution and then the mass can be deduced. Volume = 1.50 × 10^3 mL = 1.50 L Concentration = 0.100 M Amount (mol) = molarity (M) × volume (L) = 0.100 × 1.50 = 0.150 mol Molar mass of NaCl = 58.44 g mol–
ˆ mass = moles × molar mass = 0.150 × 58.44 = 8.77 g
- Hydrochloric acid is a water solution of hydrogen chloride, HCl. There are two methods by which this problem can be solved. First method : Calculate the moles of HCl in the initial solution and then using the final volume, deduce the concentration of the diluted solution. Moles of HCl in 100 mL of 0.200 M solution = molarity × volume = 12.0 × 0.100 = 1.20 mol Diluted volume = 2000 mL = 2.000 L
ˆ concentration of hydrochloric acid in the diluted solution = moles / volume
= 1.20 / 2.000 = 0.600 M
Second method : By proportion - it depends on the fact that the number of moles of HCl is constant in both the original and diluted solutions. Let the initial solution volume and molarity be represented as V 1 and M 1 respectively. Then moles of HCl present = V 1 × M 1 and this is the same number of moles as are in the diluted solution. Let the volume and molarity of the diluted solution be represented as V 2 and M 2 respectively. The number of moles of HCl in this solution is given by V 2 × M 2 , identical to the number initially present given by V 1 × M 1. Thus V 1 × M 1 = V 2 × M 2. Substituting the data, 0.100 × 12.0 = 2.000 × M 2 where M 2 is the required molarity.
ˆ M 2 = (0.100 × 12.0) / 2.000 = 0.600 M.
Note: This expression can only be applied to dilution calculations - it is not appropriate to use it in titration problems. Also note that when using this equation, the volumes do not need to be expressed in litres as long as the same units (e.g. mL) are used on both sides as the units cancel.
- Molarity = moles / volume (L) = 0.200 / 0.500 = 0.400 M
- This problem is essentially the same as those solved in Topic 9 except that the amount of potassium chromate reacting (and thus the moles of barium chromate produced) must be calculated from the volume and concentration (molarity) of the potassium chromate solution.
Balanced formula equation: BaCl 2 + K 2 CrO 4 v BaCrO 4 + 2KCl
Mole ratios: 1 mol 1 mol (Note that the amounts of the other reactant and product are not relevant to this calculation.) 1 mole of the known (K 2 CrO 4 ) produces 1 mole of the unknown (BaCrO 4 ).
ˆ equation factor = 1
Moles of known: n = V × M = 0.050 × 0.469 = 0.02345 mol Moles of unknown: As the equation factor = 1, moles of BaCrO 4 produced = 0.02345 mol Molar mass of BaCrO 4 = 253.3 g mol–
ˆ mass of barium chromate = moles × molar mass of BaCrO 4
= 0.02345 × 253.3 = 5.94 g
10. Formula equation: Al 2 (SO 4 ) 3 + 3Ba(NO 3 ) 2 v 3BaSO 4 + 2Al(NO 3 ) 3
Mole ratio of reactants: 1 mol 3 mol
ˆ Equation factor = 3
As both the concentration and volume of the aluminium sulfate are given, it is the known or standard solution and the volume of the barium nitrate solution of known concentration is the unknown quantity. Moles of aluminium sulfate in 25.0 mL of solution = M × V = 0.350 × 25.0 × 10– = 8.750 × 10–3^ mol As the equation factor = 3, then moles of barium nitrate required = 3 × 8.750 × 10–3^ mol = 2.625 × 10–2^ mol Concentration of barium nitrate = 0.280 M moles of barium nitrate, n = V × M = V × 0.280 = 2.625 × 10–2^ mol
ˆ V = (2.625 × 10–2^ ) / 0.280 = 9.38 × 10–2^ L or 93.8 mL
11. Formula equation: NaOH + HCl v NaCl + H 2 O
Mole ratio of reactants: 1 mol 1 mol
ˆ Equation factor = 1
As both the concentration and volume of the hydrochloric acid are given, it is the known or standard solution and the concentration of the sodium hydroxide is the unknown. Moles of HCl in 18.4 mL of solution = M × V = 0.452 × 18.4 × 10– = 8.317 × 10–3^ mol As the equation factor = 1, then moles of NaOH in 25.0 mL = 8.317 × 10–3^ mol Concentration of sodium hydroxide = moles / volume = 8.317 × 10–3^ / 25.0 × 10–3^ = 0.333 M
12. Formula equation: 2NaOH + H 2 SO 4 v Na 2 SO 4 + 2H 2 O
Mole ratio of reactants: 2 mol 1 mol The known (standard) reactant is the sodium hydroxide and as 1 mole of it requires 0.5 mole of sulfuric acid for complete reaction, the equation factor = 0.5. Moles of NaOH in 20.0 mL of solution = M × V = 0.493 × 20.0 × 10–3^ = 9.86 × 10–3^ mol The equation factor = 0.5, so moles of H 2 SO 4 required = 0.5 × 9.86 × 10– = 4.93 × 10–3^ mol In this problem, the molarity of the unknown sulfuric acid is given and the volume required is to be calculated. n = M × V 4.93 × 10–3^ = 0.755 × V
ˆ V = 4.93 × 10–3^ / 0.755 = 6.53 × 10–3^ L or 6.53 mL
Alternatively, using the proportion method in the last step: 1.000 L of sulfuric acid contains 0.755 mol
ˆ (1.000 / 0.755) × 4.93 × 10–3^ L contains 4.93 × 10–3^ mol
i.e. 6.53 × 10–3^ L or 6.53 mL of sulfuric acid solution are required.
- (a) Moles of KMnO 4 = mass / molar mass = 79.0 / 158. = 0.500 mol Volume of solution = 1.00 L
ˆ concentration of solution = moles / volume
= 0.500 M
(b) Moles of KMnO 4 = 0.500 mol. Diluted volume = 4.00 L
ˆ concentration of KMnO 4 in the diluted solution = moles / volume
= 0.125 M
Alternatively, using the expression V 1 × M 1 = V 2 × M 2 which can be only be applied to dilution calculations, 1.00 × 0.500 = 4.00 × M 2
ˆ M 2 = (1.00 × 0.500) / 4.
= 0.125 M
(c) Moles of KMnO 4 in 1.00 mL of 0.125 M solution = concentration × volume
= 0.125 × 1.00 × 10– = 1.25 × 10–4^ mol
The formula for potassium permanganate contains one MnO 4 –^ so moles of MnO 4 –
ions in 1.00 mL of solution is also 1.25 × 10–4^ mol.
As 1 mole of any species contains NA of that species, then 1.25 × 10–4^ mol of
MnO 4 –^ ions contains 1.25 × 10–4^ × NA MnO 4 –^ ions
= 1.25 × 10–4^ × 6.022 × 10^23 = 7.53 × 10^19 MnO 4 –^ ions.
- Using the first example, light beer, 100 mL of beer contains 3.50 mL of
ethanol. To convert this to mass of ethanol in 100 mL, use the relation
density = mass/ volume i.e. mass = density × volume. mass of ethanol in 100 mL = 0.785 × 3.50 = 2.75 g.
ˆ % m/v = 2.75 g mL!^1.
Mass of ethanol in 1000 mL = 2.75 × 10 = 27.5 g L!^1. Molarity of ethanol of formula C 2 H 5 OH = mass / molar mass = 27.5 / 46.07 = 0.60 M.
