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The amount of a substance is a measure of the number of particles of a substance, be they individual atoms, as with the noble gases, diatomic molecules, such as with oxygen and the halogens, or more complex molecules.
Since atoms are so small and weigh so little even 1 g of a compound will contain an uncountable number of particles. To rationalize the massive number particles involved the unit mole (shortened to mol) is used, where 1 mol = 6.02x10^{23} particles.
This quantity is known as Avogadro's number (symbol A) and it enables massive numbers of particles to be rendered simply.
Whilst 1 mol of any substance always contains the same number of particles, it's mass depends on the type of atoms present.
As discussed in the Introductory Unit (see the atom) the number of protons added to the number of neutrons is known as the relative atomic mass. This is the mass of 1 mol of an atom relative to the mass of 1 mol of C atoms that have 6 protons and 6 neutrons, which is taken to be 12.00 g.
There is a mathematical relationship between the mass of one mole of an atom, the actual mass of the atom present and the amount of the atom present,
or it can be expressed in a triangular relationship -
To calculate the mass of one mole of a compound, the number of each type of atom in the compound is multiplied by that atoms relative atomic mass and all those numbers added together.
This value is called the relative formula mass (or relative molecular mass or molar mass) of a compound.
(i) molar mass of NaCl | =23 g mol^{-1} + 35.5 g mol^{-1} |
= 58.5 g mol^{-1} | |
(ii) molar mass of Al_{2}(SO_{4})_{3} | =(2x27 g mol^{-1}) + (3x32 g mol^{-1}) + (12x16 g mol^{-1}) |
= 342 g mol^{-1} |
The key mathematical equation needed here is -
mass (g) = relative formula mass (g mol^{-1}) x amount (mol)
Using the triangular relationship from above if the mass section is covered over then the amount multiplied by the relative formula mass gives the mass -
(i) What is the mass of 0.25 mol of NaCl?
0.25 mol of NaCl | = 58.5 g mol^{-1} x 0.25 mol |
= 14.63 g |
(ii) What is the mass of 3 mol of Al_{2}(SO_{4})_{3}?
3 mol of Al_{2}(SO_{4})_{3} | = 342 g mol^{-1} x 3 mol |
= 1026 g or 1.026 kg |
The key mathematical equation needed here is -
amount (mol) = mass (g) / relative formula mass (g mol^{-1})
Using the triangular relationship from above if the amount section is covered over then the mass divided by the relative formula mass gives the amount -
(i) What amount is 117 g of NaCl?
117 g of NaCl | = 117 g / 58.5 g mol^{-1} |
= 2 mol |
(ii) What amount is 68.4 g of Al_{2}(SO_{4})_{3}?
68.4 g of Al_{2}(SO_{4})_{3} | = 68.4 g / 342 g mol^{-1} |
= 0.2 mol |
The key mathematical equation needed here is -
relative formula mass (g mol^{-1}) = mass (g) / amount (mol)
Using the triangular relationship from above if the molar mass section is covered over then the mass divided by the amount gives the relative formula mass -
What is the molar mass of a compound for which 0.2 mol of it has a mass of 42 g?
molar mass of compound | = 42 g / 0.2 mol |
= 210 g mol^{-1} |
Experimental sheet for balancing a chemical equation.
The importance of balancing the numbers of atoms present in a chemical equation was first mentioned in the introductory unit.
These balancing numbers have an additional meaning where these amount calculations are concerned. The balancing numbers give the numbers of moles present for each chemical involved in the reaction.
For example, sodium chloride may be prepared by the reaction of sodium hydroxide_{(aq)} and hydrochloric acid, according to the following equation -
NaOH_{(aq)} + HCl_{(aq)} NaCl_{(aq)} + H_{2}O_{(l)}
In Lower School, all this equation showed was that sodium hydroxide reacts with hydrochloric acid to give a salt (sodium chloride) and water.
However, the equation says more than just this. The equation specifies what amounts of sodium hydroxide and hydrochloric acid will react together to give the products -
1NaOH_{(aq)} + 1HCl_{(aq)} 1NaCl_{(aq)} + 1H_{2}O_{(l)}
This equation states that 1 mol of sodium hydroxide and 1 mol of hydrochloric acid will react together to give of 1 mol sodium chloride and of 1 mol water.
The ratio of NaOH : HCl : NaCl : H_{2}O is 1 : 1 : 1 : 1.
What this means is that if the amount of any one of the components in the above reaction is known then the others may be worked out from the ratio above.
Or, if the masses of chemicals reacting together are known then amounts can be calculated and the balancing numbers deduced from the amounts reacting together.
(i) What amount of sodium chloride will be produced from 0.25 mol of sodium hydroxide?
Ratio of NaOH : NaCl in the chemical equation is 1 : 1
amount of NaCl produced = 0.25 mol
(ii) What amount of hydrochloric acid is needed to make 0.5 mol if sodium chloride?
Ratio of NaCl : HCl in the chemical equation is 1 : 1
amount of HCl needed = 0.5 mol
This is all well and good; however, measuring the amount of a compound present isn't possible directly. The mass of a compound is easy to measure though. Then the mass can be converted into an amount of a compound and the mathematical process as described above can be followed.
e.g. What amount of sodium chloride will be produced from 20 g of sodium hydroxide?
molar mass of sodium hydroxide | = (23+16+1) g mol^{-1} |
= 40 g mol^{-1} | |
amount of sodium hydroxide | = 20 g / 40 g mol^{-1} |
= 0.5 mol |
Ratio of NaOH : NaCl in the chemical equation is 1 : 1
∴ amount of NaCl produced = 0.5 mol
It is then a simple matter to extend the question and ask for the mass of a compound formed/used and not its mass.
e.g. What mass of hydrochloric acid is need to give 19.5 g of sodium chloride?
molar mass of sodium chloride | = (23+35.5) g mol^{-1} |
= 58.5 g mol^{-1} | |
∴ amount of sodium chloride | = 19.5 g / 58.5 g mol^{-1} |
= 0.33 mol |
Ratio of NaCl : HCl in the chemical equation is 1 : 1
∴ amount of HCl needed | = 0.33 mol |
mass of HCl needed | = 0.33 mol x (1+35.5) g mol^{-1} |
= 12.16 g |
This is rather a simplistic example; however, it shows the general use of amounts, molar masses and chemical equations.
The calculations get harder when the chemical equation involves more complex balancing numbers.
e.g. What mass of sodium sulphate would be formed when 5 g of sodium hydroxide is reacted with excess sulphuric acid?
2NaOH_{(aq)} + H_{2}SO_{4(aq)} Na_{2}SO_{4(aq)} + 2H_{2}O_{(l)} |
Step 2 : calculate the amount of sodium hydroxide used.
molar mass of sodium hydroxide | = (23+1+16) g mol^{-1} |
= 40 g mol^{-1} | |
∴ amount of sodium hydroxide | = 5 g / 40 g mol^{-1} |
= 0.125 mol |
Step 3 : use the ratio from the chemical equation
Ratio of NaOH : Na_{2}SO_{4} in the chemical equation is 2 : 1
∴ amount of Na_{2}SO_{4} produced | = 0.125 mol / 2 |
= 0.0625 mol |
Step 4 : calculate the mass of product
mass of Na_{2}SO_{4} produced | = 0.0625 mol x ((2x23)+32+(4x16)) g mol^{-1} |
= 8.875 g |
This is the standard calculation used when you have, or may easily work out, a chemical equation for a reaction.
Experimental sheet for determining the molar volume of a gas.
Experiments discovered that gases react together in volume ratios. That is to say that 1 mol of any gas occupies a particular volume at a particular temperature.
From physics, it is known that the volume of a gas is proportional to its temperature, according to the mathematical equation -
so the higher the temperature the greater the volume a gas occupies, keeping the pressure constant, at atmospheric pressure say.
Temperature | Volume (dm^{3}) |
0 ^{o}C or 32 ^{o}F or 273 K | 22.4 |
20 ^{o}C or 68 ^{o}F or 293 K | 24.1 |
25 ^{o}C or 77 ^{o}F or 298 K | 24.5 |
The value normally used in calculations in examinations is 24 dm^{3} as this is closest to a value for room temperature.
This relationship between gas volume and amount forms another mathematical relationship -
or it can be expressed in a triangular relationship -
Using a similar method to that for the amount/mass/relative formula mass triangle, amounts of gas can be converted into volumes and volumes of gas converted into amounts
There are two uses of this type of calculation,
(i) The calculation of the percentage of an element in a compound -
e.g. What is the percentage of oxygen in carbon dioxide gas?
mass of oxygen in one mole of carbon dioxide gas | = 2x16 g |
= 32 g | |
mass of one mole of carbon dioxide | = 12 g + (2x16 g) |
= 44 g | |
∴ percentage of oxygen | = (32 g/44 g) x 100 |
= 72.7 % |
(ii) The calculation of the percentage of water of crystallization in a compound -
e.g. What is the percentage of water of crystallization in hydrated copper(II) sulphate, CuSO_{4}.5H_{2}O?
mass of water present in one mole of hydrated copper(II) sulphate | =(5x18 g) |
= 90 g | |
mass of one mole of hydrated copper(II) sulphate | = 64 g + 32 g + (4x16 g) + (5x18 g) |
= 250 g | |
∴ percentage of water of crystallization present | = (90 g/250 g) x 100 |
= 36 % |
This technique covers essentially the reverse of the above process. A typical question here involves the calculation of the lowest possible whole number ratio of atoms in an organic molecule, given the percentage elemental composition for that molecule.
e.g. An organic compound contains carbon and hydrogen only in the ratio of 85.7 % carbon to 14.3 % hydrogen. What is the empirical formula for that compound?
C | H | |
% composition | 85.7 % | 14.3 % |
mass of element in 100 g of compound | 85.7 g | 14.3 g |
∴ amount of element in 100 g of compound | ^{85.7 g}/_{12 g mol}-1 = 7.14 mol | ^{14.3 g}/_{1 g mol}-1 = 14.3 mol |
∴ lowest whole number ratio of element's amounts | ^{7.14 mol}/_{7.14 mol} = 1 | ^{14.3 mol}/_{7.14 mol} = 2 |
The empirical formula for the compound is CH_{2}.
(3) Molecular formula of a compound :The molecular formula for a compound is simply the empirical formulae multiplied by some whole number ( 1, 2, 3, etc.. ).
Knowing the molar mass of the compound enables the molecular formula of it to be found.
e.g. The actual molar mass of the compound in the previous example is 42 g mol^{-1}. What is the molecular formula for this compound?
The molar mass of CH_{2} | = (12+(2x1)) g mol^{-1}= 14 g mol^{-1} |
The molecular formula of this compound is (CH_{2})_{n}, where n is a positive whole number.
∴ The value of n | = 42 g mol^{-1} / 14 g mol^{-1} |
= 3 |
the molecular formula is (CH_{2})_{3} or more properly C_{3}H_{6}.
(4) Percentage yield for a reaction :e.g. In the reaction between iron metal and an excess of copper(II) sulphate solution, 0.56g of iron is reacted and 0.48 g of copper metal is formed. What is the yield for this reaction?
the equation - Fe_{(s)} + CuSO_{4(aq)} Cu_{(s)} + FeSO_{4(aq)}
the amount of iron used | = 0.56 g/ 56 g mol^{-1} |
= 0.01 mol |
From the equation, the maximum amount of copper metal that could be formed = the amount of iron used, as the ratio of balancing numbers is 1 : 1.
∴ the maximum possible amount of copper metal | = 0.01 mol |
∴ the maximum possible mass of copper metal formed | = 0.01 mol x 64 g mol^{-1} |
= 0.64 g | |
∴ the percentage yield of copper | =(mass produced / maximum possible mass) x 100 |
=(0.48 g/ 0.64 g) x 100 | |
= 75 % |
e.g. In the reaction between aluminium metal and an excess of copper(II) sulphate solution, 4.0 g of aluminium is reacted and 0.48 g of copper metal is formed. Assuming that all the aluminium present reacted to give copper, what is the purity of the aluminium?
2Al_{(s)} + 3CuSO_{4(aq)} 3Cu_{(s)} + Al_{2}(SO_{4})_{3(aq)}
the amount of copper made | = 4.8 g/ 64 g mol^{-1} |
= 0.075 mol |
From the equation, the maximum amount of aluminium metal that could be present = ^{2}/_{3} x the amount of copper made, as the ratio of Al : Cu is 2 : 3.
∴ the maximum possible amount of aluminium metal | =^{2}/_{3} x 0.075 mol |
=0.05 mol | |
∴ the maximum possible mass of aluminium metal present | = 0.05 mol x 27 g mol^{-1} |
=1.35 g | |
∴ the percentage purity of aluminium used | =(mass present / total mass of compound)x100 |
= (1.35 g / 4.0 g) x 100 | |
= 33.75 % |
(1) 3.5 g of lithium is heated and reacted with an excess of oxygen to make lithium oxide. What mass of lithium oxide is produced?
Take the atomic mass of lithium to be 7 g mol^{-1} and oxygen to be 16 gmol^{-1}
First balance the equation for the reaction -
Then fill in the boxes below with your answers and press check to see if you are correct -
(2) 18 g of pentane is combusted in excess oxygen to make carbon dioxide and water only. What is the minimum volume of oxygen needed?
Take the atomic mass of carbon to be 12 g mol^{-1} and hydrogen to be 1 gmol^{-1}, and the molar volume of a gas to be 24 dm^{3} mol^{-1}
First balance the equation for the reaction -
Then fill in the boxes below with your answers and press check to see if you are correct -
(3) When sodium nitrate is heated strongly it decomposes to give sodium nitrite and oxygen gas. If 60 cm^{3} of oxygen is formed from a sample of sodium nitrate, what mass of sodium nitrate must have been used?
Take the atomic mass of sodium to be 23 g mol^{-1}, nitrogen to be 14 g mol^{-1} and oxygen to be 16 gmol^{-1}, and the molar volume of a gas to be 24 dm^{3} mol^{-1}
First balance the equation for the reaction -
Then fill in the boxes below with your answers and press check to see if you are correct -
written by Dr Richard Clarkson : © Saturday, 1 November 1997
Updated : Friday, 16^{th} February, 2012
mail to: chemistryrules
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