Topic 3: Quantitative chemistry

Quantitative chemistry is the branch of the subject concerned with measuring how much. It lets you predict the mass of product a reaction will make, the amount of a chemical hidden in a solution, and how efficient a process really is. The ideas below build from the simple law that mass is conserved up to the mole calculations needed for the higher tier.

Conservation of mass and balanced equations

In a closed system no atoms are ever created or destroyed during a chemical reaction, so the total mass of the reactants always equals the total mass of the products. This is the law of conservation of mass. Because the atoms are simply rearranged, every balanced symbol equation must have equal numbers of each type of atom on both sides. For example, in the reaction of methane with oxygen, CH4 + 2O2 -> CO2 + 2H2O, count the atoms: left side has 1 carbon, 4 hydrogen and 4 oxygen; right side has 1 carbon, 4 hydrogen and 4 oxygen. The equation balances. You balance an equation by changing the big numbers (the stoichiometric coefficients) in front of formulae, never by altering the small subscript numbers inside a formula, because that would change the substance itself. Sometimes a reaction in an open container appears to change mass. If a gas is given off, such as carbon dioxide when a metal carbonate is heated, the measured mass falls because the gas escapes. If a gas is taken in, such as when a metal reacts with oxygen from the air, the measured mass rises. In both cases mass is still conserved overall once the escaping or entering gas is taken into account.

Relative formula mass

The relative formula mass, written Mr, of a compound is the sum of the relative atomic masses (Ar) of all the atoms shown in its formula. To work it out, multiply each element's Ar by the number of its atoms and add the results together. Take water, H2O: hydrogen has Ar = 1 and there are two atoms, giving 2, while oxygen has Ar = 16, so Mr = 2 + 16 = 18. For carbon dioxide, CO2: carbon is 12 and two oxygens give 32, so Mr = 12 + 32 = 44. With brackets, multiply everything inside by the number after the bracket. Calcium hydroxide, Ca(OH)2, has Mr = 40 + (16 + 1) x 2 = 40 + 34 = 74. A useful check linked to conservation of mass: in a balanced equation, the total Mr of the reactants (each multiplied by its coefficient) equals the total Mr of the products.

The mole and Avogadro (Higher)

Chemists count particles in groups called moles because atoms are far too small and numerous to count one by one. One mole of any substance contains the same number of particles as one mole of any other substance, namely 6.02 x 10^23. This figure is the Avogadro constant. The clever feature is that the mass of one mole of a substance in grams is numerically equal to its relative formula mass. So one mole of carbon (Ar = 12) has a mass of 12 g, and one mole of water (Mr = 18) has a mass of 18 g. The three quantities are linked by the relationship: amount in moles = mass in grams divided by Mr. For example, the number of moles in 88 g of carbon dioxide (Mr = 44) is 88 / 44 = 2 mol. To go the other way, mass = moles x Mr, so 0.5 mol of water has a mass of 0.5 x 18 = 9 g.

Reacting masses

A balanced equation acts as a recipe given in moles, and you can use it to calculate the mass of product formed or reactant needed. The method has three steps: convert the known mass to moles, use the equation's ratio to find the moles of the substance you want, then convert that back to a mass. Consider the thermal decomposition CaCO3 -> CaO + CO2. Suppose you heat 50 g of calcium carbonate (Mr = 100). Moles of CaCO3 = 50 / 100 = 0.5 mol. The ratio of CaCO3 to CaO is 1:1, so 0.5 mol of CaO forms. Calcium oxide has Mr = 56, so the mass of CaO = 0.5 x 56 = 28 g. The leftover 22 g is the carbon dioxide that escaped, which once again demonstrates conservation of mass.

Limiting reactants

When two reactants are mixed, they are rarely present in the exact ratio the equation demands. The reactant that runs out first is called the limiting reactant, and it controls how much product can be made; the other reactant is said to be in excess and some of it is left over. To find which is limiting, convert each reactant to moles and compare against the balanced ratio. For magnesium burning in oxygen, 2Mg + O2 -> 2MgO, imagine you have 0.10 mol of magnesium reacting with plenty of oxygen. The magnesium is limiting, so the moles of MgO are fixed at 0.10 mol by the 2:2 ratio. Adding more magnesium would make more product, but adding more oxygen once magnesium has run out would make no difference at all. The amount of product is always directly proportional to the amount of the limiting reactant.

Concentration of solutions

Concentration measures how much solute is dissolved in a given volume of solution. The simplest unit is grams per cubic decimetre, g/dm3, where one dm3 equals one litre or 1000 cm3. Concentration in g/dm3 = mass of solute in grams divided by volume of solution in dm3. If 20 g of salt is dissolved to make 0.5 dm3 of solution, the concentration is 20 / 0.5 = 40 g/dm3. A more chemically useful unit is moles per cubic decimetre, mol/dm3, found by dividing the g/dm3 value by the Mr of the solute, or directly from concentration = moles divided by volume in dm3. The more solute dissolved in a fixed volume, or the smaller the volume for a fixed mass, the more concentrated the solution. Watch your units: a volume given in cm3 must be divided by 1000 to convert it to dm3 before use.

Percentage yield and atom economy

In practice you almost never collect the maximum mass of product an equation predicts. Some product is lost during filtering or transfer, the reaction may not finish, or side reactions may occur. The percentage yield compares what you actually got with the theoretical maximum: percentage yield = (actual mass of product / theoretical mass of product) x 100. If a reaction should produce 8.0 g but you isolate 6.0 g, the yield is (6.0 / 8.0) x 100 = 75 percent. Atom economy is a different measure of efficiency that asks how much of the reactant mass ends up in the useful product rather than in waste by-products. Percentage atom economy = (Mr of desired product / total Mr of all products) x 100, with each Mr multiplied by its coefficient. A high atom economy reaction is more sustainable because it wastes fewer raw materials and produces less waste to dispose of.

Volumes of gases (Higher)

At room temperature and pressure, often abbreviated rtp (taken as 20 degrees Celsius and 1 atmosphere), one mole of any gas occupies the same volume of 24 dm3, equal to 24000 cm3. This is the molar gas volume, and it follows from the fact that equal numbers of gas molecules occupy equal volumes regardless of the gas. The link is: volume in dm3 = moles x 24. So 2 mol of carbon dioxide occupies 2 x 24 = 48 dm3, while 1.2 dm3 of hydrogen contains 1.2 / 24 = 0.05 mol. Combining this with reacting mass methods lets you predict gas volumes from solid or liquid reactants. For example, decomposing 0.5 mol of calcium carbonate gives 0.5 mol of carbon dioxide, which at rtp occupies 0.5 x 24 = 12 dm3 of gas.