Most calculation mistakes in chemistry are not arithmetic mistakes. They are unit mistakes wearing a disguise — and by far the commonest is forgetting that concentration is measured per cubic decimetre while volumes are measured in cubic centimetres.
📚 What you need to know
There are seven SI base units; everything else in chemistry is derived from them.
A prefix is a fixed power of ten, so converting between prefixed units is always a shift of the decimal point.
1 dm3 = 1000 cm3, and 1 dm3 is the same as 1 litre.
The same letter can mean several things — context and capitalisation decide which.
All non-zero digits are significant, and so are zeros between them.
Leading zeros are never significant; trailing zeros count only if there is a decimal point.
A calculated answer takes the fewest significant figures of the measured values used to get it.
The SI base units
Quantity
SI base unit
Symbol
length
metre
m
mass
kilogram
kg
time
second
s
temperature
kelvin
K
amount of substance
mole
mol
electric current
ampere
A
luminous intensity
candela
cd
Every other unit you meet is built from these. Energy in joules is a newton metre; pressure in pascals is kg m–1 s–2; concentration is mol dm–3; molar mass is g mol–1; entropy is J K–1; enthalpy change is kJ mol–1; potential difference is volts.
Note the odd one out: the base unit of mass is the kilogram, not the gram — the only base unit that already carries a prefix. Chemistry ignores this in practice and works in grams, which is fine as long as you convert when a formula demands SI units.
Prefixes are just powers of ten
Watch the capital letters. M is mega and m is milli — nine orders of magnitude apart, distinguished only by the case of a single character.
The volume conversion that costs the most marks
Concentration is quoted in mol dm–3, but nobody measures volume in cubic decimetres in a school laboratory. Pipettes and burettes read in cubic centimetres, so almost every mole calculation begins with a division by 1000.
Cubic centimetres and millilitres are identical, so no conversion is ever needed between those two. It is the step up to dm3 that catches people.
Symbols: the same letter, different jobs
The data booklet supplies chemical symbols, physical constants and the equations, but it will not tell you which meaning is intended in a given question. The letter c alone can be any of the following, and capitalisation is often the only clue.
Written as
Means
Where you see it
c
concentration
n = cV
c
specific heat capacity
Q = mcΔT
c
the speed of light
c = fλ
c
the prefix centi-
cm3
C
the coulomb
units of electrical charge
C
the element carbon
chemical formulae
State symbols are part of an equation, not decoration: (s), (l), (g) and (aq). Leaving them out of an equation that has been asked for with states will cost the mark, and (l) and (aq) are not interchangeable — one means a pure liquid, the other means dissolved in water.
Significant figures
Significant figures are the digits that carry real information about how precisely something was measured. The rules are short, and only the zeros are ever difficult.
The trailing zero is the informative one. Writing 0.00407 would claim three significant figures; writing 0.004070 claims four, and says the measurement was made to that precision.
🧩 Rounding to a given number of significant figures
Find the first significant figure — the first non-zero digit.
Count forward to the number of figures asked for.
Look at the next digit along; that is your rounder.
If it is 5 or more, increase the previous digit by 1; otherwise leave it.
In standard form, round only a. The power of ten never changes.
Carry the full calculator value all the way through and round only at the very end. Rounding in the middle introduces errors that grow with every step, and rounding an intermediate value to one significant figure is close to guaranteeing a wrong final answer.
One exception is worth knowing. Constants such as the Avogadro constant are not limited by any measurement you made — their precision is fixed by definition. Use the number of figures given in the data booklet: 6.02 × 1023 mol–1 has three.
WORKED EXAMPLE
Carry out the following conversions: (a) 25.0 cm3 into dm3; (b) 0.0450 dm3 into cm3; (c) 184 pm into metres, in standard form.
(a) cm³ to dm³Going up the staircase, so divide.25.0 ÷ 1000 = 0.0250 dm³(b) dm³ to cm³Going down, so multiply.0.0450 × 1000 = 45.0 cm³(c) picometres to metrespico- means 10⁻¹², so multiply by that, then adjust so the front number sits between 1 and 10.184 × 10⁻¹² = 1.84 × 10⁻¹⁰ mKeep the trailing zeros in (a) and (b). They are significant and they show the precision you started with.
WORKED EXAMPLE
State the number of significant figures in each value: 4107, 0.00079, 57 000, 689.0023 and 100.0.
4107The zero sits between non-zero digits, so it counts.4 significant figures0.00079Leading zeros only position the decimal point.2 significant figures57 000Trailing zeros with no decimal point are not significant.2 significant figures689.0023Every digit after the first non-zero one counts here.7 significant figures100.0The decimal point makes the trailing zeros significant.4 significant figures
WORKED EXAMPLE
A titration delivers 24.50 cm3 of a 0.0500 mol dm–3 solution. Calculate the amount in moles, giving your answer to an appropriate number of significant figures.
Step 1 — convert the volume24.50 ÷ 1000 = 0.02450 dm³Step 2 — use n = cVn = 0.0500 × 0.02450 = 1.225 × 10⁻³Step 3 — decide the precisionThe volume has 4 significant figures and the concentration has 3, so the answer takes the smaller of the two.1.23 × 10⁻³ mol“An appropriate number” is not a hint to guess. It means: count the significant figures in the data and use the fewest.
💡 Exam tip
Convert cm3 to dm3 first, before anything else in a mole calculation.
Write the unit on every answer; a bare number is not a chemistry answer.
Read the capitalisation of a symbol carefully — M and m, C and c are different things.
Give the number of significant figures the question asks for, and 3 if it is left open.
Round only at the end; keep the full value in the calculator.
Include state symbols whenever an equation is asked for with them.
⚠️ Common mix-up
Multiplying by 1000 instead of dividing when converting cm3 to dm3.
Counting leading zeros as significant figures.
Dropping trailing zeros such as the one in 0.0250, which throws away precision.
Rounding partway through a multi-step calculation.
Confusing the prefix M with m, a factor of 109.
Up next: Working with Uncertainties — significant figures are a rough statement of how precisely you measured. An uncertainty is the exact version of the same statement, and it has rules of its own for what happens when you start calculating with it.
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