CHM110H5 Chap.1 Measurement, Units and Significant Figures
Measurement, Units and Significant Figures
Why measurement opens the course
CHM110H5 begins with a block of background skills that every later topic leans on: SI and common non-SI units (grams, metres, litres, moles, atmospheres, bar and mmHg), converting between them, dimensional analysis, uncertainty, significant figures, and the difference between accuracy and precision.
None of this is glamorous, yet a lost factor of 1000 or a badly rounded answer costs marks on quizzes, midterms and lab reports alike. Treat this chapter as the toolkit you will reach for in solutions, gases, equilibrium and calorimetry.
Significant figures and what they promise
Significant figures are the digits that were actually measured.
The lecture states the core idea plainly: a calculation can never be more certain than its least accurate input. You need four counting rules: non-zero digits count, zeros trapped between significant digits count, leading zeros never count, and trailing zeros count only when a decimal point is written.
For arithmetic, sums and differences are rounded to the least precise decimal place, while products and quotients keep the smallest number of significant figures found in any factor. Exact numbers, such as defined conversions or the coefficients in a balanced equation, never limit the result.
Uncertainty in the lab
The lab manual's error analysis appendix makes these ideas concrete.
A burette graduated every 0.1 mL can be interpolated to about plus or minus 0.02 mL, and a reading is usually trusted to half of the smallest scale division. Each piece of glassware also has a printed tolerance, for example plus or minus 0.05 mL for a 50 mL burette and plus or minus 0.0001 g for an analytical balance.
When quantities are added or subtracted, absolute uncertainties combine in quadrature; when they are multiplied or divided, relative uncertainties combine instead. Precision describes how well repeat results agree with each other, while error is the distance between your result and the accepted value.
A precise set of titrations can still be wrong if a systematic fault, such as a dirty pipet, shifts every run the same way.
Dimensional analysis as a habit
Dimensional analysis means writing every conversion as a ratio equal to one and multiplying until the unwanted units cancel.
The lecture's unit map ties mass, moles, volume, density and concentration together through n = m/M, d = m/V and c = n/V, so almost every stoichiometry question becomes a chain of these factors. Carry units on every line, keep extra digits in your calculator, and round only once at the end.
The lecture even shows a salt mixture problem where two valid methods give 42% and 38% NaCl simply because two-figure data were rounded along the way. Practise reading each answer back with its units and asking whether the size of the number makes physical sense before you move on.
What this chapter covers
- 01
SI and non-SI units
- 02
Dimensional analysis chains
- 03
Counting significant figures
- 04
Rounding sums versus products
- 05
Exact numbers
- 06
Accuracy versus precision
- 07
Propagating lab uncertainty
Worked example · free
Concentration from a weighed sample, with correct figures
- 2Moles of NaCl: n = 5.846 g x (1 mol / 58.44 g) = 0.1000 mol (both values have four significant figures, so keep four).
- 2Volume in litres: 500.0 mL x (1 L / 1000 mL) = 0.5000 L; the 1000 is an exact defined factor. Concentration c = 0.1000 mol / 0.5000 L = 0.2000 mol/L.
- 1Mass in the 10.00 mL portion: n = 0.2000 mol/L x 0.01000 L = 2.000 x 10^-3 mol; m = 2.000 x 10^-3 mol x 58.44 g/mol = 0.1169 g (0.11688 rounded to four figures).
- 1Unit change: 0.2000 mol/L x (1000 mmol / 1 mol) x (1 L / 1000 mL) = 0.2000 mmol/mL; the factors are exact, so four figures remain.
Key terms
- Significant figures
- The digits in a measured or calculated value that were actually measured, including the final estimated digit; they signal how certain the number is.
- Dimensional analysis
- A problem solving method that multiplies a quantity by conversion factors written as ratios equal to one, so unwanted units cancel and the target unit remains.
- Precision
- The internal agreement among repeated measurements made the same way; a tight cluster of results is precise even if it sits far from the true value.
- Accuracy
- How close a measured or calculated result lies to the accepted or true value; poor accuracy usually points to a systematic error in the method or equipment.
- Exact number
- A counted quantity or a defined relationship, such as 1000 mL in 1 L or a stoichiometric coefficient, carrying unlimited significant figures and zero uncertainty.
- Relative uncertainty
- The absolute uncertainty of a quantity divided by the quantity itself; relative uncertainties are combined when measured values are multiplied or divided.
Measurement, Units and Significant Figures FAQ
Why do I round to decimal places when adding but to significant figures when multiplying?
Addition combines absolute uncertainties, so the measurement with the coarsest decimal place controls how far the sum can be trusted. Multiplication scales relative uncertainty, so the factor with the fewest significant figures dominates the error in the product or quotient.
Does a number like 1000 mL per litre limit my significant figures?
No. Defined conversions and counted quantities are exact and carry unlimited figures, so they never reduce precision. Only measured values, such as a balance mass or a burette volume, determine how many digits you report.
Should I round after every step of a long calculation?
Keep at least one or two extra digits through intermediate steps and round only the final reported value. Rounding early introduces drift, and the course shows two correct methods giving visibly different percentages for exactly this reason.
Exam move
Memorise the four counting rules and test yourself on ten awkward numbers such as 0.00350, 210 and 210. until you never hesitate. Redo every unit conversion in the lecture by writing each factor as a fraction and striking out cancelled units. For each lab, list the glassware you used with its tolerance from the error analysis appendix, then propagate uncertainty through one full calculation.
Finally, solve one stoichiometry problem twice, rounding early and rounding late, and compare the answers.
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