Difficulty: Introductory | Prerequisites: Basic algebra, comfort with scientific notation.
Tags: unit conversion, metric system, significant figures, sig figs, percent error, density, graduated cylinder, meniscus, lab equipment, general chemistry, Purdue CHM 11100
This material forms the quantitative backbone of everything else in general chemistry. Before you can talk sensibly about atoms, reactions or energy, you need to measure things, convert between units, and report numbers with the right precision. Most of these skills come back in every single lab report and exam for the rest of the course. If you skipped orientation week or missed the first few lectures, start here.
Learn how to convert between metric units using powers of ten, count significant figures correctly (leading zeros never count), calculate percent error, and use the density formula. Know how to read a graduated cylinder at the meniscus and identify common lab glassware.
Significant figures (sig figs)
The digits in a measurement that carry meaning, including all certain digits plus one estimated digit. Leading zeros are never significant. Trailing zeros after a decimal point are significant. In simple terms, sig figs tell you how precise a measurement is.
Percent error (% error)
A measure of how far an experimental value is from the accepted (theoretical) value, expressed as a percentage. The formula uses the absolute value of the difference divided by the theoretical value, multiplied by 100%. Think of it as: "how wrong was I, relative to what the answer should have been?"
Density
Mass per unit volume, typically in g/mL or g/cm³. 1 cm³ = 1 mL exactly. Think of it as how much stuff is packed into a given space.
Meniscus
The curved surface of a liquid in a container, caused by surface tension. For water in glass, the meniscus curves downward, and you read the volume at the bottom of the curve.
Scientific notation
A way of writing very large or very small numbers as a coefficient (between 1 and 10) multiplied by a power of ten. For example, 3,500 = 3.5 × 10³.
Metric prefixes
Standardised multipliers attached to base units. The ones you need here: kilo (10³), centi (10⁻²), milli (10⁻³), nano (10⁻⁹). These come up constantly in conversions.
1 metre = 100 centimetres = 1,000 millimetres
1 kilometre = 1,000 metres
1 metre = 10⁹ nanometres
To convert metres to centimetres, multiply by 100 (move the decimal two places right)
Example: 3.5 m × 100 = 350 cm = 3.5 × 10² cm
When comparing distances in different units, convert everything to the same unit first
Example: 100,000 cm = 1,000 m; 100 m = 100 m; 100,000,000,000 nm = 100 m; 0.1 km = 100 m. So 100,000 cm (= 1,000 m) is the greatest.
All non-zero digits are significant: 345 has 3 sig figs
Zeros between non-zero digits are significant: 3,045 has 4 sig figs
Leading zeros are never significant: 0.0386 has 3 sig figs
Trailing zeros after a decimal point are significant: 8.00 has 3 sig figs
Rounding to a specified number of sig figs: count from the first non-zero digit
Example: 0.038692836 rounded to 6 sig figs = 0.0386928 (the digits 3, 8, 6, 9, 2, 8 are the six significant figures; the leading zeros and decimal point do not count)
Multiplication and division: the answer has the same number of sig figs as the measurement with the fewest sig figs
Addition and subtraction: the answer is rounded to the same decimal place as the least precise measurement
For mixed operations, work through order of operations and apply the appropriate rule at each step
Example: 0.6931 + 8.98 + (2.0 × 2.45)
First: 2.0 × 2.45 = 4.9 (2 sig figs, limited by 2.0)
Then: 0.6931 + 8.98 + 4.9 = 14.5731, rounded to the tenths place (limited by 4.9) = 14.6
Formula: % Error = |Actual − Theoretical| / Theoretical × 100%
"Actual" is what you measured; "Theoretical" is the accepted value
Example: measured 9.5 mL of water, mass = 8.1 g, density of water = 1.00 g/mL. Theoretical mass = 9.5 × 1.00 = 9.5 g. % Error = |8.1 − 9.5| / 9.5 × 100% = 1.4 / 9.5 × 100% ≈ 15%
Formula: density = mass / volume, so volume = mass / density
Example: density = 7.8 g/cm³, mass = 39 g. Volume = 39 / 7.8 = 5.0 cm³ = 5.0 mL (because 1 cm³ = 1 mL)
Always read at the bottom of the meniscus, at eye level
Estimate one digit beyond the smallest graduation
Example: if the meniscus sits at the 43.0 mL mark and you remove 7.0 mL, the reading afterwards is 36.0 mL
A line of best fit has the equation y = mx + b, where m is the slope and b is the y-intercept
To find an unknown x value from a known y, rearrange: x = (y − b) / m
Example: f(x) = 1.1236x + 283.92. If y = 524 calories, then x = (524 − 283.92) / 1.1236 ≈ 213 calories from fat
Graduated cylinder: tall, narrow, cylindrical with volume markings along the side. Used for measuring liquid volumes precisely.
Erlenmeyer flask: conical shape with a narrow neck. Used for mixing and heating liquids.
Long-stem funnel: cone-shaped top with a long tube. Used for pouring liquids into narrow-necked containers.
Other common glassware to recognise: volumetric flask, beaker, test tube, pipet, watch glass, stir rod, hot plate.
% Error = |Actual − Theoretical| / Theoretical × 100%
% Recovery = (mass recovered / mass started with) × 100%
Density = mass / volume
Volume = mass / density
y = mx + b (linear regression)
Significant figures matter any time you report a measurement in a lab or engineering context. Saying a bridge component is 3.0 m long is meaningfully different from saying it is 3.000 m long, because the second implies much tighter precision. Density is how metallurgists identify unknown metal samples and how geologists distinguish minerals.
Students often think leading zeros are significant. They are not. In 0.0386928, the zeros before the 3 just locate the decimal point.
Students sometimes confuse accuracy (closeness to the true value) with precision (consistency of repeated measurements). Percent error measures accuracy, not precision.
The rule "1 cm³ = 1 mL" catches people off guard. It is an exact equivalence, not an approximation.
When doing mixed addition and multiplication, students often apply only one sig fig rule to the whole problem. You need to apply the multiplication rule to the multiplication step and the addition rule to the addition step.
⚠️ Sig fig questions appear on nearly every CHM 11100 exam. Know the difference between the rule for addition/subtraction and the rule for multiplication/division.
⚠️ Percent error calculations are a staple of the lab portion. Always use the absolute value in the numerator.
⚠️ Graduated cylinder readings require you to read the meniscus at the bottom of the curve. If a question provides a diagram, look carefully.
⚠️ Lab equipment identification is tested directly. Know what each piece of glassware looks like and what it is used for.
True or false: the number 0.00450 has 3 significant figures.
Fill in the blank: 2.5 km = ______ m.
True or false: when adding 12.1 + 3.456, the answer should be reported to 3 decimal places.
Fill in the blank: if density = 2.0 g/mL and mass = 10.0 g, then volume = ______ mL.
True or false: 1 cm³ and 1 mL are the same volume.
Answers: 1. True (4, 5, 0 are significant). 2. 2,500 m. 3. False (round to 1 decimal place, limited by 12.1). 4. 5.0 mL. 5. True.
Q: How many centimetres are in 3.5 metres?
A: 350 cm, or 3.5 × 10² cm. Multiply by 100.
Q: The number 0.038692836 rounded to 6 significant figures is what?
A: 0.0386928. Count six digits starting from the first non-zero digit (3, 8, 6, 9, 2, 8), then round.
Q: A student measures 9.5 mL of water with a mass of 8.1 g. If the true density of water is 1.00 g/mL, what is the percent error?
A: 15%. Theoretical mass = 9.5 g. % Error = |8.1 − 9.5| / 9.5 × 100% = 14.7%, which rounds to 15%.
Q: Which is the greatest distance: 100,000 cm, 100 m, 100,000,000,000 nm, or 0.1 km?
A: 100,000 cm = 1,000 m. The others all equal 100 m. So 100,000 cm is the greatest.
Q: If a material has a density of 7.8 g/cm³ and an object made of it has a mass of 39 g, what is its volume in millilitres?
A: 5.0 mL. Volume = 39 / 7.8 = 5.0 cm³ = 5.0 mL.
Q: Compute 0.6931 + 8.98 + (2.0 × 2.45) with correct significant figures.
A: 14.6. The multiplication gives 4.9 (2 sig figs). The sum is 14.5731, rounded to the tenths place = 14.6.
Unit conversion and sig fig rules come back in stoichiometry (Chapter 3 onwards), where you convert between grams, moles and particles. Density is used again in gas law problems. Percent error appears in every lab report for the rest of the course.
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