Difficulty: Introductory | Prerequisites: Atomic Structure notes (for periodic trends section)
This material covers the practical toolkit of general chemistry: how to report numbers with the right precision (significant figures), how to convert between metric units, how to use density and percent recovery formulas, and how periodic trends in atomic and ionic radius work. The exam also tests whether you can identify common lab glassware and read a best-fit line from a graph.
Significant figures (sig figs)
The digits in a measured number that carry meaning, including all certain digits plus one estimated digit. Rules for counting: all non-zero digits are significant; zeros between non-zero digits are significant; leading zeros are not significant; trailing zeros after a decimal point are significant.
In simple terms: sig figs tell you how precise your measurement is. More sig figs = more precise.
Density
Mass per unit volume, typically expressed as g/mL or g/cm³.
In simple terms: how much stuff is packed into a given space.
Percent recovery
The ratio of the mass of material recovered to the mass of material started with, multiplied by 100%. It measures how much of your original sample you managed to keep during a transfer or procedure.
Percent error
The absolute value of (actual value minus theoretical value), divided by the theoretical value, multiplied by 100%. It measures how far off an experimental result is from the expected value.
Atomic radius
The distance from an atom's nucleus to the outermost boundary of its electron cloud. Trends: increases going down a group (more electron shells) and decreases going left to right across a period (higher effective nuclear charge pulls electrons in).
In simple terms: atoms get bigger as you go down and to the left on the periodic table.
Ionic radius
The effective radius of an ion in a crystal lattice. Cations are smaller than their parent atoms (lost electrons, same nuclear charge pulling fewer electrons closer). Anions are larger than their parent atoms (gained electrons, electron-electron repulsion expands the cloud).
In simple terms: losing electrons shrinks the ion; gaining electrons makes it bigger.
Linear regression / best-fit line
A straight line drawn through a set of data points on a scatter plot to model the relationship between two variables. The equation y = mx + b allows you to predict values. R² tells you how well the line fits the data.
When adding or subtracting, your answer should have the same number of decimal places as the measurement with the fewest decimal places
Example: 5.6792 m + 0.6 m + 4.33 m. The least precise number (0.6 m) has one decimal place, so the answer rounds to one decimal place: 10.6092 rounds to 10.6 m
This is different from multiplication/division, where you match the fewest significant figures overall
1 L = 1000 mL, so 10.00 mL = 0.01000 L (move the decimal three places left)
1 kg = 1,000 g = 1,000,000 mg, so 0.458 kg = 458,000 mg = 4.58 x 10⁵ mg
Preserve significant figures through the conversion. 10.00 mL has four sig figs, so 0.01000 L also has four sig figs
Density = mass / volume, so volume = mass / density
Example: density = 0.793 g/mL, mass = 39 g. Volume = 39 / 0.793 = 49.2 mL (rounded to the correct number of sig figs, here 2 sig figs from "39")
% recovery = (mass recovered / mass started with) x 100%
Example: started with 125.1 g, recovered 124.0 g. % recovery = (124.0 / 125.1) x 100% = 99.12%
A result close to 100% means very little material was lost
When given an equation like y = 1.0365x + 4.4921, substitute the x value to predict y
Example: at x = 10 (weight percent phosphate), y = 1.0365(10) + 4.4921 = 10.365 + 4.4921 = 14.86 tons per acre
Use the equation, not the scatter of individual data points, for predictions
Atomic radius increases going down a group (additional electron shells)
Atomic radius decreases going left to right across a period (increasing nuclear charge pulls electrons inward)
Among Al, Cl, F, N, and Na: Na has the largest atomic radius. It is furthest left and in period 3
Cations are smaller than their neutral parent atoms (fewer electrons, same nuclear pull)
Anions are larger than their neutral parent atoms (more electrons, greater repulsion)
Among isoelectronic ions (same electron count), the one with more protons is smaller
Among Li⁺, Na⁺, K⁺, Cs⁺ (all Group IA cations): Li⁺ is smallest (fewest electron shells), Cs⁺ is largest
Given Ca²⁺, S²⁻, Cl⁻, K⁺: S²⁻ is the largest (anion with 18 electrons around only 16 protons). Ca²⁺ is the smallest (20 protons pulling on only 18 electrons)
Comparing Cl⁻ and K⁺: K⁺ is smaller than its neutral atom (lost an electron), and Cl⁻ is larger than its neutral atom (gained an electron). The ion that is smaller than the neutral element it formed from is K⁺ (ion 4)
Graduated cylinder: tall, narrow, cylindrical vessel with measurement markings along the side. Used for measuring liquid volumes
Erlenmeyer flask: conical body with a narrow neck. Used for mixing, heating, and reactions where splashing is a concern
Watch glass: a shallow, concave glass disc. Used as a cover for beakers or for evaporating small amounts of liquid
Other common glassware: beaker (wide, straight-sided), volumetric flask (flat-bottomed with a long thin neck and single volume mark), pipet (thin tube for transferring precise volumes), test tube (small cylindrical tube), stir rod (solid glass rod)
Students often apply the multiplication/division sig fig rule to addition/subtraction problems. For addition and subtraction, count decimal places, not total significant figures.
A frequent error with unit conversions is moving the decimal the wrong direction. Remember: converting to a larger unit (mL to L) makes the number smaller.
Students sometimes think cations are larger than their neutral atoms. They are not. Losing electrons means less electron-electron repulsion, so the ion shrinks.
When comparing ionic radii across a set of ions with different charges and electron counts, students often forget that anions are larger than cations with the same electron configuration. Among isoelectronic species, more protons = smaller radius.
Density = mass / volume
Volume = mass / density
% Recovery = (mass recovered / mass started with) x 100%
% Error = |actual − theoretical| / theoretical x 100%
Linear best-fit: y = mx + b (substitute the given x value to find the predicted y)
Metric prefixes: kilo (10³), centi (10⁻²), milli (10⁻³), micro (10⁻⁶)
⚠️ The sig fig addition problem is almost guaranteed. Practise identifying the least number of decimal places and rounding correctly.
⚠️ Unit conversions (mL to L, kg to mg) are quick free points if you know the conversion factors. Do not overthink them.
⚠️ Density problems require you to rearrange the formula. The exam gives you density and mass, and asks for volume.
⚠️ Percent recovery: know the formula cold. The exam gives you starting and recovered masses.
⚠️ Atomic and ionic radius trend questions appear in both multiple-choice and free-response. Be ready to rank ions by size and explain why cations shrink and anions expand.
⚠️ The best-fit line question tests whether you can substitute into y = mx + b. Use the equation, not the graph's data points.
⚠️ Lab glassware identification: learn to recognise a graduated cylinder, Erlenmeyer flask, and watch glass by sight.
Fill in the blank: when adding 5.6792 + 0.6 + 4.33, the answer should be reported to ___ decimal place(s). (Answer: 1 decimal place)
True or false: 0.458 kg = 4.58 x 10⁴ mg. (Answer: false. 0.458 kg = 4.58 x 10⁵ mg)
True or false: a cation is larger than the neutral atom it formed from. (Answer: false. Cations are smaller.)
Fill in the blank: among Li⁺, Na⁺, K⁺, and Cs⁺, the smallest ion is ___. (Answer: Li⁺)
Fill in the blank: % recovery = (mass ___ / mass ___) x 100%. (Answer: recovered, started with)
Q: Calculate 5.6792 m + 0.6 m + 4.33 m to the correct number of significant figures.
A: 10.6 m. The sum is 10.6092, but 0.6 has only one decimal place, so round to one decimal place: 10.6 m.
Q: Convert 10.00 mL to litres.
A: 0.01000 L. Divide by 1000 (since 1 L = 1000 mL). The four significant figures are preserved.
Q: Convert 0.458 kg to mg.
A: 4.58 x 10⁵ mg. 0.458 kg x 1,000 g/kg x 1,000 mg/g = 458,000 mg = 4.58 x 10⁵ mg.
Q: The density of a material is 0.793 g/mL and an object made from it has a mass of 39 grams. What is its volume in mL?
A: 49 mL (or 49.2 mL with three sig figs). Volume = mass / density = 39 / 0.793 = 49.2 mL. The number 39 has 2 significant figures, so a strict answer is 49 mL.
Q: Ken started with 125.1 g of cocoa powder and recovered 124.0 g after transfer. What is the percent recovery?
A: 99.12%. % recovery = (124.0 / 125.1) x 100% = 99.12%.
Q: Using the best-fit equation y = 1.0365x + 4.4921, what is the predicted tomato crop (tons/acre) at 10% phosphate?
A: 14.86 tons per acre. y = 1.0365(10) + 4.4921 = 14.857, which rounds to 14.86.
Q: Which of these elements has the largest atomic radius: Al, Cl, F, N, Na?
A: Na (sodium). It sits furthest to the left in period 3. Atomic radius increases to the left and down the table.
Q: Which ion has the smallest ionic radius: Li⁺, Na⁺, K⁺, Cs⁺?
A: Li⁺. It has the fewest electron shells among Group IA cations.
Q: Given ions Ca²⁺, S²⁻, Cl⁻, K⁺, which is largest?
A: S²⁻. It is an anion (gained electrons, expanded electron cloud) with 18 electrons and only 16 protons.
Q: Given ions Ca²⁺, S²⁻, Cl⁻, K⁺, which is smallest?
A: Ca²⁺. It is a cation with 20 protons pulling on only 18 electrons.
Q: Considering only Cl⁻ and K⁺, which is smaller than the neutral element it formed from?
A: K⁺. Potassium lost an electron to become K⁺, which is smaller than neutral K. (Cl⁻ gained an electron and is larger than neutral Cl.)
Significant figures and unit conversions reappear throughout the course whenever calculations are involved (stoichiometry, gas laws, solution concentrations). Periodic trends in atomic and ionic radius connect to electronegativity and ionisation energy trends, which are tested later. Density calculations come back in solution chemistry and gas law problems.
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