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Every number-crunching skill in intro chemistry: sig figs, conversions, moles, formulas, equations, stoichiometry and more.
Read the idea in Learn, then practise. Each tab has Hint, Reset and Skip, 3 levels and a crown to win.
The toolbox button (bottom right) has conversion factors, constants and a molar mass finder.
Chemistry calculations look different but follow the same habit. 1. Write down what you are given, with units. 2. Write down what you are asked for, with units. 3. Build a chain of conversion factors (or one formula) that turns the first into the second so the units cancel. 4. Check the answer: right unit, sensible size, right number of significant figures.
If the units do not cancel to the unit you want, the setup is wrong, even before you touch the calculator. Always carry units through every step.
Like the little animal, the mole works underground. You cannot walk straight from grams of one substance to grams of another, so you dig down into moles, travel through the tunnel, and come up on the other side. Learn this one tunnel and half of general chemistry opens up.
A measurement has a limit to its precision, and the digits you write should show it. Significant figures are the digits that carry meaning: all the certain digits plus one estimated digit.
Nonzero digits always count. Zeros between nonzero digits count (3004 has 4). Leading zeros never count (0.0045 has 2). Trailing zeros count only when there is a decimal point (4.50 has 3, 4500 has 2, 4500. has 4).
Multiply / divide: keep as many sig figs as the number with the fewest. Add / subtract: keep as many decimal places as the number with the fewest. Round only at the very end, never in the middle.
Exact numbers never limit your sig figs: counted items (12 eggs) and defined conversions (100 cm = 1 m, 1 in = 2.54 cm exactly). Measured conversions (1 lb = 453.6 g) do count, but we normally carry at least 4 sig figs for them.
Rounding: look only at the first digit you are dropping. 5 or more rounds up. 4 or less leaves the last digit alone. Round once, at the end.
Ending in zeros: if 5000 should show two sig figs, write it in scientific notation, 5.0 × 103. That is the only unambiguous way, and it is why the practice boxes ask for that form.
Chemists deal with huge and tiny numbers, so we write them as a coefficient from 1 to 10 times a power of 10: 602,200,000,000,000,000,000,000 = 6.022 × 1023, and 0.000 000 56 = 5.6 × 10−7. Moving the decimal left makes the exponent larger; moving it right makes it smaller.
| Prefix | Symbol | Power | Prefix | Symbol | Power |
|---|---|---|---|---|---|
| giga | G | 109 | centi | c | 10−2 |
| mega | M | 106 | milli | m | 10−3 |
| kilo | k | 103 | micro | µ | 10−6 |
| (base unit) | 100 | nano / pico | n / p | 10−9 / 10−12 |
You can only add or subtract once the exponents match. Rewrite one number: 5.0 × 103 + 2.0 × 102 = 5.0 × 103 + 0.20 × 103 = 5.2 × 103.
On a calculator, use the EE or EXP key for "× 10n". Typing 6.02 × 10^23 with the × key and the ^ key often gives wrong answers when you divide by it, so use EE and avoid extra parentheses mistakes.
A conversion factor is a fraction equal to 1. Because 1 km = 1000 m, the fraction (1000 m ÷ 1 km) equals 1. Multiplying by it changes the unit but not the amount. Arrange the fraction so the unit you start with is on the bottom; it cancels.
Squared and cubed units: convert the length unit, then square or cube the factor. 1 m3 = (100 cm)3 = 106 cm3, and 1 mL = 1 cm3.
Temperature is the exception: it is not multiplied. K = °C + 273.15 and °F = 1.8 × °C + 32.
Pressure: 1 atm = 760 mmHg = 760 torr = 101.325 kPa = 1.01325 bar = 14.7 psi. Energy: 1 cal = 4.184 J, and a food Calorie (Cal) is 1000 cal = 1 kcal. 1 kWh = 3.6 × 106 J.
Check the size: a kilometer is bigger than a meter, so the same distance is a bigger number in meters. If your answer goes the wrong way, you flipped the fraction.
The toolbox has all these factors, so you never need to memorize them.
density = mass ÷ volume, d = m / V. Solids are usually g/cm3; liquids g/mL (1 mL = 1 cm3). Rearrange it when you need the mass (m = d × V) or the volume (V = m ÷ d). Water is 1.00 g/mL, which is a handy comparison: denser things sink in water, less dense things float.
Cube = edge3. Cylinder = πr2h. Sphere = (4/3)πr3. Careful: use the radius, which is half the diameter.
Water displacement measures irregular solids: volume of the object = final water level − starting water level.
Atoms are far too small to count, so chemists count them in bunches. A mole is 6.022 × 1023 particles (Avogadro's number), the way a dozen is 12. The molar mass of a substance is the mass of one mole, in g/mol, and it comes straight from the periodic table: add up the atomic masses of every atom in the formula.
grams ⇄ moles: use molar mass.
moles ⇄ particles: use 6.022 × 1023.
Grams never go straight to particles; always pass through moles.
n = m ÷ MM (moles = mass ÷ molar mass)
N = n × NA (particles = moles × 6.022 × 1023)
mass % of an element = (mass of that element in one mole ÷ molar mass) × 100. All the percentages of a compound add up to 100.
A hydrate such as CuSO4·5H2O has water built into the crystal. Include 5 × 18.02 g/mol of water in its molar mass. Percent water = (water mass ÷ hydrate molar mass) × 100.
To count atoms of one element: particles of compound × subscript. One mole of Al2(SO4)3 holds 2 mol Al, 3 mol S and 12 mol O.
The empirical formula is the simplest whole-number ratio of atoms (CH2O). The molecular formula is the real count in one molecule (C6H12O6), always a whole-number multiple of the empirical one.
1.5 → multiply all by 2. 1.33 or 1.67 → multiply by 3. 1.25 or 1.75 → by 4. Never round 1.5 to 2; it changes the compound.
n = molar mass ÷ empirical-formula mass (a whole number). Multiply every subscript by n. If the molar mass is 180 and CH2O weighs 30, then n = 6.
Burn a compound containing C and H (and maybe O). All the carbon ends up as CO2 and all the hydrogen as H2O. So moles C = grams CO2 ÷ 44.01, and moles H = 2 × grams H2O ÷ 18.02. Oxygen is found by difference: sample mass − mass of C − mass of H. Then use the same ratio method.
The coefficients of a balanced equation are mole ratios, like a recipe. In N2 + 3 H2 → 2 NH3, 3 mol of H2 makes 2 mol of NH3. Ratios work only in moles, so the standard route is a tunnel:
The reactant that runs out first. Work out how much product each reactant could make; the smaller answer wins. The other reactant is in excess. A shortcut: moles ÷ coefficient for each reactant, smallest is limiting.
Theoretical yield is the most product possible (from the limiting reactant). Percent yield = (actual ÷ theoretical) × 100. It can never honestly exceed 100%.
Making a sandwich with 10 slices of bread and 3 slices of cheese: cheese limits you. Chemists add an excess of the cheaper reactant on purpose so the expensive one is used up completely.
NEVER work with mass ratios, only with mole ratios.
NEVER work with unbalanced reactions, always balance first.
Heat (q) flows from hot to cold. How much it changes a temperature depends on the specific heat (c): the energy to warm 1 g by 1 °C. Water's is large, 4.184 J/g·°C, which is why water heats and cools slowly.
Energy is conserved: heat lost by the hot object equals heat gained by the cold one. So qhot + qcold = 0. This lets you find a final temperature or an unknown specific heat.
While ice melts or water boils the temperature stays put. Use q = m × ΔHfus (334 J/g) or q = m × ΔHvap (2260 J/g). A heating curve is just several of these added up.
q is positive when a system takes in heat (endothermic) and negative when it gives heat off (exothermic).
In a bomb calorimeter, qrxn = −Ccal × ΔT: the calorimeter absorbs everything the reaction releases.
Hess's law: if you add reactions, you add their ΔH. Reverse a reaction → flip the sign of ΔH. Multiply a reaction → multiply ΔH by the same number.
Those topics now have their own Works, each with live models and its own crowns and certificate. They are part of the same Track 0 adventure.
The Challenge mixes every kind of problem, and first asks you to name the kind of problem. That is the exam skill: deciding which tool to use. When you have earned all the crowns, you have the number skills for an entire first-semester course.
Coming later: gas laws get their own tool (GasWorks), and equilibrium, acids and bases, and kinetics will arrive as a second level here.
CalcWorks teaches the calculations of introductory chemistry by practice. Every topic has a short lesson, a practice tab with three levels of randomly generated problems, hints and worked solutions, and a crown for mastering it. A built-in toolbox has conversion factors, constants, atomic masses and a molar mass finder.
Significant figures, scientific notation and metric prefixes, unit conversions and dimensional analysis, density, the mole and molar mass, percent composition, empirical and molecular formulas, stoichiometry with limiting reactants and percent yield, and specific heat and calorimetry. Balancing, solutions and atomic structure have their own Works.
When multiplying or dividing, keep as many significant figures as the measurement with the fewest. When adding or subtracting, keep as many decimal places as the measurement with the fewest. Round only at the end.
Convert each element to moles, divide all by the smallest number of moles, and multiply by 2, 3 or 4 if the ratios are not whole numbers. The result is the simplest whole-number ratio of atoms.
Convert each reactant to moles and divide by its coefficient in the balanced equation. The smallest result belongs to the limiting reactant, which also sets the theoretical yield.
Gas laws have their own tool, GasWorks. CalcWorks focuses on the number skills of introductory chemistry.
Yes. Earn every crown in CalcWorks and you can claim a free certificate of completion as a PDF. Finish all six Track 0 Works to also earn the Track 0 badge and certificate, which anyone can verify.
No. It runs in your browser and saves your progress on your own device.
All nine crowns earned. Enter your name exactly as it should appear on the certificate.
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