Four lines for every calculation
The exam directions ask you to show the method and steps of each calculation: with work shown, you can earn partial credit even if the final arithmetic slips; with only an answer, even a right answer may not earn the point. The safest format is four lines:
- Write the equation or relationship. For example, n = m/M, PV = nRT, or n(H₂) = n(Mg).
- Substitute the values, each with its units.
- Calculate the result, with units, rounded to the right significant figures.
- Answer the question asked. For example, “O₂ is the limiting reactant,” “the reaction is exothermic,” or “0.514 L of H₂ forms.”
Significant figures
- Multiplying and dividing: the result has as many significant figures as the least precise value.
- Adding and subtracting: the result has as many decimal places as the value with the fewest.
- Logarithms: the number of decimal places in a pH equals the number of significant figures in [H⁺]; going back, [H⁺] from a pH has as many significant figures as the pH has decimal places.
- Exact numbers (coefficients in an equation, 1 L = 1000 mL, counted items) don’t limit significant figures.
- Keep one or two extra digits in intermediate results and round only at the last step. Rounding too early lets errors build up.
Give the final answer with significant figures that match the data: a long string of calculator digits, or a single significant figure, both make a reader doubt the calculation.
Units
- Put units on every value and cancel them as you multiply and divide. If they don’t cancel, you’ve used the wrong equation or the wrong constant (4.1.2).
- Common conversions: mL → L, °C → K, J ↔ kJ, g → mol.
- Give the result in the units the question asks for: if it wants kJ/mol, don’t stop at J.
Click the lines you think would lose credit, then check:
50.0 mL of 1.00 M HCl is mixed with 50.0 mL of 1.00 M NaOH, and the temperature rises by 6.2 °C. Taking the solution’s mass as 100.0 g and its specific heat as 4.18 J/(g·°C), find ΔH of neutralization in kJ/mol.
Writing the equation first shows the reader your reasoning.
Every value is substituted with its units. Good.
Too many significant figures: ΔT = 6.2 °C has only two, so this should be 2.6 × 10³ J.
HCl and NaOH react exactly, forming 0.0500 mol of water, with units throughout.
No units and no sign: the temperature rose, so the reaction is exothermic and ΔH = −52 kJ/mol.
A 2.50 L cylinder at 25 °C holds CO₂ at 1.20 atm. Find the mass of CO₂ in the cylinder (M = 44.01 g/mol).
Equation first, then rearranged. Good.
The temperature is converted to kelvin. Good.
The setup is right, but 0.1227 mol has been rounded to 0.1 mol: rounded too early, leaving one significant figure. Keep an extra digit in intermediate results.
Using the rounded 0.1 mol makes the result about 20% too low (the correct value is about 5.40 g), and 4.401 g has four significant figures, which contradicts 0.1 mol.
Example 1: a model write-up
A 0.500 g strip of magnesium is added to excess hydrochloric acid. Find the volume of H₂ produced at 25 °C and 0.980 atm.
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Write the equation and find the relationship. Mg + 2HCl → MgCl₂ + H₂, so n(H₂) = n(Mg).
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Moles of magnesium. Keep an extra digit in this intermediate result:
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Write the gas equation and convert the temperature.
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Substitute with units. Pressure in atm and volume in L, so R = 0.08206:
Units: mol × L atm mol⁻¹ K⁻¹ × K ÷ atm = L, so they cancel cleanly. The data have three significant figures, so the answer does too.
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Answer the question. “About 0.514 L of H₂ forms at 25 °C and 0.980 atm.”
Example 2: significant figures in different situations
Write each result with the right significant figures: (a) the molarity when 2.0 × 10⁻³ mol of solute is made up to 25.00 mL; (b) 25.0 mL + 25.00 mL; (c) the pH when [H⁺] = 3.2 × 10⁻⁴ M; (d) explain why you shouldn’t round early.
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(a) Division: count significant figures. 2.0 × 10⁻³ has only two:
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(b) Addition: count decimal places. 25.0 goes to one decimal place, so 25.0 mL + 25.00 mL = 50.0 mL.
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(c) Logarithm. [H⁺] has two significant figures, so the pH gets two decimal places:
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(d) What early rounding does. In the gas problem in the tool above, the exact n is 0.1227 mol, giving a mass of 0.1227 × 44.01 = 5.40 g. Rounding n to 0.12 mol first gives 5.28 g; rounding to 0.1 mol gives only 4.40 g, about 20% low. Keep an extra digit in intermediate results and round at the end.
- Substituting T = 25 °C directly into PV = nRT
Temperature must be in K: 298 K.
- pH = −log(2.0 × 10⁻³) = 2.70
Two significant figures, so two decimal places in the pH.
- ΔH = −57.2 (no units)
It needs kJ/mol.
- n = 2.50 g ÷ 58.44 g/mol = 0.0428 mol
Units cancel, three significant figures.
- M = 0.0500 mol ÷ 250 mL = 2.00 × 10⁻⁴ M
mL wasn’t converted to L: 0.0500 ÷ 0.250 L = 0.200 M.
- q = (50.0 g)(4.18 J/(g·°C))(3.25 °C) = 679 J
Units throughout, three significant figures.
- Rounding an intermediate 0.12268 mol to 0.1 mol before going on
Rounding too early throws the result far off.
- All data have three significant figures; the answer is written as 0.513482 L
Give three, such as 0.513 L or 0.514 L.
Common mistakes
Mistake 1: “I worked it out in my head, so I’ll just write the answer.” Readers can only score what you write. Without the equation and substitution, even a right answer may miss the calculation point, and a wrong one earns nothing.
Mistake 2: “Round every step to the right significant figures.” Only the final answer needs rounding; keep one or two extra digits in between, or the errors add up and the final answer may fall outside the accepted range.
Mistake 3: “More significant figures means more precise.” Precision is limited by the measured data; copying ten calculator digits doesn’t make the answer more accurate, it just suggests you don’t know the rules.
Quick check
With correct significant figures, 2.50 × 3.1 is:
3.1 has only two significant figures, so the result has two: 7.8.
[H⁺] = 4.5 × 10⁻³ M. How should the pH be written?
Two significant figures give two decimal places in the pH.
How should 12.0 mL + 3.45 mL be written?
For addition, count decimal places: 12.0 goes to one.
After substituting into PV = nRT, the units don’t cancel. What is the most likely cause?
For example, pressure in atm with R = 8.314, or volume left in mL.
The question wants ΔH in kJ/mol. You find that 5230 J is released and n = 0.0100 mol. What do you write?
5.23 kJ ÷ 0.0100 mol = 523 kJ/mol, negative because heat is released.
Which of these is an exact number that doesn’t limit significant figures?
Defined conversions are exact; every measurement has significant figures.
AP-style practice
A 0.4085 g sample of a monoprotic weak acid HA is titrated with 0.1050 M NaOH; the equivalence point is reached after 23.85 mL of NaOH.
(a) Calculate the moles of NaOH used, showing work and units.
(b) Calculate the molar mass of HA.
(c) How many significant figures should the answer to (b) have? Explain.
(d) A student substitutes 23.85 directly in (a) without converting to liters. What would the student get in (b), and how does the result show something is wrong?
(e) If the buret isn’t rinsed with the NaOH solution before the titration, is the measured molar mass too high or too low? Explain.
Model answer and scoring
(a) n(NaOH) = M × V = 0.1050 M × 0.02385 L = 2.504 × 10⁻³ mol.
(b) n(HA) = n(NaOH) = 2.504 × 10⁻³ mol, so M = m/n = 0.4085 g ÷ (2.504 × 10⁻³ mol) = 163.1 g/mol.
(c) Four. The data 0.1050 M, 23.85 mL and 0.4085 g all have four significant figures, and a product or quotient takes the fewest, which is four.
(d) n = 0.1050 × 23.85 = 2.504 mol, so M = 0.4085 ÷ 2.504 = 0.1631 g/mol. That’s lighter than any molecule (even H₂ is 2.02 g/mol), so it’s clearly unreasonable and points to a unit error.
(e) Too low. Water left in the buret dilutes the NaOH, so more volume is needed to reach the endpoint → calculated n(NaOH) too large → n(HA) too large → M = m/n too small.
Scoring: (a) moles with units; (b) molar mass; (c) four, with a reason; (d) 0.1631 g/mol and why it’s unreasonable; (e) too low, with the chain of reasoning.
Summary
- Four lines: equation, substitution with units, result with units and suitable significant figures, answer to the question.
- Count significant figures for multiplying and dividing, decimal places for adding and subtracting; a pH has as many decimal places as [H⁺] has significant figures. Keep an extra digit in intermediate results.
- If the units don’t cancel, the equation or constant is wrong; check that the result’s units and size make sense.
AP tip: After each calculation, spend 10 seconds on four checks: are the units right, is the size reasonable, do the significant figures match the data, and did you answer the question asked?
Sources: the answering requirements follow the directions for Section II of the AP Chemistry exam (show the method and steps of each calculation); the questions in this lesson were written by this site.