📐 Concept Overview
6.1 — Density
Density is the ratio of mass to volume. It is an intensive property — it does not change with sample size.
D = m / V | units: g/mL, g/cm³, g/L
- To measure density: find mass (balance), find volume (graduated cylinder or water displacement), divide.
- Water displacement: submerge object, read volume change = volume of object.
- Objects denser than water (D > 1.00 g/mL) sink; less dense objects float.
6.2 & 6.3 — Measuring Devices and Uncertainty
- Graduated instruments (rulers, graduated cylinders, burettes): estimate one digit beyond the smallest marked scale. Uncertainty = ± half of the smallest division.
- Digital instruments (electronic balances): uncertainty = ± the last digit displayed.
- Always read a graduated cylinder at the bottom of the meniscus.
- Example: a ruler marked in mm → read to 0.1 mm (tenths of a mm) → uncertainty = ± 0.05 mm.
6.4 — Reporting Measurements
Report every measurement to the correct number of decimal places based on instrument uncertainty. Do not round or truncate beyond the instrument's precision.
6.5 — Percent Uncertainty
% Uncertainty = (uncertainty / measurement) × 100%
- Lower percent uncertainty → more trustworthy measurement.
- Example: measuring 25.0 mL with ± 0.5 mL → 2.0% uncertainty.
- A small uncertainty on a small measurement can still be a large percent uncertainty.
6.6 — Precision vs. Accuracy
- Precision: how repeatable/consistent measurements are (values close to each other). Affected by random error.
- Accuracy: how close a measurement is to the true/accepted value. Affected by systematic error.
- You can be precise but not accurate (consistent but wrong), accurate but not precise (right on average but scattered), both, or neither.
6.7 — Counting Significant Figures
- All non-zero digits are significant: 345 → 3 sig figs.
- Zeros between non-zero digits are significant: 3.04 → 3 sig figs.
- Leading zeros (before first non-zero digit) are NOT significant: 0.0034 → 2 sig figs.
- Trailing zeros with a decimal point ARE significant: 3.40 → 3 sig figs; 200. → 3 sig figs.
- Trailing zeros without a decimal point are ambiguous (usually not significant): 200 → 1 sig fig.
- Exact numbers (counted items, defined conversions) have infinite sig figs.
6.8 — Sig Fig Rules for Calculations
- Multiplication & Division: answer has as many sig figs as the measurement with the fewest sig figs.
- Addition & Subtraction: answer has as many decimal places as the measurement with the fewest decimal places.
- When doing multi-step problems, carry extra digits through and round only at the end.
Examples: 3.2 × 4.15 = 13 (2 sig figs) | 12.11 + 3.1 = 15.2 (1 decimal place)
6.9 & 6.10 — Percent Error
% Error = |Experimental − Theoretical| / Theoretical × 100%
- Theoretical (accepted) value: the known or textbook value.
- Experimental value: what you measured in the lab.
- Lower percent error = more accurate experiment.
- Ways to reduce error: use better instruments, repeat trials, control variables, improve technique.
🃏 Flashcards
Click any card to flip it.
Click to reveal
Density
Mass per unit volume. D = m/V. An intensive property — does not change with sample size. Units: g/mL or g/cm³.
Click to reveal
Uncertainty
The range of possible values for a measurement. For graduated instruments: ± half the smallest division. For digital: ± the last digit.
Click to reveal
Percent Uncertainty
(uncertainty ÷ measurement) × 100%. Measures the quality/trustworthiness of a measurement. Lower = better.
Click to reveal
Precision
How repeatable or consistent measurements are — values close together. Affected by random error. Does NOT mean correct.
Click to reveal
Accuracy
How close a measurement is to the true (accepted/theoretical) value. Affected by systematic error.
Click to reveal
Significant Figures
All digits in a measurement that are known with certainty plus one estimated digit. Reflect the precision of the instrument.
Click to reveal
Leading Zeros
Zeros before the first non-zero digit. NOT significant. Example: 0.0034 has 2 sig figs (the 3 and 4).
Click to reveal
Trailing Zeros
Zeros at the end of a number. Significant ONLY if there is a decimal point. 200 = 1 sig fig; 200. = 3 sig figs; 2.00 = 3 sig figs.
Click to reveal
Percent Error
|Experimental − Theoretical| ÷ Theoretical × 100%. Measures accuracy of an experiment. Lower = more accurate.
Click to reveal
Theoretical Value
The accepted, textbook, or known value for a measurement. Used as the denominator in percent error.
Click to reveal
Experimental Value
The value actually measured during a lab experiment. Used as the numerator in percent error calculations.
Click to reveal
Exact Number
A number with no uncertainty (counted items, defined conversions). Has infinite significant figures and does not limit calculation results. E.g., 12 eggs, 100 cm = 1 m.
Click to reveal
Meniscus
The curved surface of a liquid in a graduated cylinder. Always read from the BOTTOM of the meniscus for water-based liquids.
Click to reveal
Random Error
Unpredictable variation in measurements. Causes imprecision. Can be reduced by taking more trials and averaging.
Click to reveal
Systematic Error
Consistent, repeatable error in one direction. Causes inaccuracy. Cannot be reduced by averaging. Example: uncalibrated balance.
Click to reveal
Sig Figs: Multiply/Divide
Round answer to the same number of significant figures as the measurement with the FEWEST sig figs. Example: 3.2 × 4.15 = 13 (2 sig figs).
Click to reveal
Sig Figs: Add/Subtract
Round answer to the same number of DECIMAL PLACES as the measurement with the fewest decimal places. Example: 12.11 + 3.1 = 15.2.
Click to reveal
Water Displacement
Method to find volume of an irregular solid. Submerge in water; volume = final reading − initial reading. Useful for objects that cannot be measured geometrically.
Click to reveal
Calibration
Adjusting or checking an instrument against a known standard to ensure accuracy. Removes systematic error.
Click to reveal
Intensive Property
A property that does NOT depend on the amount of substance. Examples: density, temperature, boiling point, color. Density is intensive — a larger sample of gold has the same density as a smaller one.
Click to reveal
Resolution
The smallest increment an instrument can detect. Higher resolution → smaller uncertainty → more precise measurements.
Click to reveal
Scientific Notation
Writing a number as A × 10ⁿ where 1 ≤ A < 10. Makes sig figs unambiguous. Digits in A are all significant. Example: 2.50 × 10³ has 3 sig figs.
Click to reveal
Replicate Trials
Repeating an experiment multiple times under the same conditions. Averaging replicates reduces the effect of random error and improves reliability.
Click to reveal
Quantitative vs. Qualitative
Quantitative: measured with numbers and units (mass = 4.52 g). Qualitative: described without numbers (the liquid is blue). Significant figures apply only to quantitative measurements.
Click to reveal
Graduated Cylinder Reading
Read at eye level at the bottom of the meniscus. Estimate one digit past the smallest graduation. A 50 mL cylinder marked in 1 mL increments: report to 0.1 mL (e.g., 23.7 mL).
⚡ Quick Review
- Density formula: D = m / V (g/mL or g/cm³)
- % Uncertainty: (uncertainty ÷ measurement) × 100%
- % Error: |experimental − theoretical| ÷ theoretical × 100%
- Sig figs × ÷: fewest sig figs in the problem
- Sig figs + −: fewest decimal places in the problem
- Non-zero digits → ALWAYS significant
- Zeros between non-zeros → ALWAYS significant (e.g., 3.04 → 3 sf)
- Leading zeros → NEVER significant (e.g., 0.0034 → 2 sf)
- Trailing zeros WITH decimal → significant (e.g., 3.40 → 3 sf; 200. → 3 sf)
- Trailing zeros WITHOUT decimal → NOT significant (e.g., 200 → 1 sf)
- Exact numbers (12 eggs, 1 m = 100 cm) → infinite sig figs
- Precision = consistency (random error); Accuracy = correctness (systematic error)
- Read graduated instruments to one decimal past the last marked line
- Read digital instruments ± last displayed digit
- Read graduated cylinder at BOTTOM of meniscus, at eye level
- Lower % error = more accurate; Lower % uncertainty = more precise
- Objects with D > 1 g/mL sink in water; D < 1 g/mL float
📝 Practice Quiz
100 multiple choice + 3 short answer. Select your answers then click Check Answers.
1. How many significant figures are in the number 0.00340?
2. How many significant figures are in 1,200 (no decimal point)?
3. How many significant figures are in 1,200. (with a decimal point)?
4. How many significant figures are in 0.0050?
5. How many significant figures are in 5.030?
6. How many significant figures are in 100 (no decimal point)?
7. How many significant figures are in 100.0?
8. How many significant figures are in 6.022 × 10²³?
9. How many significant figures are in 0.1010?
10. How many significant figures are in 250.50?
11. How many significant figures are in 0.00100?
12. How many significant figures are in 80.07?
13. How many significant figures are in 0.0702?
14. How many significant figures are in 200.0?
15. How many significant figures are in 3.500 × 10²?
16. How many significant figures are in 0.500?
17. How many significant figures does the number 1000. (with decimal point) have?
18. How many significant figures are in 7.00 × 10⁻³?
19. Which of the following has exactly 4 significant figures?
20. The number of seconds in exactly 1 hour is 3,600. This number has:
Short Answer Questions
SA1. A student measures the boiling point of an ethanol/water mixture and gets 90.5°C. The accepted value is 92.0°C. (a) Calculate the percent error. (b) State whether the student's result is more or less than the accepted value and identify one lab technique that could have caused this error.
(a) % Error = |90.5 − 92.0| / 92.0 × 100% = 1.5 / 92.0 × 100% = 1.63%
(b) The experimental value (90.5°C) is lower than the accepted value. A possible source of error: the thermometer was not fully submerged in the liquid, causing a lower reading. Also possible: impurities in the sample lowered the boiling point (a real chemistry effect, not just lab error).
(b) The experimental value (90.5°C) is lower than the accepted value. A possible source of error: the thermometer was not fully submerged in the liquid, causing a lower reading. Also possible: impurities in the sample lowered the boiling point (a real chemistry effect, not just lab error).
SA2. A student records the following mass measurements for a sample: 4.52 g, 4.51 g, 4.53 g, 4.52 g. The true mass is 5.00 g. (a) Are these measurements precise? Explain. (b) Are they accurate? Explain. (c) What type of error is likely present?
(a) Yes, precise. All four values are within 0.02 g of each other — they are very consistent/repeatable.
(b) No, not accurate. All values are ~0.48 g below the true value of 5.00 g. The average (4.52 g) is far from the true value.
(c) Systematic error — likely an uncalibrated balance that reads consistently low. Random error would scatter the results; instead they are all consistently wrong in the same direction.
(b) No, not accurate. All values are ~0.48 g below the true value of 5.00 g. The average (4.52 g) is far from the true value.
(c) Systematic error — likely an uncalibrated balance that reads consistently low. Random error would scatter the results; instead they are all consistently wrong in the same direction.
SA3. Perform the following calculation and report with proper significant figures: (3.452 × 2.1) + 14.8. Show each step and explain which sig fig rule applies at each stage.
Step 1 — Multiplication: 3.452 × 2.1 = 7.2492
Rule: fewest sig figs in multiplication — 3.452 has 4, 2.1 has 2 → round to 2 sig figs = 7.2 (but carry full value for next step)
Step 2 — Addition: 7.2492 + 14.8
Rule: fewest decimal places in addition — 14.8 has 1 decimal place → answer rounds to 1 decimal place
7.2492 + 14.8 = 22.0492 → rounded to 1 decimal place = 22.0
Final answer: 22.0
Rule: fewest sig figs in multiplication — 3.452 has 4, 2.1 has 2 → round to 2 sig figs = 7.2 (but carry full value for next step)
Step 2 — Addition: 7.2492 + 14.8
Rule: fewest decimal places in addition — 14.8 has 1 decimal place → answer rounds to 1 decimal place
7.2492 + 14.8 = 22.0492 → rounded to 1 decimal place = 22.0
Final answer: 22.0