MGU Science Series — High School Chemistry
Part I: Foundations of Chemistry
SI Units · Scientific Notation · Significant Figures · Dimensional Analysis · Density
Chapter 2 of 28
MGU Science Series · High School Chemistry · Library Textbook Edition
Chapter 2:
Matter, Measurement,
and Units
SI units, metric prefixes, scientific notation, accuracy and precision, significant figures, dimensional analysis, density, and data interpretation — the measurement foundation every chemistry chapter requires
■ Part I: Foundations of Chemistry ■
■ 10 Sections • Chapter 2 of 28 ■
Follows Chapter 1: What Chemistry Studies · Prepares students for stoichiometry, gas laws, solutions, and thermochemistry
Chapter 2 — Front Matter
Purpose and Overview
Imagine you are a doctor calculating a medication dose, an engineer designing a bridge, or a chemist preparing a chemical reaction. In every case, you cannot guess. You need numbers — and those numbers must mean something precise and reliable to everyone who reads them.
Chemistry is a quantitative science. That phrase means chemistry does not just describe what happens — it measures how much, how fast, how heavy, and how hot. Every time a chemist runs an experiment, they collect numerical data. Every time they predict a result, they use numbers to check their prediction. Every time they communicate findings, they report values with units so that scientists anywhere in the world understand exactly what was measured.
This chapter builds the measurement skills that will support every chapter that follows. Without these skills, the mole concept, gas laws, stoichiometry, solution concentration, and thermochemistry would be difficult or impossible to handle confidently. With these skills, you will be able to approach any calculation in chemistry with a clear system and a reliable method.
Opening Question
In 1999, a NASA Mars orbiter was lost because one engineering team used metric units and another used customary units. The spacecraft missed its target and was destroyed. A unit error in the real world is not just a wrong answer on a test. By the end of this chapter, you will understand how chemists avoid such errors — and why a standard measurement system matters not just in chemistry, but across all of science and engineering.
10Sections
10Objectives
22Key Terms
8+Worked Examples
Learning Objectives
- 1Explain why measurement is essential in chemistry and identify the components of a valid measurement.
- 2Identify SI base units and apply metric prefixes to express quantities at different scales.
- 3Measure and describe length, mass, volume, temperature, and time in appropriate units.
- 4Write numbers in scientific notation and perform calculations using scientific notation.
- 5Distinguish between accuracy and precision and identify sources of systematic and random error.
- 6Count significant figures in a measurement and apply significant figure rules to calculations.
- 7Round calculated results correctly and record measurements with appropriate precision.
- 8Perform unit conversions using dimensional analysis, including multi-step conversions.
- 9Calculate density and use density to find mass or volume; interpret density as an intensive property.
- 10Read and interpret data tables, line graphs, and slopes in a chemistry context.
Chapter 2 — Contents
Sections at a Glance
- 2.1Why Measurement Matters in ChemistryWhat a measurement is; number plus unit; chemistry as a quantitative science; the two levels of chemistry; consequences of unit errors
- 2.2SI Units and Metric PrefixesThe International System of Units; seven base units; metric prefixes and powers of ten; derived units; scales from atomic to industrial
- 2.3Length, Mass, Volume, Temperature, and TimeDefinitions and units for each quantity; laboratory tools; mass vs. weight; temperature scales and conversions; meniscus reading
- 2.4Scientific Notation in Chemical MeasurementWhy extreme numbers need notation; format N × 10ⁿ; converting to and from standard form; multiplying, dividing, adding, subtracting
- 2.5Accuracy, Precision, and UncertaintyAccuracy vs. precision; dartboard analogy; systematic error; random error; reporting uncertainty; the estimated digit
- 2.6Significant FiguresSix rules for counting significant figures; multiplication and division rule; addition and subtraction rule; exact numbers; scientific honesty
- 2.7Rounding and Reporting MeasurementsStandard rounding rule; avoiding rounding error in multi-step calculations; components of a properly reported measurement
- 2.8Dimensional Analysis and Unit ConversionConversion factors as fractions equal to 1; setup method; single-step and multi-step conversions; why the method works
- 2.9Density as a Chemical MeasurementDensity = m/V; intensive property; density table; three calculation types; water displacement; floating and sinking
- 2.10Reading Graphs and Data TablesData table structure; line graphs; slope as a physical quantity; bar graphs; scatter plots; interpolation and extrapolation
Sections 2.1 – 2.3
Foundations of Measurement: Units and Quantities
Why Measurement Matters · SI Units · Length, Mass, Volume, Temperature, Time
Section 2.1
Why Measurement Matters in Chemistry
A measurement is more than a number. A measurement is a number combined with a unit. The unit tells you what the number means. Consider the sentence: The sample has a mass of 45. That sentence is incomplete. Forty-five what? Grams? Kilograms? Pounds? Without a unit, the number is meaningless.
Atoms and molecules are far too small to see with the naked eye. Chemists cannot directly observe what is happening at the particle level. Instead, they measure observable quantities — mass, volume, temperature, pressure, time — and use those measurements to draw conclusions about the invisible world of particles. Careful measurement is what makes this double vision possible: working simultaneously at the visible level of beakers and balances and at the invisible level of atoms and molecules.
Chemistry at Two Levels
Every measurement in chemistry connects two worlds: the observable world of instruments and numbers, and the invisible world of particles. When you measure the mass of a product formed in a reaction, you are learning something about the atoms that rearranged. Measurement is the bridge.
Real Consequences of Unit Errors
In 1999, a NASA Mars Climate Orbiter was lost because one engineering team used metric units and another used customary (imperial) units. The spacecraft missed its intended orbit and was destroyed. The cost was $327 million. A missing or wrong unit is never just a minor mistake.
Section 2.2
SI Units and Metric Prefixes
Scientists around the world use a shared measurement system called the International System of Units, abbreviated SI from the French Système International d'Unités. Because it is used internationally, measurements made by a chemist in Brazil can be understood immediately by a chemist in Japan or Canada or Egypt.
SI Base Units
| Quantity | SI Base Unit | Symbol |
|---|
| Length | meter | m |
| Mass | kilogram | kg |
| Temperature | kelvin | K |
| Time | second | s |
| Amount of substance | mole | mol |
| Electric current | ampere | A |
| Luminous intensity | candela | cd |
Metric Prefixes
A prefix is a word part added to the beginning of a unit name that changes the unit's size by a power of ten. You do not need a completely different unit every time a quantity is very large or very small — you just add a prefix to scale the base unit.
| Prefix | Symbol | Meaning | Power of Ten |
|---|
| giga- | G | 1,000,000,000 | 10⁹ |
| mega- | M | 1,000,000 | 10⁶ |
| kilo- | k | 1,000 | 10³ |
| deci- | d | 0.1 | 10⁻¹ |
| centi- | c | 0.01 | 10⁻² |
| milli- | m | 0.001 | 10⁻³ |
| micro- | μ | 0.000001 | 10⁻⁶ |
| nano- | n | 0.000000001 | 10⁻⁹ |
| pico- | p | 0.000000000001 | 10⁻¹² |
Prefixes in Context
A carbon-carbon bond is about 0.154 nm long. A hydrogen atom is about 0.1 nm in diameter. Industrial reactions may produce thousands of kilograms of product. The same prefix system handles both the atomic scale and the industrial scale using one set of units.
Section 2.3
Length, Mass, Volume, Temperature, and Time
Length
Length is the measure of distance between two points. The SI base unit is the meter (m). Common chemistry units: meter (m), centimeter (cm = 0.01 m), millimeter (mm = 0.001 m), nanometer (nm = 10⁻⁹ m). Nanometers are used for atomic and molecular dimensions — a chlorine atom has a radius of about 0.099 nm.
Mass
Mass is the measure of the amount of matter in an object. The SI base unit is the kilogram (kg). Mass is different from weight: mass does not change regardless of location, but weight depends on gravity. In chemistry, mass is always what you measure. Common units: kilogram (kg), gram (g = 0.001 kg), milligram (mg = 10⁻³ g), microgram (μg = 10⁻⁶ g). Mass is measured with a balance.
Volume
Volume is the amount of three-dimensional space a substance occupies. Practical chemistry units: liter (L), milliliter (mL = 0.001 L), cubic centimeter (cm³). Importantly, 1 mL = 1 cm³ — these units are interchangeable.
| Laboratory Tool | Use | Precision |
|---|
| Graduated cylinder | Measuring liquid volumes quickly | Moderate |
| Pipette | Transferring precise small volumes | High |
| Burette | Delivering variable volumes in titration | High |
| Volumetric flask | Preparing a precise total volume | Very high (fixed volume) |
| Beaker / Erlenmeyer flask | Mixing and reactions only | Low — not for measurement |
Reading the Meniscus
Water and most aqueous solutions form a concave meniscus — the surface curves downward in the middle. Always read the volume at the bottom of the meniscus, with your eyes level with the liquid surface. Reading from above gives a value that is too low; reading from below gives a value that is too high.
Temperature
Temperature measures the average kinetic energy of particles. Three scales matter in chemistry:
| Scale | Freezing Point of Water | Boiling Point of Water | Use in Chemistry |
|---|
| Celsius (°C) | 0 °C | 100 °C | Most laboratory work |
| Kelvin (K) | 273.15 K | 373.15 K | Required for gas laws; starts at absolute zero (0 K = −273.15 °C) |
| Fahrenheit (°F) | 32 °F | 212 °F | Rarely used in chemistry |
Temperature Conversion Formulas
K = °C + 273.15
°C = K − 273.15
°F = (°C × 9/5) + 32
°C = (°F − 32) × 5/9
Time
Time measures duration. The SI base unit is the second (s). Chemistry uses seconds (s), milliseconds (ms = 0.001 s), minutes (min = 60 s), and hours (h = 3,600 s) depending on the process being studied.
Sections 2.4 – 2.5
Scientific Notation and Measurement Quality
Scientific Notation · Accuracy vs. Precision · Sources of Error
Section 2.4
Scientific Notation in Chemical Measurement
Chemistry regularly deals with numbers that are inconveniently large or small. The number of carbon atoms in 12 grams of carbon is approximately 602,000,000,000,000,000,000,000. The mass of a single proton is approximately 0.00000000000000000000000167 kg. Writing and calculating with these numbers in standard form is error-prone. Scientific notation is the solution.
Scientific Notation Format
Every number is written as N × 10ⁿ where N is a number from 1 to <10 (exactly one non-zero digit before the decimal) and n is an integer exponent.
602,000,000,000,000,000,000,000 = 6.02 × 10²³
0.00000000000000000000000167 kg = 1.67 × 10⁻²⁴ kg
0.0000000001 m = 1.0 × 10⁻¹⁰ m
Converting to Scientific Notation
Move the decimal point until exactly one non-zero digit is to its left. Count the places moved. If you moved left, the exponent is positive. If you moved right, the exponent is negative.
Example 2.4.1
Converting to and from scientific notation
- Problem AConvert 4,580,000 to scientific notation.
- SolveMove the decimal 6 places to the left: 4.58. Exponent is +6.
4,580,000 = 4.58 × 10⁶
- Problem BConvert 0.000073 to scientific notation.
- SolveMove the decimal 5 places to the right: 7.3. Exponent is −5.
0.000073 = 7.3 × 10⁻⁵
- Problem CConvert 3.25 × 10⁴ to standard form.
- SolvePositive exponent → move decimal right 4 places.
3.25 × 10⁴ = 32,500
Calculating with Scientific Notation
Multiply: multiply coefficients, add exponents.
(3.0 × 10⁴) × (2.0 × 10³) = 6.0 × 10⁷
Divide: divide coefficients, subtract exponents.
(8.4 × 10⁶) ÷ (2.1 × 10²) = 4.0 × 10⁴
Add/Subtract: make exponents equal first, then add or subtract coefficients.
(3.5 × 10³) + (2.0 × 10²) = (3.5 + 0.20) × 10³ = 3.7 × 10³
Adjusting the Coefficient
If a calculation produces a coefficient outside the range 1 to <10, adjust it. For example, 24 × 10⁵ is not proper scientific notation. Rewrite as 2.4 × 10⁶ by moving the decimal one place left and increasing the exponent by 1.
Section 2.5
Accuracy, Precision, and Uncertainty
No measurement is perfectly exact. Every measurement has some degree of uncertainty because measuring instruments have limits, human observers have limits, and laboratory conditions are never perfectly controlled. The goal is not to eliminate uncertainty — it is to understand and honestly report it.
| Concept | Definition | Analogy |
|---|
| Accuracy | How close a measured value is to the true or accepted value. Accuracy is about correctness. | Darts clustered near the bullseye |
| Precision | How close repeated measurements are to each other. Precision is about consistency. | Darts clustered tightly together (may or may not be near the bullseye) |
Sources of Error
- Systematic error — affects every measurement in the same direction. Examples: a balance not zeroed correctly; a thermometer reading 2 °C too high; reading a meniscus from the wrong angle every time. Fix: calibrate instruments, check procedures.
- Random error — causes results to scatter unpredictably above and below the true value. Examples: slight variations in reading a scale; room temperature fluctuations. Reduce by repeating measurements and averaging.
Recording the Estimated Digit
When you record a measurement, record every digit you are certain about, plus one estimated digit. If a ruler has millimeter markings, you can estimate to the nearest tenth of a millimeter. A reading of 4.32 cm means the 3 is certain and the 2 is estimated. This practice is the foundation of significant figures.
Sections 2.6 – 2.7
Significant Figures and Rounding
Counting Sig Figs · Calculations · Rounding Rules · Reporting Measurements
Section 2.6
Significant Figures
Significant figures (also called significant digits) are the meaningful digits in a measured value. They communicate the precision of a measurement. More significant figures mean greater precision. The number of significant figures in a reported value should reflect the actual precision of the measurement — no more, no less.
Six Rules for Counting Significant Figures
- Rule 1: All non-zero digits are significant. (347 has 3 sig figs; 5.82 has 3)
- Rule 2: Zeros between non-zero digits are significant. (3,047 has 4; 20.08 has 4)
- Rule 3: Leading zeros (before first non-zero digit) are NOT significant. (0.052 has 2; 0.00340 has 3)
- Rule 4: Trailing zeros after a decimal point ARE significant. (3.40 has 3; 12.500 has 5)
- Rule 5: Trailing zeros in a whole number without a decimal point are ambiguous. (2,500 may have 2–4; write 2.5 × 10³ for 2 sig figs; 2.500 × 10³ for 4)
- Rule 6: Exact numbers (counted quantities or defined values) have unlimited significant figures and never limit the result.
Significant Figures in Calculations
Example 2.6.1
Significant figures in multiplication and division
- RuleRound to the same number of significant figures as the measurement with the fewest sig figs.
- ProblemCalculate the area of a rectangle 4.52 cm × 2.1 cm.
- Count4.52 cm → 3 sig figs. 2.1 cm → 2 sig figs. Fewest = 2.
- Calculate
4.52 × 2.1 = 9.492 cm²
- RoundRound to 2 significant figures.
Answer: 9.5 cm²
Example 2.6.2
Significant figures in addition and subtraction
- RuleRound to the same number of decimal places as the measurement with the fewest decimal places.
- ProblemAdd 12.52 g + 1.7 g + 0.821 g.
- Count12.52 → 2 decimal places. 1.7 → 1 decimal place. 0.821 → 3 decimal places. Fewest = 1.
- Calculate
12.52 + 1.7 + 0.821 = 15.041 g
- RoundRound to 1 decimal place.
Answer: 15.0 g
Why Significant Figures Matter
Reporting the correct number of significant figures is a matter of scientific honesty. If you measure mass to the nearest gram and volume to the nearest milliliter, you cannot report a density meaningful to five decimal places — that would imply precision you never achieved. Significant figures tell others exactly what you know, and what you do not know.
Section 2.7
Rounding and Reporting Measurements
Standard Rounding Rule
If the digit to be dropped is less than 5, round down — keep the preceding digit the same.
If the digit to be dropped is 5 or greater, round up — increase the preceding digit by 1.
Examples: 4.736 → 3 sig figs = 4.74 12.342 → 4 sig figs = 12.34 0.005650 → 2 sig figs = 0.0057
Multi-Step Calculations: Round Only at the End
When a calculation involves multiple steps, carry all digits through every intermediate step. Round only the final answer. Rounding early introduces rounding error that accumulates across steps and can make your final answer incorrect.
A properly reported measurement always includes: (1) a numerical value with the correct number of significant figures, (2) a unit, and sometimes (3) an indication of uncertainty (e.g., ± 0.01 g). A measurement recorded as 12.34 g communicates that the mass was determined to the nearest 0.01 gram. A measurement recorded as 12 g communicates that the mass was determined only to the nearest gram.
- State the number of significant figures in each measurement:
- a) 3.40 g
- b) 0.0052 kg
- c) 1,004 m
- d) 7.00 × 10³ mL
- e) 250 L (no decimal point)
- f) 250. L (with decimal point)
- Perform each calculation and report the answer with correct significant figures:
- a) 12.5 g + 3.14 g + 0.8 g
- b) 6.02 × 10.0
- c) 98.6 ÷ 12
- d) 1.025 m − 0.21 m
- Round each value to three significant figures: 4.7362; 0.020481; 6.995 × 10⁴
Sections 2.8 – 2.9
Dimensional Analysis and Density
Conversion Factors · Unit Conversion · Density Formula · Intensive Properties · Floating
Section 2.8
Dimensional Analysis and Unit Conversion
Dimensional analysis is a systematic method for converting measurements from one unit to another. It uses multiplication by conversion factors — fractions that equal exactly 1 — to cancel unwanted units and introduce wanted units. This method makes unit conversion reliable, verifiable, and applicable to problems of any complexity.
What a Conversion Factor Is
A conversion factor is a fraction formed from two equal quantities in different units. Because numerator and denominator are equal, the fraction = 1. Multiplying by it changes the unit without changing the value.
Since 1 km = 1,000 m, two conversion factors exist:
(1 km / 1,000 m) and (1,000 m / 1 km)
Choose the one that places the unwanted unit in the denominator so it cancels.
Example 2.8.1
Single-step unit conversion
- ProblemConvert 5.25 kilometers to meters.
- SetupThe unwanted unit is km; the wanted unit is m. Place km in the denominator.
- Solve
5.25 km × (1,000 m / 1 km) = 5,250 m = 5.25 × 10³ m
- Checkkm cancels; m remains. 3 sig figs in → 3 sig figs out. ✓
Answer: 5.25 × 10³ m
Example 2.8.2
Multi-step unit conversion
- ProblemConvert 3.5 hours to seconds.
- ThinkNo direct hours-to-seconds conversion factor. Chain through minutes.
- Solve
3.5 h × (60 min / 1 h) × (60 s / 1 min)
= 3.5 × 60 × 60 s
= 12,600 s = 1.3 × 10⁴ s
- Checkh and min both cancel; s remains. 2 sig figs → 1.3 × 10⁴ s. ✓
Answer: 1.3 × 10⁴ s
Example 2.8.3
Converting a rate: km/h to m/s
- ProblemConvert 55 miles per hour to meters per second. (1 mile = 1,609 m)
- Solve
55 mi/h × (1,609 m / 1 mi) × (1 h / 3,600 s)
= (55 × 1,609) / 3,600 m/s
= 88,495 / 3,600 m/s
= 25 m/s
Answer: 25 m/s (2 significant figures)
Solve using dimensional analysis. Show all conversion factors explicitly.
- 5.00 km to meters
- 750 mL to liters
- 2.35 kg to grams
- 86,400 s to hours
- 0.500 L to milliliters
- 3.00 × 10⁻³ km to mm
- 90.0 km/h to m/s
- 250 mg → grams and → micrograms
Section 2.9
Density as a Chemical Measurement
Density is the ratio of the mass of a substance to its volume. Density tells you how much mass is packed into a given amount of space. Density is an intensive property — it does not depend on the size of the sample. A large block of gold and a small gold nugget have the same density: approximately 19.3 g/cm³. This makes density useful for identifying unknown substances.
Densities of Common Substances
| Substance | Density (g/mL or g/cm³) |
|---|
| Water (liquid, 25 °C) | 1.00 |
| Ice (0 °C) | 0.917 |
| Ethanol | 0.789 |
| Aluminum | 2.70 |
| Iron | 7.87 |
| Gold | 19.3 |
| Air (25 °C, 1 atm) | 0.00119 |
| Oxygen gas | 0.00143 |
Ice Is Less Dense Than Liquid Water
Most solids are denser than their corresponding liquids. Water is a striking exception — ice (0.917 g/cm³) is less dense than liquid water (1.00 g/cm³), so ice floats. This has enormous consequences for life on Earth: lakes freeze from the top down, protecting aquatic life below the ice layer.
Example 2.9.1
Calculating density from mass and volume
- ProblemA piece of aluminum has a mass of 54.0 g and a volume of 20.0 cm³. What is its density?
- Solve
d = m / V = 54.0 g / 20.0 cm³ = 2.70 g/cm³
- Check3 sig figs in each measurement → 3 sig figs in result. Matches accepted density of aluminum. ✓
Answer: 2.70 g/cm³
Example 2.9.2
Finding mass from density and volume
- ProblemThe density of ethanol is 0.789 g/mL. What is the mass of 250.0 mL of ethanol?
- Rearrangem = d × V
- Solve
m = 0.789 g/mL × 250.0 mL = 197 g
Answer: 197 g
Example 2.9.3
Water displacement — measuring volume of an irregular solid
- ProblemAn irregular copper sample has a mass of 89.6 g. When added to a graduated cylinder containing 40.0 mL of water, the level rises to 50.0 mL. Find the density.
- Volume
V = 50.0 mL − 40.0 mL = 10.0 mL
- Density
d = 89.6 g / 10.0 mL = 8.96 g/mL
- CheckAccepted density of copper is 8.96 g/cm³. Result is accurate. ✓
Answer: 8.96 g/mL
Floating and Sinking
An object floats in a fluid if its density is less than the density of the fluid. An object sinks if its density is greater than the density of the fluid.
Wood floats on water (density ≈ 0.5–0.9 g/mL < 1.00 g/mL). A steel nail sinks (density ≈ 7.8 g/mL > 1.00 g/mL). A steel ship floats because the hollow hull makes the average density of ship + air less than 1.00 g/mL.
- A sample of iron has a mass of 78.7 g and a volume of 10.0 cm³. Calculate its density and identify the substance using the table above.
- The density of gold is 19.3 g/cm³. What is the volume of a 250.0 g gold bar?
- A graduated cylinder reads 35.0 mL before an object is added, and 48.5 mL after. The object has a mass of 17.2 g. What is its density?
- A liquid has a density of 1.26 g/mL. What mass does 75.0 mL of this liquid have? Based on the table above, what substance is this?
- Will a block of wood with density 0.65 g/cm³ float or sink in ethanol (d = 0.789 g/mL)? Explain your reasoning.
Section 2.10
Reading Graphs and Data Tables
Data Tables · Line Graphs · Slope · Interpolation · Extrapolation
Section 2.10
Reading Graphs and Data Tables
Chemistry generates large amounts of numerical data. Data tables and graphs are tools that organize, display, and reveal patterns in that data. A graph can show trends, relationships, and anomalies that would be invisible in a list of numbers. A table displays precise values that would be hard to read from a graph. Both are essential tools in chemistry.
Reading Data Tables
Read the column headings carefully, including units. Identify the independent variable (what was changed or controlled) and the dependent variable (what was measured as a result). Look for patterns and note any values that seem unusual.
| Sample | Mass (g) | Volume (mL) | Density (g/mL) |
|---|
| 1 | 5.60 | 7.10 | 0.789 |
| 2 | 11.2 | 14.2 | 0.789 |
| 3 | 22.4 | 28.4 | 0.789 |
| 4 | 33.6 | 42.6 | 0.789 |
The density is constant regardless of sample size — confirming that density is an intensive property. The liquid is ethanol (d = 0.789 g/mL).
Reading Line Graphs and Calculating Slope
A line graph shows how one variable changes as another changes. The horizontal axis (x-axis) shows the independent variable; the vertical axis (y-axis) shows the dependent variable. Read axis labels and units carefully. Note whether the relationship is linear (straight line) or nonlinear (curve).
Example 2.10.1
Calculating slope from a mass vs. volume graph
- ProblemA graph of mass vs. volume passes through (0 mL, 0 g) and (50.0 mL, 39.5 g). Calculate the slope.
- Solve
slope = (39.5 − 0) g / (50.0 − 0) mL = 0.790 g/mL
- InterpretThe slope equals the density of the liquid: 0.790 g/mL. Compare to the table to identify it as ethanol.
Answer: slope = 0.790 g/mL = density of the liquid
Interpolation vs. Extrapolation
Interpolation: reading a value from a graph that falls between measured data points. Generally reliable when the trend is clear.
Extrapolation: reading a value beyond the measured data range by extending the line. Less reliable — the trend may not continue the same way. Always treat extrapolated values with caution.
Use the data table from the section above (ethanol samples) to answer these questions.
- Calculate the density for each of the four trials. Are the results consistent? What does this tell you about density as an intensive property?
- What is the average density calculated from the four trials?
- Sketch a properly labeled graph with volume on the x-axis and mass on the y-axis. What does the slope of the best-fit line represent?
- A gas volume vs. temperature graph (in Kelvin) passes through (200 K, 4.00 L) and (400 K, 8.00 L). Calculate the slope. What are its units? What is the physical meaning of this slope?
- A student reads a value at 250 mL from a graph where measurements only went up to 200 mL. Is this interpolation or extrapolation? Should the student trust this value? Explain.
Chapter 2 — Back Matter
Chapter Summary
- Sec 2.1A measurement is a number combined with a unit. Without a unit, a number is meaningless. Chemistry is a quantitative science that connects observable measurements to the invisible world of particles.
- Sec 2.2The SI system provides international standard units. Metric prefixes scale any base unit by powers of ten, allowing chemists to work from nanometers to kilometers within one coherent system.
- Sec 2.3The five core chemistry quantities are length (m), mass (g/kg), volume (L/mL), temperature (°C or K), and time (s). Mass differs from weight. Kelvin is required for gas laws. 1 mL = 1 cm³. Read the meniscus at its bottom.
- Sec 2.4Scientific notation expresses any number as N × 10ⁿ where 1 ≤ N < 10. Multiply: multiply coefficients, add exponents. Divide: divide coefficients, subtract exponents. Add/subtract: match exponents first.
- Sec 2.5Accuracy is closeness to the true value. Precision is consistency among repeated measurements. Systematic errors shift all results in one direction; random errors scatter results unpredictably. Reduce error by calibrating, repeating, and averaging.
- Sec 2.6Significant figures communicate measurement precision. Six rules determine which digits are significant. For multiplication/division, round to the fewest sig figs. For addition/subtraction, round to the fewest decimal places. Exact numbers have unlimited sig figs.
- Sec 2.7Round down when the dropped digit is < 5; round up when it is ≥ 5. In multi-step calculations, round only the final answer. A reported measurement must include a value, a unit, and appropriate precision.
- Sec 2.8Dimensional analysis converts units by multiplying by conversion factors (fractions = 1). Place the unwanted unit in the denominator so it cancels. Chain multiple factors for multi-step conversions. Units cancel algebraically.
- Sec 2.9Density = m/V. It is an intensive property — independent of sample size. Use density to identify substances, to find mass or volume, and to predict floating or sinking. Volume of irregular solids is measured by water displacement.
- Sec 2.10Data tables organize measurements by row and column; read headings and units carefully. On a mass vs. volume line graph, the slope equals density. Interpolation (between data points) is reliable; extrapolation (beyond data range) requires caution.
Key Terms Review
accuracy
How close a measured value is to the true or accepted value. Accuracy is about correctness, not consistency.
conversion factor
A fraction formed from two equal quantities in different units, equal to 1, used to convert measurements between units.
density
The ratio of mass to volume (d = m/V). An intensive property — it does not depend on the amount of substance.
derived unit
A unit formed by combining base units, such as g/mL for density or m/s for speed.
dimensional analysis
A method for converting measurements between units by multiplying by appropriate conversion factors so unwanted units cancel.
exact number
A number obtained by counting or defined by a relationship. Exact numbers have unlimited significant figures and never limit calculation results.
extrapolation
Estimating a value beyond the range of measured data. Less reliable than interpolation because the trend may not continue.
intensive property
A property that does not depend on the amount of substance. Density, temperature, and boiling point are intensive properties.
interpolation
Estimating a value between measured data points. Generally reliable when the data trend is clear.
kelvin (K)
The SI base unit of temperature. 0 K = absolute zero = −273.15 °C. Required for gas law calculations.
liter (L)
Common volume unit in chemistry. 1 L = 1,000 mL. 1 mL = 1 cm³.
mass
The measure of the amount of matter in an object. Unlike weight, mass does not depend on gravity.
meniscus
The curved surface of a liquid in a container. For water, the meniscus is concave; volume is read at its lowest point.
metric prefix
A word element added to an SI unit name that scales the unit by a power of ten (e.g., kilo- = 10³, milli- = 10⁻³).
precision
How close repeated measurements are to each other — the reproducibility of a measurement. Not the same as accuracy.
random error
Measurement error that scatters results unpredictably above and below the true value. Reduced by repeating measurements.
scientific notation
A way of expressing any number as N × 10ⁿ where 1 ≤ N < 10 and n is an integer exponent.
SI
The International System of Units (Système International d'Unités). The globally accepted measurement system for science.
significant figures
The meaningful digits in a measured value, reflecting the precision of the measurement. Rules govern which zeros count.
slope
The rate of change on a graph, calculated as Δy/Δx. On a mass vs. volume graph, the slope equals the density of the substance.
systematic error
Measurement error that consistently shifts results in one direction due to a flaw in the instrument or procedure.
volume
The amount of three-dimensional space a substance occupies. Measured in liters or milliliters in chemistry. 1 mL = 1 cm³.
Formula Review
Worked Example Review
Worked Example A
Significant figures in multiplication
- ProblemA rectangle measures 5.34 cm × 2.1 cm. What is the area?
- Count5.34 → 3 sig figs. 2.1 → 2 sig figs. Fewest = 2.
- Solve
5.34 × 2.1 = 11.214 cm² → round to 2 sig figs
Answer: 11 cm²
Worked Example B
Temperature conversion
- ProblemConvert 37.0 °C (normal body temperature) to Kelvin.
- Solve
K = °C + 273.15 = 37.0 + 273.15 = 310.15 K
Answer: 310.2 K (to appropriate precision)
Worked Example C
Scientific notation multiplication
- ProblemCalculate (2.50 × 10³) × (4.00 × 10⁵).
- Solve
Coefficients: 2.50 × 4.00 = 10.0
Exponents: 10³ × 10⁵ = 10⁸
Result: 10.0 × 10⁸ = 1.00 × 10⁹
Answer: 1.00 × 10⁹ (adjust coefficient to keep between 1 and 10)
Worked Example D
Multi-step unit conversion
- ProblemA sample of gas occupies 4.50 L. Convert to milliliters.
- Solve
4.50 L × (1,000 mL / 1 L) = 4,500 mL = 4.50 × 10³ mL
Answer: 4.50 × 10³ mL (3 significant figures preserved)
Measurement Practice
- Write 0.000450 in scientific notation
- Write 67,200 in scientific notation
- Write 0.00000082 in scientific notation
- Write 1,020,000 in scientific notation
- Convert 3.7 × 10⁴ to standard form
- Convert 2.05 × 10⁻³ to standard form
- Convert 9.0 × 10⁶ to standard form
- Convert 1.11 × 10⁻¹ to standard form
- 25.0 °C to Kelvin
- 373 K to Celsius
- 100.0 °C to Fahrenheit
- −40 °C to Kelvin
- 0 K to Celsius
- 98.6 °F to Celsius
Error Analysis
Each problem contains an error. Identify it and write the correct answer.
Problem 1
A student records a mass as "15 g" even though the balance reads to the nearest 0.01 g.
What is wrong with this record? What should have been written instead?
Problem 2
3.5 kg × (1 kg / 1,000 g) = 0.0035 g
The student set up the conversion factor backwards. What is the correct answer? Show the correct setup.
Problem 3
d = V / m = 25.0 mL / 10.5 g = 2.38 mL/g
Identify two errors in this density calculation.
Problem 4
Temperature is 298 °C. Kelvin temperature = 298 − 273 = 25 K.
What went wrong? What is the correct Kelvin temperature?
Problem 5
4.25 × 3.1 = 13.175
Which significant figure rule was violated? What should the reported answer be?
Problem 6
A student reads the volume from the top of the meniscus rather than the bottom in every trial.
Will the recorded volume be too high or too low? Is this systematic error or random error? Explain.
Mixed Review
- Express the density of gold (19.3 g/cm³) in kg/m³. (Hint: convert both g to kg and cm³ to m³ — this is a two-unit conversion.)
- A solution is prepared by dissolving 5.00 g of salt in water to make 250.0 mL of solution. Calculate the concentration in g/mL. Express in scientific notation.
- Which sample has greater mass: 500 mL of water (d = 1.00 g/mL) or 500 mL of ethanol (d = 0.789 g/mL)? Explain without arithmetic — what property of density explains this?
- A student records these repeated mass measurements: 10.45 g, 10.44 g, 10.46 g, 10.45 g. The true mass is 10.50 g. Are these measurements precise? Are they accurate? Explain both answers.
- Calculate the volume occupied by 85.0 g of aluminum (d = 2.70 g/cm³).
- An iron rod has density 7.87 g/cm³ and volume 12.5 cm³. What is its mass?
- Convert 0.500 μg to grams. Express in scientific notation.
- What is the mass of 0.750 L of ethanol (d = 0.789 g/mL)?
- A substance is heated from 25.0 °C to 75.0 °C. What is the change in temperature in Kelvin? What does this tell you about temperature changes in Celsius vs. Kelvin?
- Explain in your own words why the result of a calculation cannot be more precise than the least precise measurement used. Give a numerical example to support your explanation.
Chapter 2 Checkpoint
Chapter 2 Checkpoint
Complete without assistance — assesses all key skills and concepts from Chapter 2
Section 1 — Vocabulary and Concepts
- What is the difference between accuracy and precision? Give a laboratory example of a measurement that is precise but not accurate.
- Explain why 1 mL = 1 cm³. When would you use each unit?
- Why must temperature be expressed in Kelvin (not Celsius) when using gas laws?
- What is an intensive property? Give two examples from this chapter.
Section 2 — SI Units and Scientific Notation
- Write the SI base unit and symbol for: (a) mass, (b) temperature, (c) length, (d) amount of substance.
- Write each in scientific notation: (a) 0.000560, (b) 8,030,000, (c) 0.0000000142
- Calculate and express in proper scientific notation: (4.0 × 10⁵) × (2.5 × 10³)
- Convert 298 K to Celsius and to Fahrenheit.
Section 3 — Significant Figures
- State the number of significant figures: (a) 0.00408 g, (b) 12.500 mL, (c) 3,600 s (no decimal), (d) 3.600 × 10³ s
- Calculate with correct significant figures: (a) 14.35 g + 2.1 g + 0.887 g (b) 9.82 cm × 3.1 cm
Section 4 — Dimensional Analysis and Density
- Convert 4.50 km to centimeters using dimensional analysis. Show all conversion factors.
- A liquid sample has a mass of 64.5 g and occupies 81.8 mL. Calculate its density and identify it using the table in Section 2.9.
- The density of iron is 7.87 g/cm³. What volume does 250.0 g of iron occupy?
- A wood block (d = 0.72 g/cm³) is placed in a container of ethanol (d = 0.789 g/mL). Will it float or sink? Explain.
Section 5 — Error Analysis
- A student calculates density as d = V/m. What conceptual error did the student make? What is the correct formula?
- A student multiplies 3.5 × 2.14 and reports the answer as 7.490. Identify the significant figure error and write the correct answer.
Looking Ahead
Chapter 3: Classification of Matter
Chapter 2 gave you the measurement tools chemists use to describe and quantify matter. Now you are ready to ask: what kinds of matter exist?
Chapter 3 examines how chemists classify matter. Not all matter is the same — some samples are pure substances with a fixed composition, while others are mixtures of two or more substances combined in variable amounts. Among pure substances, some are elements while others are compounds. Mixtures can be uniform throughout or visibly non-uniform. Understanding these categories helps chemists decide how to identify a substance, predict its behavior, and choose how to separate it from other materials.
You will encounter separation methods chemists use to sort matter: filtration, distillation, chromatography, and evaporation. Each method exploits a physical property difference between substances. The measurement skills from this chapter — density, mass, and volume especially — will appear immediately in Chapter 3 as tools for classifying and separating matter.
MGU Library Connection
- Appendix A: Mathematics for Chemistry — scientific notation, unit conversion, solving equations
- Appendix B: Measurement and Laboratory Reference — SI units, metric prefixes, significant figures, lab equipment
- Appendix H: Gas Law and Thermochemistry Formula Sheet — temperature conversion, gas law formulas
- Appendix I: Chemistry Safety Reference — safe handling of glassware, thermometers, and balances
- MGU Mathematics Series — Algebra I, Chapter 1: Variables and Expressions; Chapter 2: Real Numbers
- MGU Dictionary entries: measurement, SI, density, significant figures, dimensional analysis, accuracy, precision