Essentials: Units, Measurement, & Problem Solving – Study Notes

Essentials: Units, Measurement, & Problem Solving

Big Ideas

Chemistry relies on accurate, precise, and reproducible measurements. Understanding units, significant figures, and uncertainty is fundamental to scientific problem solving in chemistry.

• Chemistry as an Experimental Science: Chemistry depends on measurements that must be accurate and precise to ensure reliable results.

• Uncertainty in Measurements: All measurements have some degree of uncertainty, typically reflected in the number of significant figures reported. The last digit in a measurement is always an estimate.

Interpreting and Using Scientific Notation

Scientific notation is a method for expressing very large or very small numbers in a compact form, which is essential in chemistry for clarity and precision.

• Scientific Notation: Numbers are written as the product of a coefficient (between 1 and 10) and a power of ten. For example, 0.00052 is written as $5.2 \times 10^{-4}$.

• Decimal and Scientific Notation: Converting between these forms is necessary for calculations and reporting results.

• Calculator Use: Always enter numbers in scientific notation to avoid errors with very large or small values.

• Example: $3,400,000 = 3.4 \times 10^6$

SI Units and Prefixes

The International System of Units (SI) provides standard units for scientific measurements. Prefixes indicate multiples or fractions of these units.

• Base SI Units: Common units include meter (m) for length, kilogram (kg) for mass, and second (s) for time.

• SI Prefixes: Prefixes such as kilo- (k, $10^3$), centi- (c, $10^{-2}$), and milli- (m, $10^{-3}$) are used to express different scales.

• Converting Prefixes: To convert between units, multiply or divide by the appropriate power of ten.

• Example: $1 \text{ km} = 1000 \text{ m}$

Temperature Scales

Temperature can be measured in Celsius, Kelvin, or Fahrenheit. Chemistry commonly uses Celsius and Kelvin.

• Celsius (°C): Water freezes at 0°C and boils at 100°C.

• Kelvin (K): The SI unit for temperature. $0 \text{ K}$ is absolute zero.

• Conversion Formulas:

  • $K = ^\circ C + 273.15$

  • $^\circ C = K - 273.15$

• Example: $25^\circ C = 298.15 \text{ K}$

Accuracy, Precision, and Significant Figures

Understanding the difference between accuracy and precision is crucial for evaluating measurements. Significant figures communicate the certainty of a measurement.

• Accuracy: How close a measurement is to the true value.

• Precision: How close repeated measurements are to each other.

• Significant Figures: Digits in a measurement that are known with certainty plus one estimated digit.

• Rules for Counting Significant Figures:

  • All nonzero digits are significant.

  • Zeros between nonzero digits are significant.

  • Leading zeros are not significant.

  • Trailing zeros are significant only if there is a decimal point.

• Example: 0.00450 has three significant figures.

Uncertainty and Reporting Measurements

All measurements have uncertainty, which is reflected in the number of significant figures reported. Calculations must account for this uncertainty.

• Exact Numbers: Have no uncertainty (e.g., counting numbers, defined quantities).

• Measured Numbers: Have uncertainty; report using the correct number of significant figures.

• Rounding Rules: Round to the correct number of significant figures based on the operation performed.

• Example: $2.345 \times 1.2 = 2.8$ (rounded to two significant figures)

Density and Volume

Density is a physical property that relates the mass and volume of a substance. It is useful for identifying substances and solving problems involving mass and volume.

• Density Formula:

  • $\text{Density} = \frac{\text{Mass}}{\text{Volume}}$

• Units: Commonly expressed in $\text{g/cm}^3$ or $\text{g/mL}$ for solids and liquids.

• Applications: Predicting whether a substance will float or sink in another substance.

• Example: If a block has a mass of 10 g and a volume of 2 cm3, its density is $5 \text{ g/cm}^3$.

Unit Conversions and Dimensional Analysis

Unit conversions are essential for solving problems in chemistry. Dimensional analysis uses conversion factors to change from one unit to another.

• Conversion Factor: A ratio that expresses how many of one unit are equal to another unit.

• Dimensional Analysis: Multiply by conversion factors so that units cancel appropriately.

• Example: Convert 5.0 km to meters: $5.0 \text{ km} \times \frac{1000 \text{ m}}{1 \text{ km}} = 5000 \text{ m}$

Exponent Rules in Calculations

When performing calculations involving units raised to a power, apply exponent rules to both the numerical value and the unit.

• Example: $(2 \text{ cm})^3 = 8 \text{ cm}^3$

• Application: Used in calculations involving volume, area, or other derived units.

Summary Table: Key Concepts in Measurement

Concept	Definition	Example
Scientific Notation	Expressing numbers as a coefficient times a power of ten	$6.02 \times 10^{23}$
SI Unit	Standard unit of measurement in science	Meter (m), Kilogram (kg)
Significant Figures	Digits that reflect the precision of a measurement	0.00450 (3 significant figures)
Density	Mass per unit volume	$\frac{10 \text{ g}}{2 \text{ cm}^3} = 5 \text{ g/cm}^3$
Conversion Factor	Ratio used to convert between units	$1 \text{ in} = 2.54 \text{ cm}$