Chapter E: Essentials—Units, Measurement, and Problem Solving (General Chemistry Study Notes)

Chapter E: Essentials—Units, Measurement, and Problem Solving

Introduction

This chapter covers the foundational concepts of measurement in chemistry, including the use of scientific notation, units of measurement, significant figures, unit conversions, and key definitions. Mastery of these topics is essential for accurate problem solving in general chemistry.

Scientific Notation

Definition and Usage

• Scientific notation is a method of expressing very large or very small numbers in the form a × 10n, where a is a number between 1 and 10, and n is an integer.

• Examples:

  • $1.50 \times 10^2$

  • $0.0100 = 1.00 \times 10^{-2}$

Standard Units of Measurement

SI Base Units

• Mass: kilogram (kg)

• Temperature: kelvin (K)

• Time: second (s)

• Length: meter (m)

• Amount of substance: mole (mol)

• Volume: cubic centimeter (cm3), milliliter (mL), or liter (L)

• Density: grams per cubic centimeter (g/cm3) or grams per milliliter (g/mL)

Prefix Multipliers

Common SI Prefixes

Prefix multipliers are used to express units that are much larger or smaller than the base unit. The following table summarizes common prefixes:

Prefix	Symbol	Multiplier
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)

Temperature Conversions

Celsius and Kelvin Scales

• To convert Celsius to Kelvin: $K = ^\circ C + 273$

• To convert Kelvin to Celsius: $^\circ C = K - 273$

• 0°C = 273 K and 100°C = 373 K

• A change of 10 K is the same as a change of 10°C

• 0°C to 100°C is the same change as 273 K to 373 K

Significant Figures

Rules for Counting Significant Figures

• All nonzero digits are significant.

  • Example: 1.234 has 4 significant figures.

• Zeros between nonzero digits are significant.

  • Example: 1.024 has 4 significant figures.

• Zeros to the left of the first nonzero digit are never significant.

  • Example: 0.003 has 1 significant figure.

• Zeros to the right of the decimal point and after nonzero digits are significant.

  • Example: 0.0200 has 3 significant figures.

• Zeros to the right of a nonzero digit in a whole number may or may not be significant, depending on context.

  • Example: 22000 may have 2, 3, 4, or 5 significant figures depending on the measurement's precision.

Significant Figures in Calculations

• When adding or subtracting, the result should have the same number of decimal places as the measurement with the fewest decimal places.

• When multiplying or dividing, the result should have the same number of significant figures as the measurement with the fewest significant figures.

• When rounding, if the first digit to be dropped is 5 or greater, round up by 1.

• Always check the question to see if a specific number of significant figures is required.

Measurement and Uncertainty

Exact vs. Measured Numbers

• Exact numbers have no uncertainty and are counted or defined values (e.g., 12 students, 1 dozen = 12).

• Measured numbers are obtained using instruments and have uncertainty, which is reflected in significant figures.

Unit Conversions and Conversion Factors

General Approach

• Use conversion factors to change from one unit to another.

• General formula for a one-step unit conversion:

  • $\text{initial unit} \times \frac{\text{desired unit}}{\text{initial unit}} = \text{desired unit}$

• Example: Convert 657 cm to meters:

  • $657 \text{ cm} \times \frac{1 \text{ m}}{100 \text{ cm}} = 6.57 \text{ m}$

• Multiple factors can be used in a single calculation. For example, converting grams to volume using density:

• Example:

  • Given: 0.035 kg methanol, density = 0.792 g/mL

  • Convert to mL:

    • $0.035 \text{ kg methanol} \times \frac{1000 \text{ g}}{1 \text{ kg}} \times \frac{1 \text{ mL methanol}}{0.792 \text{ g methanol}} = 44 \text{ mL methanol}$

Important Equations

• Density: $\text{density} = \frac{\text{mass}}{\text{volume}}$ or $\rho = \frac{m}{V}$

Key Definitions

• Accuracy: The closeness of a measured value to the true or accepted value.

• Precision: The closeness of repeated measurements to each other, regardless of their accuracy.

• Density: The mass per unit volume of a substance, typically expressed in g/cm3 or g/mL.

• Conversion factor: A ratio used to convert from one unit to another, derived from the equivalence between different units.