Measurement and Problem Solving in Chemistry: Scientific Notation, Significant Figures, and Unit Conversions

Measurement and Problem Solving

Introduction

Accurate measurement and problem solving are foundational skills in chemistry. This section covers scientific notation, significant figures, units of measurement, and strategies for converting between units, all of which are essential for reporting and interpreting scientific data.

Scientific Notation

Structure of Scientific Notation

Scientific notation is a method for expressing very large or very small numbers in a compact form. It consists of two parts:

• Decimal part: A number between 1 and 10.

• Exponential part: 10 raised to an exponent, n.

For example: $1.2 \times 10^{-10}$

Converting Numbers to Scientific Notation

• Move the decimal point to obtain a number between 1 and 10.

• Multiply that number by $10^n$, where n is the number of places the decimal was moved.

• If the decimal point is moved to the left, the exponent is positive.

• If the decimal point is moved to the right, the exponent is negative.

Example:

• $5983 = 5.983 \times 10^3$

• $0.00034 = 3.4 \times 10^{-4}$

Significant Figures

Definition and Identification

Significant figures (sig figs) indicate the precision of a measured quantity. The rules for identifying significant figures are:

• Nonzero digits are always significant.

• Interior zeros (between nonzero digits) are significant.

• Trailing zeros after a decimal point are significant.

• Trailing zeros before a decimal point are significant only if the decimal point is shown.

• Leading zeros (to the left of the first nonzero digit) are not significant.

• Trailing zeros at the end of a number without a decimal point are ambiguous.

Exact Numbers

• Exact numbers have an unlimited number of significant figures.

• Examples include counted objects (e.g., 7 pennies), defined quantities (e.g., 1 inch = 2.54 cm), and integral numbers in equations.

Counting Significant Figures: Examples

• 0.035 → 2 significant figures

• 180.0 → 4 significant figures

• 71.00 → 4 significant figures

• 7.00 × 103 → 3 significant figures

• 1 dozen = 12 (unlimited significant figures)

• 0.00000 → ambiguous

Significant Figures in Calculations

Rounding Rules

• Round only the final answer in multi-step calculations.

• Round down if the last digit dropped is 4 or less; round up if it is 5 or more.

Multiplication and Division Rule

• The result should have the same number of significant figures as the factor with the fewest significant figures.

Example:

• $2.02 \times 89.665 \times 0.10 = 18.13$ (rounded to 2 significant figures: 18)

Addition and Subtraction Rule

• The result should have the same number of decimal places as the quantity with the fewest decimal places.

Example:

• $5.892 + 6.10 = 12.00$ (rounded to 2 decimal places)

Mixed Operations

• Perform steps in parentheses first, determine significant figures for intermediate results, and round only the final answer.

• Example: $3.489 \times (5.67 - 2.3) = 3.489 \times 3.4 = 12$ (rounded to 2 significant figures)

Units of Measurement

SI Base Units

The International System of Units (SI) is the standard for scientific measurements. The main SI base units are:

Quantity	Unit	Symbol
Length	meter	m
Mass	kilogram	kg
Time	second	s
Temperature	kelvin	K

Mass vs. Weight

• Mass: Measure of the quantity of matter in an object; independent of gravity.

• Weight: Measure of the gravitational pull on an object; depends on gravity.

SI Prefix Multipliers

Prefix multipliers are used to express units in powers of ten.

Symbol	Meaning	Multiplier
T	trillion	1,000,000,000,000
G	billion	1,000,000,000
M	million	1,000,000
k	thousand	1,000
h	hundred	100
da	ten	10
d	tenth	0.1
c	hundredth	0.01
m	thousandth	0.001
μ	millionth	0.000001
n	billionth	0.000000001
p	trillionth	0.000000000001
f	quadrillionth	0.000000000000001

Derived Units

• Derived units are formed from combinations of base units.

• Volume is a derived unit: $1 \text{ m}^3$, $1 \text{ cm}^3$, $1 \text{ mm}^3$.

• Volume is often measured in liters (L) or milliliters (mL).

Dimensional Analysis and Unit Conversion

Dimensional Analysis

• Units are treated as algebraic quantities and can be multiplied, divided, and canceled.

• Always write numbers with their units and include units in calculations.

• Conversion problems use the format: Given unit × Conversion factor = Desired unit

Solution Maps

• A solution map is a visual outline of the steps required to solve a problem, focusing on unit conversions.

• Each step should have a conversion factor with the previous unit in the denominator and the next unit in the numerator.

Converting Units Raised to a Power

• When converting units raised to a power, the conversion factor must also be raised to that power.

• Example: $1 \text{ in}^3 = (2.54 \text{ cm})^3 = 16.387 \text{ cm}^3$

Density

Definition and Calculation

• Density is the ratio of mass to volume.

• Formula: $d = \frac{m}{V}$

• Common units: g/cm3 or g/mL

Example: A sample with mass 27.2 g and volume 2.5 mL has density: $d = \frac{27.2 \text{ g}}{2.5 \text{ mL}} = 10.88 \text{ g/mL}$

Density as a Conversion Factor

• Density can be used to convert between mass and volume.

• Example: For a liquid with density 1.32 g/mL, to find the volume for 68.4 g: $V = \frac{68.4 \text{ g}}{1.32 \text{ g/mL}} = 51.8 \text{ mL}$

Densities of Common Substances

Substance	Density (g/cm3)
Charcoal, oak	0.57
Ethanol	0.789
Aluminum	2.7
Titanium	4.50
Iron	7.86
Copper	8.96
Lead	11.4
Gold	19.3
Platinum	21.4

General Problem-Solving Strategy

• Sort: Organize the information given in the problem.

• Strategize: Create a solution map to outline the steps needed.

• Solve: Perform calculations, paying attention to significant figures and units.

• Check: Ensure the answer makes physical sense and the units are correct.

Chapter Review

• Measured quantities have associated units.

• SI units: meter (length), kilogram (mass), second (time).

• Prefix multipliers (e.g., kilo-, milli-) are used with base units.

• Volume is a derived unit (length cubed); liters and milliliters are common.

• Density is mass divided by volume: $d = \frac{m}{V}$, usually in g/cm3 or g/mL.

• Density is a fundamental property and varies between substances.

Additional info: Some content was inferred and expanded for clarity and completeness, including the full SI prefix table and density calculation examples.