Measurement and Problem Solving: Scientific Notation, Significant Figures, and Units

Measurement and Problem Solving

Scientific Notation

Scientific notation is a method used to express very large or very small numbers in a concise format, making calculations easier and reducing errors. It is commonly used in chemistry to represent quantities such as atomic sizes or astronomical distances.

• Format: A number is written as the product of a coefficient (between 1 and 10) and a power of ten. For example, 94,500,000 miles becomes $9.45 \times 10^7$ miles.

• Examples:

  • Distance from Earth to Sun: $94,500,000$ miles = $9.45 \times 10^7$ miles

  • Size of a hydrogen atom: $0.000000000053$ meters = $5.3 \times 10^{-11}$ meters

• Conversion Practice: Numbers such as 582, 0.076, 4,700.0, and 25,800,000.0 should be written in scientific notation without changing the number of digits.

• Important Note: Never place a dash (-) in front of a number unless indicating a negative value.

Measurement and Uncertainty

Measurements in chemistry are fundamental to experimentation and problem solving. Every measurement consists of two parts: a numerical value and a unit. Measurements are never exact and always have some degree of uncertainty.

• Numerical Value: Indicates the magnitude of the measurement.

• Unit: Specifies what is being measured (e.g., kilograms, meters).

• Uncertainty: All measurements have uncertainty due to limitations in the measuring instrument and estimation.

• Precision: The place of uncertainty indicates the precision of the measurement.

• Example: Is 8 inches equal to 8.0 inches? No, because 8.0 inches is more precise.

Recording Measurements

When recording a measurement, include all digits known with certainty plus one digit that is estimated. These digits are called significant figures.

• Significant Figures: Digits in a measured number that are known for certain, plus one estimated digit.

• Example: If an object extends past 60 cm but not quite to 70 cm, estimate the ones place for the uncertain digit.

Significant Figures: Rules

Significant figures are essential for expressing the precision of measurements. The following rules help identify significant figures in a number:

• All nonzero digits are significant.

• Interior zeros (between two numbers) are significant.

• Trailing zeros after a decimal point are significant.

• Trailing zeros before a decimal point are significant.

• Leading zeros (before the first nonzero digit) are not significant.

• Trailing zeros at the end of a number, but before an implied decimal point, are ambiguous and should be avoided.

Identifying Exact Numbers

Exact numbers have an unlimited number of significant figures. They include:

• Counting discrete objects (e.g., 10 pencils).

• Integral numbers in equations (e.g., radius = diameter/2).

• Defined quantities (e.g., 1 in = 2.54 cm).

Significant Figures in Scientific Notation

When converting a number to scientific notation, the number of significant figures remains unchanged.

• Example: $6.023 \times 10^{23}$ has 4 significant figures.

• Example: $1.25 \times 10^{-3}$ has 3 significant figures.

Significant Figures in Calculations

When performing calculations, the result must reflect the precision of the data. Rounding rules and significant figure rules differ for multiplication/division and addition/subtraction.

• Rounding: Round only the final answer, not intermediate steps.

• Multiplication/Division: The result has the same number of significant figures as the factor with the fewest significant figures.

• Addition/Subtraction: The result has the same number of decimal places as the quantity with the fewest decimal places.

Addition/Subtraction with Scientific Notation

When adding or subtracting numbers in scientific notation:

• Align exponents (rewrite numbers to share the same power of 10).

• Add or subtract the decimal numbers as usual.

• Identify the least precise number (fewest decimal places).

• Round the result to match the decimal place of the least precise number.

• Verify the final answer is in proper scientific notation.

Complex Calculations: Order of Operations

For calculations involving both multiplication/division and addition/subtraction, follow the standard order of operations (PEMDAS): Parentheses, Exponents, Multiplication/Division, Addition/Subtraction. Determine significant figures for intermediate answers without rounding until the final step.

Logs and Antilogs

When taking logarithms or antilogarithms:

• Logarithms: The number of significant figures in the original number determines the number of decimal places in the result.

• Antilogs: The number of decimal places in the original number determines the number of significant figures in the result.

• Example: $\log(3.245 \times 10^{-9})$ (4 significant figures) yields a result with 4 decimal places: -8.4887.

• Example: $\text{antilog}(-3.45)$ (2 decimal places) yields a result with 2 significant figures: $3.5 \times 10^{-4}$.

The Basic Units of Measurement

International System of Units (SI)

The SI system is the standard for scientific measurements. It is based on the metric system and includes base units for length, mass, volume, and other quantities.

Metric Units of Length

The metric system uses multiples of ten for easy conversion between units. Common units include meter (m), centimeter (cm), millimeter (mm), and kilometer (km).

Unit	Abbreviation	Meter equivalent	Exponential equivalent
Kilometer	km	1000 m	$10^3$ m
Meter	m	1 m	$10^0$ m
Decimeter	dm	0.1 m	$10^{-1}$ m
Centimeter	cm	0.01 m	$10^{-2}$ m
Millimeter	mm	0.001 m	$10^{-3}$ m
Micrometer	μm	0.000001 m	$10^{-6}$ m
Nanometer	nm	0.000000001 m	$10^{-9}$ m
Angstrom	Å	0.0000000001 m	$10^{-10}$ m

Metric Units of Mass

Unit	Abbreviation	Gram equivalent	Exponential equivalent
Kilogram	kg	1000 g	$10^3$ g
Gram	g	1 g	$10^0$ g
Decigram	dg	0.1 g	$10^{-1}$ g
Centigram	cg	0.01 g	$10^{-2}$ g
Milligram	mg	0.001 g	$10^{-3}$ g
Microgram	μg	0.000001 g	$10^{-6}$ g

Metric Units of Volume

The SI unit for volume is the cubic meter ($m^3$), but for liquids and gases, the liter (L) is more commonly used.

• 1 L = 1000 cm3 = 1 dm3

• 1 mL = 1 cm3

Metric Conversions

Conversions between metric units are based on powers of ten. Moving from a larger unit to a smaller unit, multiply; from a smaller unit to a larger unit, divide.

Problem Solving and Dimensional Analysis

Unit Conversion Problems

Unit conversion is a fundamental skill in chemistry. Conversion factors are used to relate different units and solve problems such as "How many days in 3 weeks?" or "How many inches in 4 feet?"

• Conversion Factors: Equalities such as 2.54 cm = 1 in, 12 in = 1 foot, 5280 ft = 1 mile.

• Dimensional Analysis: Use conversion factors to cancel units and solve for the desired quantity.

• Example: To convert the length of a marathon (26.218 miles) to meters, use the appropriate conversion factors.

Density

Definition and Calculation

Density is a physical property defined as mass per unit volume. It is temperature dependent because volume changes with temperature.

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

• Example: If a metal has a mass of 427 g and a volume of 35.0 mL, its density is $\frac{427\,\text{g}}{35.0\,\text{mL}} = 12.2\,\text{g/mL}$.

• Density as a Conversion Factor: The density of gold is 19.3 g/cm3. A 25.0 cm3 volume of gold has a mass of $19.3 \times 25.0 = 482.5$ g.

Additional info: Density is often used to identify substances and to convert between mass and volume in laboratory calculations.