Chemistry and Measurements: Scientific Notation, Significant Figures, and Unit Conversions

Chemistry and Measurements

Scientific Notation

Scientific notation is a method used to express very large or very small numbers in a concise format, making them easier to read and write in chemistry.

• Format: A number is written as coefficient × 10n, where the coefficient is between 1 and 10, and n is an integer (positive or negative).

• Base: Always 10.

• Exponent (n): Indicates how many places the decimal point is moved. Positive n means a large number; negative n means a small number.

• Examples:

  • $2.5 \times 10^3 = 2,500$

  • $1.34 \times 10^{-2} = 0.0134$

  • $-7.8 \times 10^6 = -7,800,000$

  • $2.0 \times 10^{-4} = 0.0002$

• Standard Notation: The usual way of writing numbers without exponents.

Additional info: Scientific notation is especially useful for expressing measurements in chemistry, such as Avogadro's number or atomic masses.

Measurements and Units

Measurements in chemistry require both a number and a unit. The unit indicates the scale or standard being used.

• Volume: The amount of space an object occupies (e.g., liters, milliliters).

• Length: The distance between two points (e.g., meters, centimeters, millimeters).

• Mass: The amount of matter in an object (e.g., grams, kilograms, milligrams).

• Temperature: Measured in degrees Celsius (°C), Kelvin (K), or Fahrenheit (°F).

• Time: Measured in seconds (s), minutes (min), hours (hr), etc.

Example: Measuring the length of a table as 1.1 meters.

Measured vs. Exact Numbers

Numbers in chemistry can be classified as measured or exact, which affects how calculations are performed.

• Exact Numbers: Values known with complete certainty, often from counting or defined relationships (e.g., 1 inch = 2.54 cm).

• Measured Numbers: Obtained using measuring tools; always involve some uncertainty.

• Example: Counting 10 test tubes (exact); measuring 1.1 meters (measured).

Significant Figures (Sig Figs)

Significant figures reflect the precision of a measured number. The rules for determining significant figures are essential for reporting scientific data accurately.

• All nonzero digits are significant. (e.g., 2.2 has two sig figs)

• Zeros between nonzero digits are significant. (e.g., 507 has three sig figs)

• Leading zeros (at the beginning) are not significant. (e.g., 0.003 has one sig fig)

• Trailing zeros:

  • If there is a decimal point, trailing zeros are significant. (e.g., 5000. has four sig figs)

  • If there is no decimal point, trailing zeros are not significant. (e.g., 5000 has one sig fig)

• In scientific notation, only the coefficient determines the number of significant figures. (e.g., $3.20 \times 10^{-9}$ has three sig figs)

Examples:

• 0.000880 kg: 3 significant figures

• 2.0 × 10-3 m: 2 significant figures

• 28.80: 4 significant figures

Rounding Rules

When rounding numbers to a certain number of significant figures:

• If the first digit to be dropped is 5 or greater, round up.

• If the first digit to be dropped is less than 5, drop it without rounding up.

Examples:

• 19.7616 rounded to five sig figs: 19.762

• 19.7616 rounded to two sig figs: 20 (must show as 20. to indicate two sig figs)

• 35.7823 rounded to three sig figs: 35.8

Significant Figures in Calculations

The number of significant figures in the final answer depends on the operation performed:

• Multiplication/Division: The result should have the same number of significant figures as the measurement with the fewest significant figures.

• Addition/Subtraction: The result should have the same number of decimal places as the measurement with the fewest decimal places.

Examples:

• 6.8 × 0.37 = 2.5 (rounded to two sig figs)

• 104 + 7.8 = 112 (rounded to zero decimal places)

• 153.247 - 14.8 = 138.4 (rounded to one decimal place)

SI Prefixes and Scientific Notation

SI prefixes are used to indicate multiples or fractions of units. Knowing these prefixes and their corresponding powers of ten is essential for unit conversions.

Prefix	Symbol	Power of Ten
peta	P	$10^{15}$
tera	T	$10^{12}$
giga	G	$10^{9}$
mega	M	$10^{6}$
kilo	k	$10^{3}$
deci	d	$10^{-1}$
centi	c	$10^{-2}$
milli	m	$10^{-3}$
micro	\mu	$10^{-6}$
nano	n	$10^{-9}$
pico	p	$10^{-12}$
femto	f	$10^{-15}$

Example: 1 kilometer (km) = $10^3$ meters (m)

Unit Conversions and Conversion Factors

Unit conversions are performed using conversion factors, which are ratios that express how many of one unit are equal to another unit.

• Conversion Factor: A ratio derived from the equality between two different units (e.g., 1 inch = 2.54 cm).

• Dimensional Analysis: A method to convert units by multiplying by appropriate conversion factors.

Examples:

• 1 inch = 2.54 cm

• 1 meter = 100 centimeters

• 1 kilogram = 1000 grams

Additional info: Always match the units so that unwanted units cancel, leaving the desired unit.

Practice Problems and Applications

• Convert 65 miles per hour to kilometers per hour using appropriate conversion factors.

• Calculate the density of water: $\text{Density} = \frac{\text{mass}}{\text{volume}}$

• Determine body fat percentage if body fat is 35% of total body weight.