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

Measurement and Problem Solving

Scientific Notation

Scientific notation is a method used to express very large or very small numbers in a concise form, which is especially useful in chemistry for reporting measurements and calculations.

• Decimal Part: A number between 1 and 10.

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

Example: $1.2 \times 10^{-10}$

Reading Scientific Notation

• A positive exponent means multiplying 1 by 10 n times.

• A negative exponent means dividing 1 by 10 n times.

Examples:

• $10^0 = 1$

• $10^1 = 10$

• $10^2 = 100$

• $10^3 = 1000$

• $10^{-1} = 0.1$

• $10^{-2} = 0.01$

• $10^{-3} = 0.001$

Converting Numbers to Scientific Notation

1. Determine the decimal part of the number and find the exponent.

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

3. Multiply the decimal part by 10 raised to the power to compensate for moving the decimal point.

Examples:

• $5983 = 5.983 \times 10^3$ (decimal moved left, exponent positive)

• $0.00034 = 3.4 \times 10^{-4}$ (decimal moved right, exponent negative)

Uncertainty in Measurement

Measurements in science are reported to reflect their precision. The more digits reported, the greater the precision. The uncertainty in a measurement is indicated by the last reported digit.

• Precision: More digits indicate higher precision.

• Uncertainty: The last digit is always estimated.

Example: Reporting a temperature increase of 0.6°C means $0.6 \pm 0.1$°C, so the actual increase could be between 0.5°C and 0.7°C.

Significant Figures

Significant figures reflect the precision of a measurement. The rules for determining significant figures are:

1. All nonzero digits are significant.

2. Interior zeros (between nonzero digits) are significant.

3. Trailing zeros after a decimal point are significant.

4. Trailing zeros before a decimal point are significant.

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

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

Examples:

• 0.0035: two significant figures

• 1.080: four significant figures

• 2371: four significant figures

• $2.97 \times 10^5$: three significant figures

• 1 dozen = 12: unlimited significant figures (exact number)

• 100.00: five significant figures

• 100,000: ambiguous

Exact Numbers

Exact numbers have an unlimited number of significant figures. These 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 Calculations

Rounding Rules

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

• Use only the last digit being dropped to decide rounding direction.

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

Examples:

• 2.33 rounds to 2.3 (two significant figures)

• 2.37 rounds to 2.4 (two significant figures)

Multiplication and Division Rule

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

Example:

• $5.02 \times 89.667 \times 0.10 = 45.0118 \rightarrow 45$ (two significant figures)

Addition and Subtraction Rule

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

Example:

• $5.74 + 0.823 + 2.651 = 9.214 \rightarrow 9.21$ (two decimal places)

• $4.8 - 3.965 = 0.835 \rightarrow 0.8$ (one decimal place)

Mixed Operations

• Do steps in parentheses first.

• Determine significant figures in the intermediate answer without rounding.

• Complete the remaining steps and round the final answer.

Example:

• $3.489 \times (5.67 - 2.3) = 3.489 \times 3.37 = 11.758 \rightarrow 12$ (two significant figures)

SI Units and Prefixes

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

Standards of Measurement

• Meter: Distance light travels in vacuum in $1/299,792,458$ seconds.

• Kilogram: Defined using Planck's constant, $6.62607015 \times 10^{-34} \text{kg} \cdot \text{m}^2 \cdot \text{s}^{-1}$.

• Second: Duration of $9,192,631,770$ periods of radiation from cesium-133 atom.

Weight vs. Mass

• Mass: Quantity of matter in an object; does not depend on gravity.

• Weight: Gravitational pull on an object; depends on gravity.

SI Prefix Multipliers

Prefix	Symbol	Meaning	Multiplier
tera-	T	trillion	$10^{12}$
giga-	G	billion	$10^9$
mega-	M	million	$10^6$
kilo-	k	thousand	$10^3$
hecto-	h	hundred	$10^2$
deca-	da	ten	$10^1$
deci-	d	tenth	$10^{-1}$
centi-	c	hundredth	$10^{-2}$
milli-	m	thousandth	$10^{-3}$
micro-	μ	millionth	$10^{-6}$
nano-	n	billionth	$10^{-9}$
pico-	p	trillionth	$10^{-12}$
femto-	f	quadrillionth	$10^{-15}$

Summary

• Scientific notation simplifies the expression of very large and small numbers.

• Significant figures communicate the precision of measurements.

• SI units and prefixes standardize scientific measurements.

• Rules for calculations ensure correct reporting of significant figures.

Additional info: These notes are based on textbook slides and cover the essential concepts of Chapter 2: Measurement and Problem Solving, suitable for introductory college chemistry students.