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 format, making calculations and comparisons easier in chemistry.

• Decimal Part: A number between 1 and 10.

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

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

• A positive exponent means multiplying 1 by 10 n times: $10^3 = 1 \times 10 \times 10 \times 10 = 1000$

• A negative exponent means dividing 1 by 10 n times: $10^{-2} = \frac{1}{10 \times 10} = 0.01$

Converting Numbers to Scientific Notation

• Determine the decimal part and the exponent.

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

• Multiply by $10^n$ to compensate for the movement.

Example: $5983 = 5.983 \times 10^3$; $0.00034 = 3.4 \times 10^{-4}$

• Moving the decimal left: positive exponent.

• Moving the decimal right: negative exponent.

Uncertainty in Measurement

All measurements in science have some degree of uncertainty, which is reflected in how numbers are reported.

• Precision: More digits indicate greater precision.

• Uncertainty: The last reported digit is uncertain.

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

Significant Figures

Significant figures indicate the precision of a measurement. The rules for identifying 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, are ambiguous.

Exact numbers (e.g., counted objects, defined quantities) have an unlimited number of significant figures.

Examples of Significant Figures

Number	Significant Figures
0.0035	2
1.080	4
2371	4
2.97 × 105	3
1 dozen = 12	Unlimited
100.00	5
100,000	Ambiguous

Significant Figures in Calculations

Rounding Rules

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

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

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

Multiplication and Division Rule

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

Example: $5.02 \times 89.667 \times 0.10 = 45.0118 \rightarrow 45$ (2 significant figures)

Addition and Subtraction Rule

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

Example: $5.74 + 0.823 + 2.651 = 9.214 \rightarrow 9.21$ (2 decimal places)

Mixed Operations

• Do parentheses first, determine significant figures for intermediate results, then complete remaining steps.

Example: $3.489 \times (5.67 - 2.3) = 3.489 \times 3.37 = 11.758 \rightarrow 12$ (2 significant figures)

SI Units and Prefix Multipliers

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

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.

• Weight: Gravitational pull on that matter; 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	$10^3$
centi-	c	hundredth	$10^{-2}$
milli-	m	thousandth	$10^{-3}$
micro-	μ	millionth	$10^{-6}$
nano-	n	billionth	$10^{-9}$
pico-	p	trillionth	$10^{-12}$

Additional info:

• These notes cover the essential concepts of measurement, scientific notation, significant figures, and SI units, which are foundational for all subsequent topics in introductory chemistry.