Measurement, Significant Figures, Metric System, and Dimensional Analysis in General Chemistry

Measurement and Significant Figures

Significant Figures: Definition and Importance

Significant figures are the digits in a measurement that carry meaning regarding its precision. They help communicate the certainty of a measurement and are essential in reporting scientific data accurately.

• All nonzero digits are significant.

• Zeros between nonzero digits are significant.

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

• Trailing zeros are not significant unless they are followed by a decimal point.

• Exact numbers (such as counted items or defined quantities) have unlimited significant figures.

Example: 60,000 meters has one significant figure; 6.0000 meters has five significant figures.

Rules for Mathematical Operations

When performing calculations, the number of significant figures in the result depends on the operation:

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

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

Example: $13.3806$ (result rounded according to significant figure rules).

Scientific Notation

Purpose and Application

Scientific notation is a method used to express very large or very small numbers in a compact form using powers of ten. It is commonly used in chemistry to simplify calculations and to clearly indicate the precision of measurements.

• Scientific notation replaces metric prefixes or removes leading zeros.

• Example: 60,000 meters = $6 \times 10^4$ meters; 0.0000589 = $5.89 \times 10^{-5}$.

Accuracy vs. Precision

Definitions and Differences

Accuracy and precision are terms used to describe the quality of measurements:

• Accuracy: How close a measurement is to the true or accepted value.

• Precision: How close repeated measurements are to each other.

Example: If the true value is 98 cm, measurements of 100 cm, 95 cm, and 97 cm show varying accuracy. Measurements of 98 mL, 96 mL, 100 mL, and 99 mL show precision if they are close together, regardless of their accuracy.

Note: It is possible to have precision without accuracy, and vice versa.

The Metric System

Base Units

The metric system is the standard system of measurement in science. It uses base units for different physical quantities:

Quantity	Unit	Symbol
Mass	Kilogram	kg
Length	Meter	m
Volume	Liter	L
Time	Second	s
Temperature	Celsius/Kelvin	°C/K

Temperature Conversion:

$\text{Kelvin} = \text{°C} + 273.15$

Metric Prefixes

Metric prefixes are used to indicate multiples or fractions of base units:

Prefix	Notation	Symbol
Kilo	1,000 or $10^3$	k
Centi	0.01 or $10^{-2}$	c
Milli	0.001 or $10^{-3}$	m
Micro	0.000001 or $10^{-6}$	μ
Nano	0.000000001 or $10^{-9}$	n
Pico	0.000000000001 or $10^{-12}$	p
Angstrom	0.0000000001 or $10^{-10}$	Å

Dimensional Analysis

Unit Conversion Method

Dimensional analysis is a systematic method for converting one unit of measurement to another using conversion factors.

General Formula:

$A \times \frac{B}{A} \times \frac{C}{B} \times \frac{D}{C} \ldots$

Example: Converting 600 microliters ($\mu$L) to milliliters (mL):

$600 \mu L \times \frac{1\,mL}{1,000\,\mu L} = 0.6\,mL$

Density

Definition and Calculation

Density is a physical property that relates an object's mass to its volume. It is commonly used to identify substances and assess purity.

• Mass: The amount of matter within an object (measured in grams, kilograms, etc.).

• Volume: The total amount of space an object occupies (measured in liters, cubic centimeters, etc.).

Formula:

$\text{Density} = \frac{\text{Mass}}{\text{Volume}}$

Example: If a cube has a mass of 3.5176 g and a length of 2.0 cm per side, its volume is $2.0\,cm \times 2.0\,cm \times 2.0\,cm = 8.0\,cm^3$, so its density is $\frac{3.5176\,g}{8.0\,cm^3} = 0.4397\,g/cm^3$.

Additional info: Volume of a cube is calculated as $\text{Length} \times \text{Width} \times \text{Height}$.