Matter and Measurements: Mathematical Concepts, Scientific Notation, and Metric Conversions

Matter and Measurements

Mathematical Concepts in Chemistry

Understanding basic mathematical operations is essential for solving chemical problems, including calculations involving measurements, conversions, and chemical quantities. This section reviews positive and negative numbers, orders of operations, and algebraic manipulation as applied in chemistry.

• Positive Numbers: Numbers greater than zero, often written with a positive sign (+), though the sign is usually omitted. Examples: 1, 3, 8, +100

• Negative Numbers: Numbers less than zero, written with a negative sign (−). Examples: −1, −3, −8, −100

• Multiplication and Division: Multiplying or dividing two positive or two negative numbers yields a positive result. Multiplying or dividing a positive and a negative number yields a negative result. Examples: $2 \times 3 = 6$ $-2 \times 3 = -6$ $6 \div 3 = 2$ $-6 \div 3 = -2$

• Addition and Subtraction: Adding two positive numbers or two negative numbers yields a result with the same sign. Adding a positive and a negative number yields a result with the sign of the larger absolute value. Examples: $3 + 4 = 7$ $9 + 3 = 12$ $5 + (-3) = 2$ $12 + (-15) = -3$ $12 - (-5) = 17$

Algebraic Manipulation

Equations can be rearranged to solve for unknown variables, a skill necessary for solving chemical equations and conversions.

• Isolating Variables: Move all terms involving the variable to one side and constants to the other. Example: Solve $2x + 8 = 14$ Subtract 8 from both sides: $2x = 6$ Divide both sides by 2: $x = 3$

• Solving for a Variable in a Formula: Example: $P_1 V_1 = P_2 V_2$ Divide both sides by $V_1$: $P_1 = \frac{P_2 V_2}{V_1}$

Mathematical Operations and Functions

Scientific Notation

Scientific notation is used to express very large or very small numbers in chemistry. It consists of a coefficient (between 1 and 9.999...) and a power of 10.

• Format: $a \times 10^n$ where $a$ is the coefficient and $n$ is the exponent.

• Example: $2,400 = 2.4 \times 10^3$

• Example: $0.00086 = 8.6 \times 10^{-4}$

• Steps to Convert:

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

  2. Express the number of places moved as the power of 10.

  3. Write the product of the coefficient and the power of 10.

Table: Positive Powers of 10

Number	Multiples of 10	Scientific Notation
1,000	$10 \times 10 \times 10$	$1 \times 10^3$
100	$10 \times 10$	$1 \times 10^2$
10	$10$	$1 \times 10^1$

Table: Negative Powers of 10

Number	Multiples of 10	Scientific Notation
0.1	$\frac{1}{10}$	$1 \times 10^{-1}$
0.01	$\frac{1}{10 \times 10}$	$1 \times 10^{-2}$
0.001	$\frac{1}{10 \times 10 \times 10}$	$1 \times 10^{-3}$

Application: Scientific notation is essential for expressing measurements such as the diameter of the Earth ($1.28 \times 10^7$ m) or the mass of a bacterium ($1 \times 10^{-13}$ g).

Metric System and Conversion Factors

International System of Units (SI)

The metric system is the standard system of measurement in science, using SI units for length, mass, temperature, and time.

• Base Units:

  • Length: meter (m)

  • Mass: kilogram (kg)

  • Temperature: Kelvin (K)

  • Time: second (s)

• Prefixes: Used to indicate multiples or fractions of base units. Examples: kilo- ($10^3$), centi- ($10^{-2}$), milli- ($10^{-3}$), micro- ($10^{-6}$), nano- ($10^{-9}$)

Table: Metric Prefixes

Prefix	Symbol	Value	Scientific Notation
kilo	k	1,000	$1 \times 10^3$
centi	c	0.01	$1 \times 10^{-2}$
milli	m	0.001	$1 \times 10^{-3}$
micro	μ	0.000001	$1 \times 10^{-6}$
nano	n	0.000000001	$1 \times 10^{-9}$

Conversion Factors and Dimensional Analysis

Conversion factors are ratios that express the relationship between two units and are used to convert measurements from one unit to another.

• Equality and Conversion: $1 \text{ km} = 1000 \text{ m}$ can be written as $\frac{1 \text{ km}}{1000 \text{ m}}$ or $\frac{1000 \text{ m}}{1 \text{ km}}$ depending on which unit needs to be cancelled.

• Example: Convert 5.3 km to meters: $5.3 \text{ km} \times \frac{1000 \text{ m}}{1 \text{ km}} = 5300 \text{ m}$

• Example: Convert 240 mL to liters: $240 \text{ mL} \times \frac{1 \text{ L}}{1000 \text{ mL}} = 0.240 \text{ L}$

Accuracy and Precision

Accuracy and precision are important concepts in measurement.

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

• Precision: How reproducible measurements are when repeated under unchanged conditions.

• Significant Figures: Used to communicate the certainty of measurements.

• Visual Example: A target diagram can illustrate good accuracy and precision (points close to the center and to each other), poor accuracy (points far from the center), and poor precision (points far from each other).

Practice Problems and Applications

• Scientific Notation: Write 432 as $4.32 \times 10^2$; 0.0000007934 as $7.934 \times 10^{-7}$

• Metric Conversion: Convert 3.2 x $10^9$ to decimal: 3,200,000,000

• Dimensional Analysis: Always check which units need to cancel and ensure the answer makes sense in context.

Additional info: These foundational mathematical and measurement concepts are essential for all subsequent topics in GOB Chemistry, including chemical reactions, solution calculations, and laboratory procedures.