Mathematical Operations and Functions: Pre-Course Math Review for General Chemistry

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Mathematical Operations and Functions

Algebra: Simplifying Expressions

Algebraic manipulation is essential for solving chemical equations and understanding quantitative relationships in chemistry. Simplifying expressions involves reducing them to their simplest form.

Simplifying Expressions: Combine like terms and use the distributive property to reduce expressions.
Example: Simplify

Simplifying Algebraic Expressions Steps:

Distribute constants/variables into parentheses.
Combine like terms by adding/subtracting.
Combine the terms by adding/subtracting.

Exponents in Expressions

Exponents represent repeated multiplication and are commonly used in chemical calculations, such as determining concentrations and reaction rates.

General Form: where is the base and is the exponent.
Exponent Rules:

Rule	Description	Example
	Add exponents when multiplying like bases
	Subtract exponents when dividing like bases
	Multiply exponents when raising a power to a power
	Distribute exponent to each factor
	Any nonzero number to the zero power is 1
	Negative exponent means reciprocal

Algebra: Solving Equations

Solving equations is fundamental for determining unknowns in chemical problems, such as concentrations or reaction yields.

Use operations (+, -, ×, ÷) to isolate the variable.
Example: Solve

Solving Linear Equations Steps:

Distribute as necessary.
Combine like terms.
Isolate the variable by adding/subtracting.
Check the solution by substituting into the original equation.

Graphing by Plotting Points

Graphing is used to visualize relationships between variables, such as concentration vs. time in kinetics.

Identify x- and y-values.
Calculate y for each x-value.
Plot points and connect them.

Systems of Equations: Solving

Systems of equations are used to solve for multiple unknowns, such as in stoichiometry or equilibrium problems.

Substitution Method: Solve one equation for one variable, substitute into the other, and solve.
Example: Substitute into the second equation:

Slopes of Lines

The slope of a line represents the rate of change, which is important in rate laws and graphical analysis in chemistry.

Formula:
Example: For points (1,2) and (3,6):

Graphing Linear Equations

Linear equations are graphed to show direct relationships between variables.

Identify the y-intercept and slope.
Plot the y-intercept, use the slope to find another point, and draw the line.

Quadratic Equations: Solving

Quadratic equations appear in chemical equilibrium and kinetics problems.

Square Root Property:
Quadratic Formula:

Quadratic Equations: Graphing

Quadratic graphs (parabolas) are used to analyze reaction rates and energy profiles.

Vertex: is the maximum or minimum point.
Axis of symmetry:
Opens upward if , downward if .

Proportional Reasoning

Understanding proportional relationships is crucial for interpreting chemical equations and reaction stoichiometry.

Directly Proportional	Inversely Proportional	Jointly Proportional
As ↑, ↑	As ↑, ↓	For constant , For constant ,

Trigonometry: Sine, Cosine, Tangent

Trigonometric functions are used in molecular geometry and vector analysis in chemistry.

Sine (SOH)	Cosine (CAH)	Tangent (TOA)

Pythagorean Theorem:

Calculus: Derivatives

Derivatives represent instantaneous rates of change, such as reaction rates in chemistry.

Definition: The derivative of a function is its instantaneous rate of change.
Common Derivatives:

Function	Derivative	Example

	$0$

Calculus: Integrals

Integrals are used to calculate areas under curves, such as total concentration over time in kinetics.

Definition: The integral of a function is the area under its curve.
Definite Integral:
Common Integrals:

Function	Integral	Example

Additional info: These mathematical concepts are foundational for success in General Chemistry, especially in quantitative problem-solving, stoichiometry, kinetics, and thermodynamics.