Chemistry

Basics

Given some element

Where

Cations: Positively （＋） Charged
Anions: Negatively （−） Charged

Mnemonic

Cations are Pawsitive

Conventions on homework

When a question says, determine the energy of 1 ㏖ of photons, the unit will be ᴶ/㏖.

SI Prefixes

Value	Prefix	Symbol
	deca
	hecto
	kilo
	mega
	giga
	tera

Value	Prefix	Symbol
	deci
	centi
	mili
	micro
	nano
	pico

Classification of Matter

Overview

Mixtures

Heterogeneous mixture

Where the prefix Hetero- means different

The Atom and Nuclear Chemistry

Isotopes and Subatomic Particles

Electrons and Quantum Mechanics

Average Atomic Mass

If the average atomic mass of boron is , what is the percent abundance of boron-11 （mass of ） if the only other isotope is boron-10 （mass of ）? Given the formula for average atomic mass:

Therefore in summary, we are given the following known quantities

With the following unknown quantities

And asked to find the percent abundance of boron-11 （）. Therefore our equation is

We have two unknowns, but luckily we can use the following fact/relation and therefore express the percent abundance of boron-10 in terms of the percent abundance of boron-11

Therefore

But we aren't done, we have to compute significant figures.

Therefore, we round our answer to 2 sig figs, yielding

Molecules and Compounds

Terms

Electronegativity: Can be used to to determine type of bond based on electronegativity difference.
Ionic Bond
Covalent Bond
Polar Covalent Bond: Sharing of electrons.
Hydrogen Bond: Hydrogen is bonded with an electronegative element.

Prefixes

Prefix	Value
mono-
di-
tri-
tetra-
penta-

Prefix	Value
hexa-
hepta-
octa-
nona-
deca-

Formula Summary

Formal Charge （Option 1）

Given some element which has number of valance electrons, and a lewis diagram of bonded （） electrons and non-bonded （） electrons, the formal charge for the given element in the lewis structure is

For the given quantities

Formal Charge （Option 2）

Given some element which has number of valance electrons, and a lewis diagram of some number of bonds （） and dots （）, the formal charge for the given element in the lewis structure is

For the given quantities

Note: each bond is 2 electrons, but counts as one bond.

Polyatomic Ions

Remembering the number of oxygens

General

Given a set of oxygens in increasing order

Where

For example

Given some element or compound ionically bonded with such

For example

The following table denotes the naming conventions therein

Prefix	Prefix Meaning （relative to suffix）	Suffix	Oxygen Order	Example
Prefix	Prefix Meaning （relative to suffix）	Suffix	Oxygen Order	Formula	Name
Per-	More than	-ate	Highest #		Perchlorate
		-ate	Higher #		Chlorate
		-ite	Lower #		Chlorite
Hypo-	less than	-ite	Lowest #		Hypochlorite

Oxygen vs No-Oxygen Comparison

Prefix	Suffix	Meaning	Example	Name
Per-	-ate		or	Persulfate
	-ate			Sulfate
	-ite			Sulfite
Hypo-	-ite			Hyposulfite
	-ide	No oxygennon-metal		Sulfide

In summary

Per*ate

Greater than

*ate
*ite
Hypo*ite

*ate

Less than

Per*ate

Greater than

*ite
Hypo*ite

*ite

Less than

Per*ate
*ate

Greater than

Hypo*ite

Hypo*ite

Less than

Per*ate
*ate
*ite

Generally

per-ate	Used in the ion with the largest number of oxygen atoms
-ate-ite	Polyatomic ions of oxygen
hypo-ite	Used in the ion with the lowest number of oxygen atoms
-ide	Non-metal, no oxygens

Determining The Charge^†

Warning

Phosphate （） is the only one that violates this rule!

Example for Hydrogen Carbonate

†

Source

Notes

When naming an ionic compound, the name of the cation is followed by the name of the anion. Monatomic anions are named with the ending -ide.

Periodic Properties of the Elements

Note, electron affinity is not the same as electronegativity!

Terms

Isoelectronic: Atoms with the same number of electrons.
Ionization energy
Predict Metallic Character Based on Periodic Trends

Expanded Octet （Exceptions to the Octet Rule）

All non-metals from period 3 to period 8 of the Periodic Table, can have expanded octets.

Quantum Mechanical Models of the Atom

Terms

Pauli Exclusion Principle: No two electrons in an atom can have the same four quantum numbers.; Pauli’s Principle prevents two electrons with the same spin from existing in the same subshell, each subshell will be filled with one spin direction before they are filled with the opposite spin. This is the second of Hund’s Rules.
Aufbau Principle: This pattern of orbital filling is known as the aufbau principle （the German word aufbau means “build up”）.
Hund’s rule: When filling degenerate orbitals, electrons fill them singly first, then with parallel spins.; I.e. start by filling boxes with single 'upward' arrows, and then once all of such boxes are maxed out, then you add double arrows pointing in opposite directions.; Pauli’s Principle prevents two electrons with the same spin from existing in the same subshell, each subshell will be filled with one spin direction before they are filled with the opposite spin. This is the second of Hund’s Rules.
Coulomb’s Law

Pauli's Exclusion Principle

Each election has a unique set of four quantum numbers （i.e. see quantum numbers）. They are

Overview

Values

Name	Symbol	Unit	Description	Range
Wavelength		Any unit for distance	Distance between two analogous points	Always Positive
Frequency	or （nu）		Number of cycles	Always Positive
Energy		（joule）	Amount of energy （） in a light packet

Constants

Name	Symbol	Unit	Value
Speed of Light
Planck's constant		Energy multiplied by time joule-seconds

Other Formulas

Heisenberg's Uncertainty Principle

Where

is the uncertainty in position.
is the uncertainty in velocity.
is the mass of the particle.
is the plank's constant.

In general it states that the more you know about an electrons position, the less you know about it's velocity.

Energy of an Electron in an Orbital with Quantum Number in a Hydrogen Atom

Energy of an Electron in an Orbital with Quantum Number for any atom

Where is the atomic number of the given element.

Change in Energy That Occurs in an Atom When It Undergoes a Transition between Levels （Further Details）

and

If is negative, energy is being released.
If is positive, energy is being absorbed.

Ionization Energy

Where

is the electron number
the point on the table where you see the highest difference/delta.

Atomic Spectroscopy

The Principal Quantum Number （n）（Hydrogen Atom）

For the hydrogen atom, the energy of an electron in an orbital with quantum number is given by

Therefore the difference in energy is given by the following

The Principal Quantum Number （n）（Any Atom）

For the hydrogen atom, the energy of an electron in an orbital with quantum number is given by

TODO

Where is the atomic number of the given element.

Electron Configuration

Examples

Electron configuration for

Since the electron configuration for Argon is

Electron configuration for

Beginning with the electron configuration for

Remove the electrons from the term with the higher electron state. Warning! Do not just remove the electrons from the rightmost term since the rightmost term may be a lower electron state. For instance given

is in a higher electron state
is in a lower electron state

Alternatively, to compute the lowest energy orbital, add the principle quantum number （） to the The angular momentum quantum number （） to get the orbital with the lowest energy. Therefore

Given and

NVM this is an exception to the rule...

TODO move this somewhere else...

As shown

Therefore the electron configuration for is:

Electron configuration for

It would appear that the electron configuration for would be

But this is wrong! It's actually

How-tos

What are the valence electrons?

Given

The valance electrons will be the ones in the highest energy state. Therefore

Therefore there are valence electrons.

Given

The valance electrons will be the ones in the highest energy state. Therefore

Therefore there are valence electrons.

Quantum Numbers

Overview

Symbol	Description
	The principle quantum number
	The angular momentum quantum number
	The magnetic quantum number
	The spin quantum number

The Principle Quantum Number （）

Value of	Value of	Orbital Sublevel

Angular Momentum Quantum Number

Value	Result

Value of	Value of

Summary

Useful Formulas

The equation for a maximum number of electrons a given energy level can hold given some value for

How many orbitals are possible given some value for

Examples

Light

Interference and Diffraction

Constructive Interference

If two waves of equal amplitude are in phase when they interact—that is, they align with overlapping crests—a wave with twice the amplitude results. This is called constructive interference.

Destructive Interference

If two waves are completely out of phase when they interact—that is, they align so that the crest from one overlaps with the trough from the other—the waves cancel by destructive interference.

Mathematics

Algebra

Miscellaneous

Functional Utilities & Notation Conveniences

Right to Left Evaluation

Left to Right Evaluation

Derivative Shorthand

For this notation, the derivative with respect to a given variable, is implicit.

Radians & Radian Conversion

Constants

Conversion

Constants

ℯ （Euler's number）

Algebra

Properties

Trigonometry

The Unit Circle & Special Angles In Trig

Warning

Never use Pi （）! It makes （thinking in terms of） radians confusing, Tao （） is what the enlightened trigonometer uses, and won't screw you over.

To easily memorize the special angles in trig, notice the repeating patterns on the above angles.

For values on the x-axis, anything over and under will be negative
For values on the y-axis, anything over will be negative
Diagonals will be , For ratios of on the sides, i.e. , , , , , , , and . Draw a circle and dot the point where it occurs （which is pretty easy since the above are simple ratios of a circle when expressed in terms of ）. Then with regards to the and axis values:
- The longer size will be
- The shorter side will be
See the above examples.

Trigonometric Identities

Pythagorean Identities

Sum and Difference Identities

Cofunction Identities

Ratio Identities

Double-Angle Identities

Half-Angle Identities

Power-Reducing Identities

Product-to-Sum Identities

Sum-to-Product-Identities

Trigonometric Equations

Euler's Formula

Coordinate & Number Systems

Polar Coordinate System

Given

Then

Properties

Given

Then

De Moivre’s Theorem

De Moivre’s Theorem For Finding Roots

Trigonometric form of a complex number

Vectors

Quick Facts

Two vectors are equal if they share the same magnitude and direction.
Initial points don't matter.
You can define a vector with two points.
Do not divide vectors!

Vector Operations

Dot Product

Cross Product

Length of a Vector

Definition of Vector Addition

If and are positioned so the initial point of is at the terminal point of , then the sum is the vector from the initial point of to the terminal point of .

Given some vectors and , the vector is the vector that points from the head of to the head of

Standard Basis Vectors

Orthogonal

Two vectors are orthogonal if and only if

The Unit Vector

If is the angle between the vectors and , then

If is the angle between the nonzero vectors and , then

Two nonzero vectors and are parallel if and only if

Properties of the Dot Product

Direction Cosines & Direction Angles of a Vector

Where

Direction Cosines

Direction Angles

Theorem

Proof

Given

Therefore

Vector Relations

Parallel Vectors

When two vectors are parallel; they never intersect （duh）.

Given some vectors

The vectors and are parallel if and only if they are scalar multiples of one another.

Alternatively

Orthogonal Vectors

When two vectors are orthogonal; they meet at right angles.

Given some vectors

Two vectors are orthogonal if and only if

Reparameterization of the position vector in terms of length

We can parametrize a curve with respect to arc length; because arc length arises naturally from the shape of the curve and does not depend on any coordinate system.

The Arc Length Function

Given

We can redefine in terms of arc length between two endpoints

That is, is the length of the curve （） between and .

Furthermore from the adjacent definition; we can simply the above to

The Arc Length Function

That is

Vectors Derived From Some Curve Defined by

The Unit Vector

The Unit Tangent Vector

The Unit Normal Vector

The Binormal Vector

Therefore, the binormal vector is orthogonal to both the tangent vector and the normal vector.
The plane determined by the normal and binormal vectors N and B at a point P on a curve C is called the normal plane of C at P.
The plane determined by the vectors T and N is called the osculating plane of C at P. The name comes from the Latin osculum, meaning “kiss.” It is the plane that comes closest to containing the part of the curve near P. （For a plane curve, the osculating plane is simply the plane that contains the curve.）

Kappa - Curvature of a Vector

Tangential & Normal Components of the Acceleration Vector of the Curve

When we study the motion of a particle, it is often useful to resolve the acceleration into two components, one in the direction of the tangent and the other in the direction of the normal.

Specifically

Vector Calculus

The Position Vector

（Original Function）

The Velocity Vector

（First Derivative）

The velocity vector is also the tangent vector and points in the direction of the tangent line.
The speed of the particle at time t is the magnitude of the velocity vector, that is,

The Acceleration Vector

（Second Derivative）

Matrices

Reference

The Determinant of A Matrix

Only works for square matrices.

The Cross Product

Geometry

The Circle

Definition of a Line

Vector Equation of a Line

Given

We can define a vector between and

Therefore

The equation of a line in 3D space or can be defined VIA the following options

That is

Parametric Equation of a Line

Essentially

That is, is the scaling factor. In a way, it's like it's a function of , but also similar to the slope （） in , except （i.e. ） is parameterized.

Sometimes this will be （confusingly） denoted as

Symmetric Equation of a Line

Therefore

Rationale

We rewrite in terms of .

That is

Parameterizations of a curve

Parametrized curve: A curve in the plane is said to be parameterized if the set of coordinates on the curve, （x,y）, are represented as functions of a variable t.; A parametrized Curve is a path in the xy-plane traced out by the point as the parameter ranges over an interval .; A parametrized Curve is a path in the xyz-plane traced out by the point as the parameter ranges over an interval .

Curvature Properties

Length of a Curve

The Arc Length Function

Suppose

Given some curve defined by some vector in
where is continuous and is traversed exactly once as increases from to

We can define it's arc length function VIA

Calculus

Derivative Tables

Integration Tables

Riemann Sums

Given

Left Riemann Sum

Right Riemann Sum

Midpoint Riemann Sum

We can also do away with the index notation and simplify things.

Trapezoidal Riemann Sum

Simpson's Rule

Improper Integrals

Infinite Sequences

Infinite Sequence

Helpful Theorem

Example

Given

Therefore

Example

Given

Therefore

Infinite Series

Note that the limit of every convergent series is equal to zero. But the inverse isn't always true. If the limit is equal to zero, it may not be convergent.

For example, does diverge; but it's limit is equal to zero.

If the limit is equal to zero; the test is inconclusive.

Geometric Series

Given

Alternatively

Tests

Furthermore

The Integral Test

Constraints on

Continuous
Positive
Decreasing （i.e. use derivative test）

P-Series -or- Harmonic Series

Note: the Harmonic series is the special case where

Comparison Test

Limit Comparison Test

Warning

If , this only means that the limit comparison test can be used. You still need to determine if either or converges or diverges.
Therefore, this does not apply to any arbitrary rational function.

Notes

For many series, we find a suitable comparison, , by keeping only the highest powers in the numerator and denominator of .

Estimating Infinite Series

Differential Equations

Separable Differential Equations

Growth and Decay Models

The above states that all solutions for are of the form .

Where

Exponential growth occurs when , and exponential decay occurs when .

The Law of Natural Growth:

The Logistic Model of Population Growth:

Where

Solving the Logistic Equation

Via partial fraction decomposition

Rewriting the differential equation

Second Order Homogeneous Linear Differential Equations with Constant Coefficients

Properties

If and are solutions; then is also a solution. Therefore, the most general solution to some second order homogeneous linear differential equations with constant coefficients would be .
This is why, when you find two solutions to the characteristic equation and respectively, we write it like so.

Given some:

We can presume that is of the form , and therefore:

Substituting this back into the original equation, we have:

Where:

So therefore:

Where the general solution is of the form:

Parametric Equations

First Derivative Formula

To find the derivative of a given function defined parametrically by the equations and .

Second Derivative Formula

To find the second derivative of a given function defined parametrically by the equations and .

Given

Therefore

The above shows different ways of representing . （I.e. it doesn't correspond to some final solution.）

Arc Length

Formula for the arc length of a parametric curve over the interval .

Physics

Kinematic Equations in 1D

Conventions

Graphical Representation

Standard Equations

Summary

Formula	Missing	Quantities Present

Basics

Constant Velocity

Uniform acceleration

Miscellaneous

Deriving Displacement Formulas

Displacement when object moves with constant velocity

Deriving

Displacement when object accelerates from rest

Deriving

Displacement when object accelerates with initial velocity

Deriving

Deriving The Other Kinematic Formulas

Deriving

Given

We can rearrange from equation （1） like so

Therefore

Deriving

Given

from equation （1） can be rearranged as

from equation （2） can be rearranged like so

Using the following equations from above

from equation （3）
from equation （4）

Rearranging equation （5）

Rearrange again to obtain the more common form

TODO

Two-dimensional Projectile Motion

Conventions

Summary

It's easy to see in the above visualization that and increase linearly, while is non-linear.

Formulas

Displacement & Projectile Position

Generalized

In general （without respect to any or axis values）

Where the distance traveled or displaced is

In terms of and axis values

With respect to the axis

The displacement of a given projectile in terms of the axis is

Since

Which can be read as （in terms of the axis）

With respect to the axis

The displacement of a given projectile in terms of the axis is

Note that （because there is no force acting on the projectile in the horizontal direction）, and therefore the initial and final velocities are the same. I.e. it's constant throughout. Therefore in summary

and therefore we will simple refer to the velocity vector as as .

Therefore we can simplify equation （1） considerably

Solving Projectile Motion Problems

Projectile Motion

In terms of the axis

TODO

In terms of the axis

TODO

In Summary

Initial Quantities

Derived expressions

Solutions

Projectile Motion from an initial height, with given initial velocity and angle

Given

A projectile angle
The initial height
The initial velocity

We can therefore derive the the initial velocities for and in terms of the given angle and initial velocity.

Given the general formulas for displacement and velocity

Which this information, we will derive specific equations in terms of the and axes governing the projectile.

In terms of the axis

Deriving displacement as a function of time

Using the general formula from above in terms of as a function of time.

Which we can simplify using the following facts

From the given depiction of the problem, we know that .
There is no acceleration along the axis, so .
as shown above.

Therefore

Deriving velocity

In terms of the axis

Deriving displacement as a function of time

Using the general formula from above in terms of as a function of time.

Which we can simplify using the following facts

Initial height is given to us which we will represent as , for the sake of generality.
Acceleration along the axis is the constant for gravity, so .
as shown above.

Therefore

Deriving velocity

In summary

To find the range

We know that at the moment of impact , therefore we can use equation

Rearranging a bit and setting , we can see that solving for will yield the time at which .

Therefore

Plugging the solution for （and ignoring the negative or non-real solutions for ） into will yield the horizontal displacement （range） at the time . Therefore:

To find the maximum vertical displacement （i.e. peak height）

We begin with equation

We know that at the moment our projectile crests its trajectory, the vertical component of our projectile will be zero. Therefore . To find the time, we simply solve for .

Therefore, knowing the time at which our projectile crests its trajectory, we simply plugin our solution for into the function given in equation . I.e:

To find the velocity at a given moment of time

Given some time which we will denote as , to find the velocity we simply plug in our given values for and into equations and . I.e.

With the given value for , yielding the vector at time , which we will denote as

To define the vector in terms of engineering notation, （i.e. ）

To define the vector in terms of magnitude （which we will denote as ） and direction （which we will denote as ）

Range

The distance a projectile travels is called its range.

Only applies in situations where the projectile lands at the same elevation from which it was fired.

Reasoning About Projectile Motion

Notes

An object is in free fall when the only force acting on it is the force of gravity.

Question

Based on the figure, for which trajectory was the object in the air for the greatest amount of time?

Answer

Trajectory A

Explanation

All that matters is the vertical height of the trajectory, which is based on the component of the initial velocity in the vertical direction （）. The higher the trajectory, the more time the object will be in the air, regardless of the object's range or horizontal velocity.

Problems

The function in this graph represents an object that is speeding up, or accelerating at a constant rate.

When you throw a ball directly upward, what is true about its acceleration after the ball has left your hand?

Answer: The ball’s acceleration is always directed downward.

Wrong: The ball’s acceleration is always directed downward, except at the top of the motion, where the acceleration is zero.

Question

As an object moves in the x-y plane, which statement is true about the object’s instantaneous velocity at a given moment?

Answer

The instantaneous velocity is tangent to the object’s path

Wrong

The instantaneous velocity is perpendicular to the object’s path.
The instantaneous velocity can point in any direction, independent of the object’s path.

Explanation

As an object moves in the x-y plane the instantaneous velocity is tangent to the object‘s path at a given moment. This is because the displacement vector during an infinitesimally small time interval is always directed along the object’s path and the velocity vector always has the same direction as the displacement vector.

Relative Motion

Galilean transformation of velocity

The velocity of some object P as seen from a stationary frame must be the sum of and

Where

Symbol	Description
	Velocity as measured in a stationary frame
	Velocity of an object measured in the moving frame relative to the moving frame
	velocity of the moving frame - with respect to the stationary frame

Galilean transformation of velocity （alternate notation）

Given two reference frames \text｛ A ｝ and and some object . The velocity of the object can be defined in terms of or as shown

Symbol	Description
	The velocity of relative to
	The velocity of relative to
	The velocity of relative to
	The velocity of relative to . It locates the origin of relative to the origin of .

Therefore

Rotational Motion & Kinematics

Basics

Auxiliary Formula Reference

Formula	Missing	Quantities Present

A particle moves with uniform circular motion if and only if its angular velocity V is constant and unchanging.

Uniform Circular Motion

Uniform means content speed

Forces and Newton's laws of motion

Newton's laws of motion

Normal force and contact force

Balanced and unbalanced forces

Inclined planes and friction

Tension

Test Page

Hello world

Math

Drawings

Top-Level Title

Top-level Title

Subsection Title

Some Drawing Entry

Some Other Drawing Entry

Hello world

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Lorem ipsum dolor sit amet, consectetur adipiscing elit. Etiam a nisi eu sapien bibendum scelerisque. Vestibulum ante ipsum primis in faucibus orci luctus et ultrices posuere cubilia curae; Fusce id blandit lacus, at interdum massa.
Lorem ipsum dolor sit amet, consectetur adipiscing elit. Etiam a nisi eu sapien bibendum scelerisque. Vestibulum ante ipsum primis in faucibus orci luctus et ultrices posuere cubilia curae; Fusce id blandit lacus, at interdum massa.
Lorem ipsum dolor sit amet, consectetur adipiscing elit. Etiam a nisi eu sapien bibendum scelerisque. Vestibulum ante ipsum primis in faucibus orci luctus et ultrices posuere cubilia curae; Fusce id blandit lacus, at interdum massa.
Lorem ipsum dolor sit amet, consectetur adipiscing elit. Etiam a nisi eu sapien bibendum scelerisque. Vestibulum ante ipsum primis in faucibus orci luctus et ultrices posuere cubilia curae; Fusce id blandit lacus, at interdum massa.
Lorem ipsum dolor sit amet, consectetur adipiscing elit. Etiam a nisi eu sapien bibendum scelerisque. Vestibulum ante ipsum primis in faucibus orci luctus et ultrices posuere cubilia curae; Fusce id blandit lacus, at interdum massa.
Lorem ipsum dolor sit amet, consectetur adipiscing elit. Etiam a nisi eu sapien bibendum scelerisque. Vestibulum ante ipsum primis in faucibus orci luctus et ultrices posuere cubilia curae; Fusce id blandit lacus, at interdum massa.

Morbi iaculis porta diam, quis rhoncus orci dictum tempor. Pellentesque viverra quam erat, sagittis luctus nulla condimentum in. Morbi dui ligula, laoreet eget urna sed, accumsan semper metus. Orci varius natoque penatibus et magnis dis parturient montes, nascetur ridiculus mus. Mauris ex enim, dapibus vitae sollicitudin vitae, placerat et lorem. Class aptent taciti sociosqu ad litora torquent per conubia nostra, per inceptos himenaeos. Etiam vel viverra ante. Ut ac lacus erat. Nunc eleifend elementum ex a posuere.