Modern Atomic Theory

The Atom

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The Atom Theory

The Atom

Nucleus

The small dense core of an atom consisting of neutrons and protons. Neutrons and protons have approximately the same mass.

Neutron

A neutral particle located in the nucleus of an atom.

Proton

A positively charged particle located in the nucleus of an atom.

Electron

A negatively charged particle located in the volume of space surrounding the nucleus of an atom.

Charge

Electrons and protons have charge that is equal and opposite, because of this we can say the relative charge in atomic units is one.

A neutral atom has the same number of protons as electrons.

Physical Properties of Sub-atomic Particles

Particle	Charge (Coulombs)	Mass (kilograms)	Relative Charge	Mass (unified atomic mass units)
Proton	$+1.6$ x $10^{-19}\,C$	$1.673$ x $10^{-27}\,kg$	+1	1.0073 $\,u$
Neutron	No Charge	$1.675$ x $10^{-27}\,kg$	0	1.0087 $\,u$
Electron	$-1.6$ x $10^{-19}\,C$	$9.109$ x $10^{-31}\,kg$	-1	$5.4858$ x $10^{-4}\,u$

Unified atomic mass units (u):

$\frac{1}{12}$ the mass of a carbon atom containing six protons and six neutrons.

Using this measurement, the mass of a proton and neutron is approximately $1 \,u$.

The masses of all elements are defined using unified atomic mass units.

$1\,u = 1.661$x$10^{-27}\,kg = \frac{1}{12}$ the mass of a carbon atom

Atom Terminology

Z = number of protons = atomic number

A = number of protons + number of neutrons = mass number

Element

One or more atoms of the same type.

Chemical Symbol (X)

A one or two letter abbreviation assigned to each element.

Atomic Number (Z)

The number of protons in the nucleus of an atom. The type of atom/element is determined by the number of protons.

Mass Number (A)

The sum of the number of neutrons and protons in the nucleus of an atom.

The number of protons defines the type of atom or element!

Isotopes

Isotopes are atoms of the same type that have the same number of protons and varying numbers of neutrons.

We know:

The number of protons determines the type of atom or element.
Neutrons and protons comprise the nucleus of an atom and have the same approximate mass.

Isotopes:

The number of protons determines the type of atom or element.
Isotopes of elements are distinguished by the mass number (A).

Isotopes Example:

Atoms of an element that have a different number of neutrons.

28-14=14 neutrons

29-14=15 neutrons

30-14=16 neutrons

Isotope Notation

Including the atomic number is redundant when representing isotopes. It is common to use the following notation: X-A.
For example: Si-28, Si-29, Si-30

Atomic Mass

We know:

Isotopes are atoms of the same type that have a different number of neutrons.
The mass of an atom is dictated by the mass of protons and neutrons because the mass of an electron is negligible in comparison.

Atomic Mass:

Is the weighted average of isotopic masses based on the natural abundance of each isotope.
The natural abundance is the relative amount of each isotope found in a natural sample of any element. Natural abundance is relatively constant and unique for each element.

The atomic mass of an element as seen on the periodic table is a weighted average of isotopic masses of that element.

Atomic Mass Example:

This Copper (Cu) sample will contain approximately 30.83 % of Cu-65 and 69.17 % of Cu-63

Atomic Mass Calculations:

Atomic mass for an element can be calculated with the following formula:

$$\sum_{n} \text{(fraction of isotope n) x (mass of isotope n)}$$

In simple terms this formula is telling us that atomic mass can be calculated by:

1) Multiplying each isotopic mass of an element by its relative abundance.

2) Adding the resulting masses of the isotopes together.

An atomic mass calculation is included in the example section!

The Mole

A measurement similar to a pair, dozen or score. A pair is 2 objects, a dozen is 12 objects and a score is 20 objects.

1 mole $= 6.022$x$10^{23}$ objects = Avogadro's number

One mole of people is the Earth’s population multiplied by $9$x$10^{13}$.

One mole of bricks is equivalent to $2$x$10^{14}$ Great Walls of China.

One mole of sand is the entire Sahara Desert!

Defining the Mole:

The mole is defined using the C-12 isotope.

One mole = number of atoms in exactly 12 g C-12.

12 g of C-12 = 1 mole of C-12 atoms = exactly $6.02$x$10^{23}$ C-12 atoms

The mass of one mole of C-12 atoms is 12 g

1 mole of carbon (C) = $6.02$x$10^{23}$ atoms

Molar Mass:

Using C-12 as a standard for the mole gives a relationship between mass, number of atoms, and unified atomic mass units.

The mole can be used to count atoms based on their mass.

The molar mass of an element is the mass of one mole of each element.

1 mole of carbon (C) $6.02$x$10^{23}$ atoms

1 mole of sulfur (S) $6.02$x$10^{23}$ atoms

1 mole of gold (Au) $6.02$x$10^{23}$ atoms

Molar Mass of an Element:

The molar mass of an element is the mass of one mole of each element.

Molar mass $= \frac{\text{grams}}{\text{mole}} = \frac{\text{g}}{\text{mol}} =$ the mass of the $6.02$ x $10^{23}$ atoms of each element.

Molar mass is numerically equivalent to an element’s mass in unified atomic mass units $u$ .

Molar Mass $\frac{\text{g}}{\text{mol}}$ is numerically equivalent to atomic mass $u$

The Periodic Table of Elements

Metals, Non-Metals and Metalloids

Periodic Trends

Metals:

Generally good conductors of heat and electricity

Tend to lose electrons in a chemical reaction

Non-Metals:

Generally poor conductors of heat and electricity

Tend to gain electrons in a chemical reaction

Metalloids:

Have properties of both metals and non-metals.

Main Group Elements and Transition Metals

Periodic Trends

Main Group:

Elements that have predictable properties

Transition Metals:

Elements that have unpredictable properties.

Lanthanoids and Actinoids:

Elements that are placed separately for a more compact periodic table. They have similar properties to the elements lanthanum and actinium.

Groups and Periods of the Periodic Table

Periodic Trends

Each column in the periodic table is called a group and numbered 1-18

Each row in the periodic table is called a row and numbered 1-7

The elements in a group have similar properties.

For example: The noble gases in group 18 are all relatively unreactive.

Ions

Common Main Group Ions:

We know:

The number of protons determines the type of atom or element.
Neutrons and protons comprise the nucleus of an atom, and have the same approximate mass.

Ions:

One or more atoms that has gained or lost an electron(s).
Ions are charged particles.
The periodic table can help predict which ions will form.

Ions and Groups:

A main group non-metal will generally gain electrons.
A main group metal will generally lose electrons.

Ions are charged particles that have gained or lost an electron(s).

Cation (+):

One or more atoms that has lost an electron(s).

A main group METAL will LOSE electrons to form a CATION with the same number of electrons as the nearest noble gas.

Anion (-):

One or more atoms that has gained an electron(s).

A main group NONMETAL will GAIN electrons to form an ANION with the same number of electrons as the nearest noble gas.

Ion Notation:

Ions are represented as the chemical symbol of the ion with the charge of the ion noted as a superscript to the right.

$\text{X}^{\text{charge}}$

Lithium ion $=\text{Li}^{\text{+}}$, Oxygen ion $=\text{O}^{2-}$

Examples

Example Problem 1

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The following problem is broken down into steps that can be seen by clicking on the tabs below.

Problem:

Calculate the average atomic mass of silicon given the information in the table.

Isotope	Percent Natural Abundance (%)	Mass (u)
Si-28	92.2	27.9769
Si-29	4.7	28.9764
Si-30	3.1	29.9738

What is the problem asking?

Atomic mass of silicon = u = ?

What do you have?

Si-28: % abundance = 92.2%, mass = 27.9769 u

Si-29: % abundance = 4.7%, mass = 28.9764 u

Si-30: % abundance = 3.1%, mass = 29.9738 u

What do you need?

Convert percentages to fractions:

$\frac{(\%\,abundance)}{100} = \text{(fraction of isotope)}$

Atomic Mass Formula:

$$\sum_{n} \text{(fraction of isotope n) x (mass of isotope n)}$$

Set up the problem

Step 1:

$Si-28 = \frac{(\%\,abundance)}{100} = \frac{92.2}{100} = 0.922$ $Si-29 = \frac{(\%\,abundance)}{100} = \frac{4.7}{100} = 0.047$ $Si-30 = \frac{(\%\,abundance)}{100} = \frac{3.1}{100} = 0.031$

Step 2:

$$Atomic Mass_{Si} = \sum_{n} \text{(fraction of isotope n) x (mass of isotope n)}$$ $= 0.922(27.9769\,u) + 0.047(28.9764\,u) + 0.031(29.9739\,u)$

Calculate your answer

$= 0.922(27.9769\,u) + 0.047(28.9764\,u) + 0.031(29.9739\,u)$ $= 28.0858$ $= 28.086\,u$

Check your answer.

The atomic mass of silicon is 28.086 $u$, this answer makes sense and can be easily checked using the periodic table.

Example Problem 2

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The following problem is broken down into steps that can be seen by clicking on the tabs below.

Problem:

These gold earrings have a mass of 1.00 g. Calculate the moles of gold in the earrings, how many gold atoms do the earrings contain? Assume the earrings are pure gold.

What is the problem asking?

Moles of Au = mol = ?
Number of Au atoms = atoms = ?

What do you have?

Mass of earrings = 1.00 g

What do you need?

Avogadro’s number:

$= 6.022$x$10^{23}\frac{\text{atoms}}{\text{mol}}$

Molar mass of Au:

$= 196.97\frac{\text{g}}{\text{mol}}$

Set up the problem

Step 1:

$1.00\,g\,Au$ x $\frac{1\,mol\,Au}{196.97\,g\,Au} =\, ?\,mols\, Au$

Step 2:

$mols\, Au$ x $\frac{{6.022}{\text{x}}10^{23}{Au\,atoms}}{1\,mol\,Au}$ = $?\,Au\,atoms$

Calculate your answer

Step 1:

$1.00\,g\,Au$ x $\frac{1\,mol\,Au}{196.97\,g\,Au} =\,0.005077\,mols\,Au$

Step 2:

$0.005077mols\,Au$ x $\frac{{6.022}{\text{x}}10^{23}\,Au\,atoms}{1\,mol\,Au} = 3.06$x$10^{21} Au\,atoms$

There are $5.08$x$10^{-3}$ moles of gold and $3.06$x$10^{21}$ gold atoms in the earrings.

Check your answer.

There are $5.08$x$10^{-3}$ moles of gold and $3.06$x$10^{21}$ gold atoms in the earrings. This answer is reasonable, the earrings have a smaller mass than 1 mole of gold, and since atoms are so small it makes sense that there would be a lot of atoms in one pair of earrrings!

Problem Set

The Atom Problem Set

Below are two documents. One is practice problems, the second is the same problems with solutions.
They can be downloaded and changed to suit your needs.

Click or Tap the images below to view.

Practice Problems

Problem Solutions

Quiz

Modern Atomic Theory

Atomic Structure

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Atomic Structure Theory

Atom Models

Bohr Model of an Atom

An electron's position described as a trajectory.

Quantum Mechanical Model of an Atom

An electron's position described as a probability distribution.

Schrodinger - Quantum Mechanical Models

Quantum Mechanical Model of an Atom. An electron's position described as a probability distribution.

Orbital:

The likely position of an electron in an atom described by a probability distribution map.

Quantum Numbers = Electron's "Address"

Quantum Numbers

Principal Quantum Number (n)

Orbital’s size and energy level of an electron.

A whole number 1,2,3,4…..

n increases with an electron’s energy and distance from the nucleus.

Each level is called a shell.

Angular Momentum Quantum Number (l)

The shape of an orbital.

0,1,2,3,…,n-1

Each value for l is a subshell

Each number value of l is assigned a letter s,p,d,f,g….

Magnetic Quantum Number ($m_{l}$)

Orientation of orbital in 3D space.

Value= integer values from –l…l.

There are 2(l)+1 values of $m_{l}$ for each value of l.

Orbital Shapes

Orbital Shapes = Angular Momentum Number (l)

Values of (l) angular momentum
Number	Letter
0	s
1	p
2	d
3	f
4	g

Letter values continue according to the alphabet after f, with the exception of J.

s,p,d,f,g,h,i,k,….. Sober Physicists Don’t Find Giraffes Hiding In Kitchens

s Orbital:

l=0, $m_{l} = 0$

One allowed value for $m_{l}$ = one type of s orbital for each value of n.

p Orbitals:

l=0, $m_{l} = -1, 0, 1$

Three allowed values for $m_{l}$ = three types of p orbitals for n values greater than and equal to two.

d Orbitals:

l=0, $m_{l} = -2, -1, 0, 1, 2$

Five allowed values for $m_{l}$ = five types of d orbitals for n greater than and equal to three.

Spin Quantum Number $m_{s}$

Spin is intrinsic to all electrons, it is quantized, with only two orientations; spin up or spin down.

Value $=+\frac{1}{2}$ OR $-\frac{1}{2}$

Pauli Exclusion Principle

No two electrons in an atom can have the same four quantum numbers, however, Any three quantum numbers can be the same.

This implies: One orbital may contain two electrons of opposite spin orientation!

Aufbau Principle

In a one electron system the subshells are degenerate; meaning s,p,d,f,.., all have the same energy level in each shell. However, in a multi-electron atom the subshells have different energy levels.

Single electron atom

Multi-electron atom

The Aufbau Principle:

That means that we want to know about a system in which the lower energy orbitals fill before the higher energy orbitals. This idea is referred to as the Aufbau principle, and the general order in which electrons fill orbitals in an atom to achieve the ground state configuration is: 1s,2s,2p,3s,3p,4s,3d,4p,5s,4d,5p,6s,4f,5d,6p,7s,5f,6d,7p

Hund’s Rule

Electrons will singly occupy degenerate orbitals until they are half full before pairing up with other electrons. Electrons in half full orbitals will have the same spin.

Electronic Configurations

A notation that represents the orbitals occupied by electrons in an atom.

Write the value of n, followed by the number of electrons allowed in each orbital until all electrons in the atom are assigned to an orbital.

Example Carbon: 6 electrons = $1s^{2}2s^{2}2p^{2}$

Core Electrons

Electrons in a filled inner shell.

Valence Electrons

Electrons in the outermost shell (highest n value).

Example Carbon:
6 electrons = $\stackrel{1s^{2}}{Core}$ $\stackrel{2s^{2}2p^{2}}{Valence}$

Short-hand Electronic Configuration

An electronic configuration using a noble gas to represent the filled shells and only writing out the valence electrons.

Example Carbon:
6 electrons = $\stackrel{He}{Core}$ $\stackrel{2s^{2}2p^{2}}{Valence}$

Orbital Diagrams

A diagram that represents the filled orbitals in an atom by denoting the electron as an arrow.

Each subshell is represented as a line or a circle, electrons are represented as arrows pointing up or down according to the electron's spin.

Example Carbon: 6 electrons = $1s^{2}2s^{2}2p^{2}$

Examples

Problem Set

Atomic Structure Problem Set

Below are two documents. One is practice problems, the second is the same problems with solutions.
They can be downloaded and changed to suit your needs.

Click or tap the images below to view.

Practice Problems

Problem Solutions

Quiz

A)	n=2	l=1	\(m_{l}=0\)	\(m_{s}=1\)
B)	n=1	l=0	\(m_{l}=0\)	\(m_{s}=+\frac{1}{2}\)
C)	n=3	l=3	\(m_{l}=-1\)	\(m_{s}=-\frac{1}{2}\)
D)	n=4	l=2	\(m_{l}=-2\)	\(m_{s}=-\frac{1}{2}\)

A)	n=1	l=0	\(m_{l}=0\)	\(m_{s}=+\frac{1}{2}\)
B)	n=4	l=2	\(m_{l}=-2\)	\(m_{s}=-\frac{1}{2}\)

n=1	l=0	\(m_{l}=0\)	= 1 orbital	s orbital
n=2	l=0	\(m_{l}=0\)	= 1 orbital	s orbital
	l=1	\(m_{l}=-1\)	= 1 orbital	\(p_{x}\) orbital
	l=1	\(m_{l}=0\)	= 1 orbital	\(p_{y}\) orbital
	l=1	\(m_{l}=1\)	= 1 orbital	\(p_{z}\) orbital
n=3	l=0	\(m_{l}=0\)	= 1 orbital	s orbital
	l=1	\(m_{l}=-1\)	= 1 orbital	\(p_{x}\) orbital
	l=1	\(m_{l}=0\)	= 1 orbital	\(p_{y}\) orbital
	l=1	\(m_{l}=1\)	= 1 orbital	\(p_{z}\) orbital
	l=2	\(m_{l}=-2\)	= 1 orbital	\(d_{xy}\) orbital
	l=2	\(m_{l}=-1\)	= 1 orbital	\(d_{xz}\) orbital
	l=2	\(m_{l}=0\)	= 1 orbital	\(d_{yz}\) orbital
	l=2	\(m_{l}=1\)	= 1 orbital	\(d_{z^{2}}\) orbital
	l=2	\(m_{l}=2\)	= 1 orbital	\(d_{x^{2}y^{2}}\) orbital
			= 14 orbitals

n=4	l=0	\(m_{l}=0\)	\(m_{s}=+\frac{1}{2}\)	\(-\frac{1}{2}\)	= 2 electrons
	l=1	\(m_{l}=-1\)	\(m_{s}=+\frac{1}{2}\)	\(-\frac{1}{2}\)	= 2 electrons
	l=1	\(m_{l}=0\)	\(m_{s}=+\frac{1}{2}\)	\(-\frac{1}{2}\)	= 2 electrons
	l=1	\(m_{l}=1\)	\(m_{s}=+\frac{1}{2}\)	\(-\frac{1}{2}\)	= 2 electrons
	l=2	\(m_{l}=-2\)	\(m_{s}=+\frac{1}{2}\)	\(-\frac{1}{2}\)	= 2 electrons
	l=2	\(m_{l}=-1\)	\(m_{s}=+\frac{1}{2}\)	\(-\frac{1}{2}\)	= 2 electrons
	l=2	\(m_{l}=0\)	\(m_{s}=+\frac{1}{2}\)	\(-\frac{1}{2}\)	= 2 electrons
	l=2	\(m_{l}=1\)	\(m_{s}=+\frac{1}{2}\)	\(-\frac{1}{2}\)	= 2 electrons
	l=2	\(m_{l}=2\)	\(m_{s}=+\frac{1}{2}\)	\(-\frac{1}{2}\)	= 2 electrons
	l=3	\(m_{l}=-3\)	\(m_{s}=+\frac{1}{2}\)	\(-\frac{1}{2}\)	= 2 electrons
	l=3	\(m_{l}=-2\)	\(m_{s}=+\frac{1}{2}\)	\(-\frac{1}{2}\)	= 2 electrons
	l=3	\(m_{l}=-1\)	\(m_{s}=+\frac{1}{2}\)	\(-\frac{1}{2}\)	= 2 electrons
	l=3	\(m_{l}=0\)	\(m_{s}=+\frac{1}{2}\)	\(-\frac{1}{2}\)	= 2 electrons
	l=3	\(m_{l}=1\)	\(m_{s}=+\frac{1}{2}\)	\(-\frac{1}{2}\)	= 2 electrons
	l=3	\(m_{l}=2\)	\(m_{s}=+\frac{1}{2}\)	\(-\frac{1}{2}\)	= 2 electrons
	l=3	\(m_{l}=3\)	\(m_{s}=+\frac{1}{2}\)	\(-\frac{1}{2}\)	= 2 electrons
					= 32 electrons

Particle	Charge (Coulombs)	Mass (kilograms)	Relative Charge	Mass (unified atomic mass units)
Proton	\(+1.6\) x \(10^{-19}\,C\)	\(1.673\) x \(10^{-27}\,kg\)	+1	1.0073 \(\,u\)
Neutron	No Charge	\(1.675\) x \(10^{-27}\,kg\)	0	1.0087 \(\,u\)
Electron	\(-1.6\) x \(10^{-19}\,C\)	\(9.109\) x \(10^{-31}\,kg\)	-1	\(5.4858\) x \(10^{-4}\,u\)

Modern Atomic Theory

The Atom

Apply your Knowledge

The Atom

Nucleus

Neutron

Proton

Electron

Charge

A neutral atom has the same number of protons as electrons.

Physical Properties of Sub-atomic Particles

Unified atomic mass units (u):

\(1\,u = 1.661\)x\(10^{-27}\,kg = \frac{1}{12}\) the mass of a carbon atom

Atom Terminology

Z = number of protons = atomic number A = number of protons + number of neutrons = mass number

Element

Chemical Symbol (X)

Atomic Number (Z)

Mass Number (A)

The number of protons defines the type of atom or element!

Isotopes

Isotopes are atoms of the same type that have the same number of protons and varying numbers of neutrons.

We know:

Isotopes:

Isotopes Example:

Atoms of an element that have a different number of neutrons.

Isotope Notation

Including the atomic number is redundant when representing isotopes. It is common to use the following notation: X-A. For example: Si-28, Si-29, Si-30

Atomic Mass

We know:

Atomic Mass:

The atomic mass of an element as seen on the periodic table is a weighted average of isotopic masses of that element.

Atomic Mass Example:

This Copper (Cu) sample will contain approximately 30.83 % of Cu-65 and 69.17 % of Cu-63

Atomic Mass Calculations:

Atomic mass for an element can be calculated with the following formula:

In simple terms this formula is telling us that atomic mass can be calculated by:

1) Multiplying each isotopic mass of an element by its relative abundance.

2) Adding the resulting masses of the isotopes together.

An atomic mass calculation is included in the example section!

The Mole

A measurement similar to a pair, dozen or score. A pair is 2 objects, a dozen is 12 objects and a score is 20 objects.

1 mole \(= 6.022\)x\(10^{23}\) objects = Avogadro's number

Defining the Mole:

The mass of one mole of C-12 atoms is 12 g

1 mole of carbon (C) = \(6.02\)x\(10^{23}\) atoms

Molar Mass:

Using C-12 as a standard for the mole gives a relationship between mass, number of atoms, and unified atomic mass units. The mole can be used to count atoms based on their mass. The molar mass of an element is the mass of one mole of each element.

Molar Mass of an Element:

Molar Mass \(\frac{\text{g}}{\text{mol}}\) is numerically equivalent to atomic mass \(u\)

The Periodic Table of Elements

Metals, Non-Metals and Metalloids

Periodic Trends

Metals:

Generally good conductors of heat and electricity Tend to lose electrons in a chemical reaction

Non-Metals:

Generally poor conductors of heat and electricity Tend to gain electrons in a chemical reaction

Metalloids:

Have properties of both metals and non-metals.

Main Group Elements and Transition Metals

Periodic Trends

Main Group:

Elements that have predictable properties

Transition Metals:

Elements that have unpredictable properties.

Lanthanoids and Actinoids:

Elements that are placed separately for a more compact periodic table. They have similar properties to the elements lanthanum and actinium.

Groups and Periods of the Periodic Table

Periodic Trends

Each column in the periodic table is called a group and numbered 1-18

Each row in the periodic table is called a row and numbered 1-7

The elements in a group have similar properties.

For example: The noble gases in group 18 are all relatively unreactive.

Ions

Common Main Group Ions:

We know:

Ions:

Ions and Groups:

Ions are charged particles that have gained or lost an electron(s).

Cation (+):

Z = number of protons = atomic number

A = number of protons + number of neutrons = mass number

Including the atomic number is redundant when representing isotopes. It is common to use the following notation: X-A.
For example: Si-28, Si-29, Si-30

Using C-12 as a standard for the mole gives a relationship between mass, number of atoms, and unified atomic mass units.

The mole can be used to count atoms based on their mass.

The molar mass of an element is the mass of one mole of each element.

Generally good conductors of heat and electricity

Tend to lose electrons in a chemical reaction

Generally poor conductors of heat and electricity

Tend to gain electrons in a chemical reaction

Click or tap the grey boxes to the left and right
to scroll through the examples.

Click or tap the grey boxes to the left and right
to scroll through the examples.

Below are two documents. One is practice problems, the second is the same problems with solutions.
They can be downloaded and changed to suit your needs.

Your current score is

out of 5

Letter values continue according to the alphabet after f, with the exception of J.

s,p,d,f,g,h,i,k,….. Sober Physicists Don’t Find Giraffes Hiding In Kitchens

l=0, \(m_{l} = 0\)

One allowed value for \(m_{l}\) = one type of s orbital for each value of n.

l=0, \(m_{l} = -1, 0, 1\)

Three allowed values for \(m_{l}\) = three types of p orbitals for n values greater than and equal to two.

l=0, \(m_{l} = -2, -1, 0, 1, 2\)

Five allowed values for \(m_{l}\) = five types of d orbitals for n greater than and equal to three.

Spin is intrinsic to all electrons, it is quantized, with only two orientations; spin up or spin down.

Value \(=+\frac{1}{2}\) OR \(-\frac{1}{2}\)