Quantum Numbers and Electron Configurations

Quantum Numbers

The Bohr model was a one-dimensional model that used one quantum number to describe the distribution of electrons in the atom. The only information that was important was the size of the orbit, which was described by the n quantum number. Schrödinger's model allowed the electron to occupy three-dimensional space. It therefore required three coordinates, or three quantum numbers , to describe the orbitals in which electrons can be found.

The three coordinates that come from Schrödinger's wave equations are the principal ( n ), angular ( l ), and magnetic ( m ) quantum numbers. These quantum numbers describe the size, shape, and orientation in space of the orbitals on an atom.

The principal quantum number ( n ) describes the size of the orbital. Orbitals for which n = 2 are larger than those for which n = 1, for example. Because they have opposite electrical charges, electrons are attracted to the nucleus of the atom. Energy must therefore be absorbed to excite an electron from an orbital in which the electron is close to the nucleus ( n = 1) into an orbital in which it is further from the nucleus ( n = 2). The principal quantum number therefore indirectly describes the energy of an orbital.

The angular quantum number ( l ) describes the shape of the orbital. Orbitals have shapes that are best described as spherical ( l = 0), polar ( l = 1), or cloverleaf ( l = 2). They can even take on more complex shapes as the value of the angular quantum number becomes larger.

There is only one way in which a sphere ( l = 0) can be oriented in space. Orbitals that have polar ( l = 1) or cloverleaf ( l = 2) shapes, however, can point in different directions. We therefore need a third quantum number, known as the magnetic quantum number ( m ), to describe the orientation in space of a particular orbital. (It is called the magnetic quantum number because the effect of different orientations of orbitals was first observed in the presence of a magnetic field.)

Rules Governing the Allowed Combinations of Quantum Numbers

- The three quantum numbers ( n , l , and m ) that describe an orbital are integers: 0, 1, 2, 3, and so on.
- The principal quantum number ( n ) cannot be zero. The allowed values of n are therefore 1, 2, 3, 4, and so on.
- The angular quantum number ( l ) can be any integer between 0 and n - 1. If n = 3, for example, l can be either 0, 1, or 2.
- The magnetic quantum number ( m ) can be any integer between - l and + l . If l = 2, m can be either -2, -1, 0, +1, or +2.

Shells and Subshells of Orbitals

Orbitals that have the same value of the principal quantum number form a shell . Orbitals within a shell are divided into subshells that have the same value of the angular quantum number. Chemists describe the shell and subshell in which an orbital belongs with a two-character code such as 2 p or 4 f . The first character indicates the shell ( n = 2 or n = 4). The second character identifies the subshell. By convention, the following lowercase letters are used to indicate different subshells.

Although there is no pattern in the first four letters ( s , p , d , f ), the letters progress alphabetically from that point ( g , h , and so on). Some of the allowed combinations of the n and l quantum numbers are shown in the figure below.

The third rule limiting allowed combinations of the n , l , and m quantum numbers has an important consequence. It forces the number of subshells in a shell to be equal to the principal quantum number for the shell. The n = 3 shell, for example, contains three subshells: the 3 s , 3 p , and 3 d orbitals.

Possible Combinations of Quantum Numbers

There is only one orbital in the n = 1 shell because there is only one way in which a sphere can be oriented in space. The only allowed combination of quantum numbers for which n = 1 is the following.

There are four orbitals in the n = 2 shell.

There is only one orbital in the 2 s subshell. But, there are three orbitals in the 2 p subshell because there are three directions in which a p orbital can point. One of these orbitals is oriented along the X axis, another along the Y axis, and the third along the Z axis of a coordinate system, as shown in the figure below. These orbitals are therefore known as the 2 p x , 2 p y , and 2 p z orbitals.

There are nine orbitals in the n = 3 shell.

There is one orbital in the 3 s subshell and three orbitals in the 3 p subshell. The n = 3 shell, however, also includes 3 d orbitals.

The five different orientations of orbitals in the 3 d subshell are shown in the figure below. One of these orbitals lies in the XY plane of an XYZ coordinate system and is called the 3 d xy orbital. The 3 d xz and 3 d yz orbitals have the same shape, but they lie between the axes of the coordinate system in the XZ and YZ planes. The fourth orbital in this subshell lies along the X and Y axes and is called the 3 d x 2 - y 2 orbital. Most of the space occupied by the fifth orbital lies along the Z axis and this orbital is called the 3 d z 2 orbital.

The number of orbitals in a shell is the square of the principal quantum number: 1 2 = 1, 2 2 = 4, 3 2 = 9. There is one orbital in an s subshell ( l = 0), three orbitals in a p subshell ( l = 1), and five orbitals in a d subshell ( l = 2). The number of orbitals in a subshell is therefore 2( l ) + 1.

Before we can use these orbitals we need to know the number of electrons that can occupy an orbital and how they can be distinguished from one another. Experimental evidence suggests that an orbital can hold no more than two electrons.

To distinguish between the two electrons in an orbital, we need a fourth quantum number. This is called the spin quantum number ( s ) because electrons behave as if they were spinning in either a clockwise or counterclockwise fashion. One of the electrons in an orbital is arbitrarily assigned an s quantum number of +1/2, the other is assigned an s quantum number of -1/2. Thus, it takes three quantum numbers to define an orbital but four quantum numbers to identify one of the electrons that can occupy the orbital.

The allowed combinations of n , l , and m quantum numbers for the first four shells are given in the table below. For each of these orbitals, there are two allowed values of the spin quantum number, s .

Summary of Allowed Combinations of Quantum Numbers

The Relative Energies of Atomic Orbitals

Because of the force of attraction between objects of opposite charge, the most important factor influencing the energy of an orbital is its size and therefore the value of the principal quantum number, n . For an atom that contains only one electron, there is no difference between the energies of the different subshells within a shell. The 3 s , 3 p , and 3 d orbitals, for example, have the same energy in a hydrogen atom. The Bohr model, which specified the energies of orbits in terms of nothing more than the distance between the electron and the nucleus, therefore works for this atom.

The hydrogen atom is unusual, however. As soon as an atom contains more than one electron, the different subshells no longer have the same energy. Within a given shell, the s orbitals always have the lowest energy. The energy of the subshells gradually becomes larger as the value of the angular quantum number becomes larger.

Relative energies: s < p < d < f

As a result, two factors control the energy of an orbital for most atoms: the size of the orbital and its shape, as shown in the figure below.

A very simple device can be constructed to estimate the relative energies of atomic orbitals. The allowed combinations of the n and l quantum numbers are organized in a table, as shown in the figure below and arrows are drawn at 45 degree angles pointing toward the bottom left corner of the table.

The order of increasing energy of the orbitals is then read off by following these arrows, starting at the top of the first line and then proceeding on to the second, third, fourth lines, and so on. This diagram predicts the following order of increasing energy for atomic orbitals.

1 s < 2 s < 2 p < 3 s < 3 p <4 s < 3 d <4 p < 5 s < 4 d < 5 p < 6 s < 4 f < 5 d < 6 p < 7 s < 5 f < 6 d < 7 p < 8 s ...

Electron Configurations, the Aufbau Principle, Degenerate Orbitals, and Hund's Rule

The electron configuration of an atom describes the orbitals occupied by electrons on the atom. The basis of this prediction is a rule known as the aufbau principle , which assumes that electrons are added to an atom, one at a time, starting with the lowest energy orbital, until all of the electrons have been placed in an appropriate orbital.

A hydrogen atom ( Z = 1) has only one electron, which goes into the lowest energy orbital, the 1 s orbital. This is indicated by writing a superscript "1" after the symbol for the orbital.

H ( Z = 1): 1 s 1

The next element has two electrons and the second electron fills the 1 s orbital because there are only two possible values for the spin quantum number used to distinguish between the electrons in an orbital.

He ( Z = 2): 1 s 2

The third electron goes into the next orbital in the energy diagram, the 2 s orbital.

Li ( Z = 3): 1 s 2 2 s 1

The fourth electron fills this orbital.

Be ( Z = 4): 1 s 2 2 s 2

After the 1 s and 2 s orbitals have been filled, the next lowest energy orbitals are the three 2 p orbitals. The fifth electron therefore goes into one of these orbitals.

B ( Z = 5): 1 s 2 2 s 2 2 p 1

When the time comes to add a sixth electron, the electron configuration is obvious.

C ( Z = 6): 1 s 2 2 s 2 2 p 2

However, there are three orbitals in the 2 p subshell. Does the second electron go into the same orbital as the first, or does it go into one of the other orbitals in this subshell?

To answer this, we need to understand the concept of degenerate orbitals . By definition, orbitals are degenerate when they have the same energy. The energy of an orbital depends on both its size and its shape because the electron spends more of its time further from the nucleus of the atom as the orbital becomes larger or the shape becomes more complex. In an isolated atom, however, the energy of an orbital doesn't depend on the direction in which it points in space. Orbitals that differ only in their orientation in space, such as the 2 p x , 2 p y , and 2 p z orbitals, are therefore degenerate.

- One electron is added to each of the degenerate orbitals in a subshell before two electrons are added to any orbital in the subshell.
- Electrons are added to a subshell with the same value of the spin quantum number until each orbital in the subshell has at least one electron.

When the time comes to place two electrons into the 2 p subshell we put one electron into each of two of these orbitals. (The choice between the 2 p x , 2 p y , and 2 p z orbitals is purely arbitrary.)

C ( Z = 6): 1 s 2 2 s 2 2 p x 1 2 p y 1

The fact that both of the electrons in the 2 p subshell have the same spin quantum number can be shown by representing an electron for which s = +1/2 with an

arrow pointing up and an electron for which s = -1/2 with an arrow pointing down.

The electrons in the 2 p orbitals on carbon can therefore be represented as follows.

When we get to N ( Z = 7), we have to put one electron into each of the three degenerate 2 p orbitals.

Because each orbital in this subshell now contains one electron, the next electron added to the subshell must have the opposite spin quantum number, thereby filling one of the 2 p orbitals.

The ninth electron fills a second orbital in this subshell.

The tenth electron completes the 2 p subshell.

There is something unusually stable about atoms, such as He and Ne, that have electron configurations with filled shells of orbitals. By convention, we therefore write abbreviated electron configurations in terms of the number of electrons beyond the previous element with a filled-shell electron configuration. Electron configurations of the next two elements in the periodic table, for example, could be written as follows.

Na ( Z = 11): [Ne] 3 s 1

Mg ( Z = 12): [Ne] 3 s 2

The aufbau process can be used to predict the electron configuration for an element. The actual configuration used by the element has to be determined experimentally. The experimentally determined electron configurations for the elements in the first four rows of the periodic table are given in the table in the following section.

The Electron Configurations of the Elements

(1st, 2nd, 3rd, and 4th Row Elements)

Exceptions to Predicted Electron Configurations

There are several patterns in the electron configurations listed in the table in the previous section. One of the most striking is the remarkable level of agreement between these configurations and the configurations we would predict. There are only two exceptions among the first 40 elements: chromium and copper.

Strict adherence to the rules of the aufbau process would predict the following electron configurations for chromium and copper.

The experimentally determined electron configurations for these elements are slightly different.

In each case, one electron has been transferred from the 4 s orbital to a 3 d orbital, even though the 3 d orbitals are supposed to be at a higher level than the 4 s orbital.

Once we get beyond atomic number 40, the difference between the energies of adjacent orbitals is small enough that it becomes much easier to transfer an electron from one orbital to another. Most of the exceptions to the electron configuration predicted from the aufbau diagram shown earlier therefore occur among elements with atomic numbers larger than 40. Although it is tempting to focus attention on the handful of elements that have electron configurations that differ from those predicted with the aufbau diagram, the amazing thing is that this simple diagram works for so many elements.

Electron Configurations and the Periodic Table

When electron configuration data are arranged so that we can compare elements in one of the horizontal rows of the periodic table, we find that these rows typically correspond to the filling of a shell of orbitals. The second row, for example, contains elements in which the orbitals in the n = 2 shell are filled.

There is an obvious pattern within the vertical columns, or groups, of the periodic table as well. The elements in a group have similar configurations for their outermost electrons. This relationship can be seen by looking at the electron configurations of elements in columns on either side of the periodic table.

The figure below shows the relationship between the periodic table and the orbitals being filled during the aufbau process. The two columns on the left side of the periodic table correspond to the filling of an s orbital. The next 10 columns include elements in which the five orbitals in a d subshell are filled. The six columns on the right represent the filling of the three orbitals in a p subshell. Finally, the 14 columns at the bottom of the table correspond to the filling of the seven orbitals in an f subshell.

## Quantum Numbers, Atomic Orbitals, and Electron Configurations

## Quantum Numbers and Atomic Orbitals

- Principal Quantum Number ( n ): n = 1, 2, 3, …, ∞ Specifies the energy of an electron and the size of the orbital (the distance from the nucleus of the peak in a radial probability distribution plot). All orbitals that have the same value of n are said to be in the same shell ( level ). For a hydrogen atom with n =1, the electron is in its ground state ; if the electron is in the n =2 orbital, it is in an excited state . The total number of orbitals for a given n value is n 2 .
- Angular Momentum (Secondary, Azimunthal) Quantum Number ( l ): l = 0, ..., n -1. Specifies the shape of an orbital with a particular principal quantum number. The secondary quantum number divides the shells into smaller groups of orbitals called subshells ( sublevels ). Usually, a letter code is used to identify l to avoid confusion with n :

The subshell with n =2 and l =1 is the 2 p subshell; if n =3 and l =0, it is the 3 s subshell, and so on. The value of l also has a slight effect on the energy of the subshell; the energy of the subshell increases with l ( s < p < d < f ).

- Magnetic Quantum Number ( m l ): m l = - l , ..., 0, ..., + l . Specifies the orientation in space of an orbital of a given energy ( n ) and shape ( l ). This number divides the subshell into individual orbitals which hold the electrons; there are 2 l +1 orbitals in each subshell. Thus the s subshell has only one orbital, the p subshell has three orbitals, and so on.
- Spin Quantum Number ( m s ): m s = +½ or -½. Specifies the orientation of the spin axis of an electron. An electron can spin in only one of two directions (sometimes called up and down ). The Pauli exclusion principle (Wolfgang Pauli, Nobel Prize 1945) states that no two electrons in the same atom can have identical values for all four of their quantum numbers . What this means is that no more than two electrons can occupy the same orbital, and that two electrons in the same orbital must have opposite spins . Because an electron spins, it creates a magnetic field, which can be oriented in one of two directions. For two electrons in the same orbital, the spins must be opposite to each other; the spins are said to be paired . These substances are not attracted to magnets and are said to be diamagnetic . Atoms with more electrons that spin in one direction than another contain unpaired electrons. These substances are weakly attracted to magnets and are said to be paramagnetic .

## Table of Allowed Quantum Numbers

Writing electron configurations.

1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f

examples will be added at a later date

## Properties of Monatomic Ions

- The Group IIA and IIIA metals also tend to lose all of their valence electrons to form cations.
- The Group IV and V metals can lose either the electrons from the p subshell, or from both the s and p subshells, thus attaining a pseudo-noble gas configuration .
- The Group IV - VII non-metals gain electrons until their valence shells are full (8 electrons).
- The Group VIII noble gases already possess a full outer shell, so they have no tendency to form ions.
- Transition metals (B-group) usually form +2 charges from losing the valence s electrons, but can also lose electrons from the highest d level to form other charges.

Chapter 8. Electronic Structure

## Quantum Numbers for Electrons

Where n = 3, 4, 5, 6,…, but they could not explain why this was so.

Thus, for a given value of n , there are different possible values of ℓ, as shown in Table 8.2.

The following explicitly lists the possible values of m ℓ for the possible values of ℓ:

Of the set of quantum numbers { n , ℓ, m ℓ , m s }, which are possible and which are not allowed?

- {3, 2, 1, +½}
- {2, 2, 0, −½}
- {3, −1, 0, +½}
- The principal quantum number n must be an integer, which it is here. The quantum number ℓ must be less than n , which it is. The m ℓ quantum number must be between −ℓ and ℓ, which it is. The spin quantum number is +½, which is allowed. Because this set of quantum numbers follows all restrictions, it is possible.
- The quantum number n is an integer, but the quantum number ℓ must be less than n , which it is not. Thus, this is not an allowed set of quantum numbers.
- The principal quantum number n is an integer, but ℓ is not allowed to be negative. Therefore, this is not an allowed set of quantum numbers.

## Test Yourself

- {4, 2, −2, 1}
- {3, 1, 0, −½}
- Spin must be either +½ or −½, so this set of quantum number is not allowed.

## Chemistry Is Everywhere: Neon Lights

- Electrons in atoms have quantized energies.
- The state of electrons in atoms is described by four quantum numbers.
- Differentiate between a continuous spectrum and a line spectrum.
- Under what circumstances is a continuous spectrum formed? Under what circumstances is a line spectrum formed?
- What is the wavelength of light from the hydrogen atom spectrum when n = 3?
- What is the wavelength of light from the hydrogen atom spectrum when n = 5?
- What are the restrictions on the principal quantum number?
- What are the restrictions on the angular momentum quantum number?
- What are the restrictions on the magnetic quantum number?
- What are the restrictions on the spin quantum number?
- What are the possible values for ℓ when n = 5?
- What are the possible values for ℓ when n = 1?
- What are the possible values for m ℓ when ℓ = 3?
- What are the possible values for m ℓ when ℓ = 6?
- Describe the shape of an s orbital.
- Describe the shape of a p orbital.
- {4, 1, −2, +½}
- {2, 0, 0, −½}
- {5, 2, −1, −½}
- {3, −1, −1, −½}
- A continuous spectrum is a range of light frequencies or wavelengths; a line spectrum shows only certain frequencies or wavelengths.
- 6.56 × 10 −7 m, or 656 nm
- The principal quantum number is restricted to being a positive whole number.
- The absolute value of m ℓ must be less than or equal to ℓ: | m ℓ | ≤ ℓ.
- ℓ can be 0, 1, 2, 3, or 4.
- m ℓ can be −3, −2, −1, 0, 1, 2, or 3.
- An s orbital is spherical in shape.
- Because | m ℓ | must be less than ℓ, this set of quantum numbers is not allowed.

## Media Attributions

- “Prisms and Light” by David W. Ball © CC BY-NC-SA (Attribution-NonCommercial-ShareAlike)
- “Hydrogen Spectrum” by David W. Ball © CC BY-NC-SA (Attribution-NonCommercial-ShareAlike)
- “Bohr’s Model of the Hydrogen Atom” by David W. Ball © CC BY-NC-SA (Attribution-NonCommercial-ShareAlike)
- “Electron Orbitals” by David W. Ball © CC BY-NC-SA (Attribution-NonCommercial-ShareAlike)
- “Neon Internet Cafe” by Justin Cormack © CC BY-SA (Attribution-ShareAlike)

An image that contains all colours of light.

An index that corresponds to a property of an electron, like its energy.

When a quantity is restricted to having only certain values.

The theory of electrons that treats them as a wave.

The index that largely determines the energy of an electron in an atom.

A term used to describe electrons with the same principal quantum number.

An index that affects the energy and the spatial distribution of an electron in an atom.

A term used to describe electrons in a shell that have the same angular momentum quantum number.

The index that determines the orientation of the electron’s spatial distribution.

The specific set of principal, angular momentum, and magnetic quantum numbers for an electron.

The index that indicates one of two spin states for an electron.

## Share This Book

n = 1, 2, 3, and so on.

ℓ = 0, 1, 2, . . . , n − 1

m ℓ starts at negative ℓ, runs by whole numbers to zero and then goes by whole numbers to positive ℓ.

"On the basis of these results one is also led to a general classification of every electron in the atom by the principal quantum number n and two auxiliary quantum numbers k 1 and k 2 to which is added a further quantum number m ℓ in the presence of an external field. In conjunction with a recent paper by E. C. Stoner this classification leads to a general quantum theoretical formulation of the completion of electron groups in atoms."

after the n, ℓ and m ℓ to be used have been determined, assign the value + 1 ⁄ 2 to one electron, then assign − 1 ⁄ 2 to the next electron, while using the same n, ℓ and m ℓ values.

Quantum Number Symbol Possible Values Principal n 1, 2, 3, 4, . . . Angular Momentum ℓ 0, 1, 2, 3, . . . , (n − 1) Magnetic m ℓ −ℓ, . . . , −1, 0, 1, . . . , ℓ Spin m s + 1 ⁄ 2 , − 1 ⁄ 2

## Fifteen Examples

(a) 3, 2, 2, − 1 ⁄ 2 (b) 4, 3, 3, + 1 ⁄ 2 (c) 2, 1, −3, + 1 ⁄ 2 (d) 2, 0, 0, − 1 ⁄ 2 (e) 1, 0, 1, − 1 ⁄ 2

(a) All n values are correct. (b) All m s values are correct.

All the n and ℓ pairs are correct. For example, look at (c). When n = 2, the possible ℓ values are 0, 1, and 2. (c) has a correct n, ℓ pair.

Choice (c) is incorrect. When ℓ = 1, the possible m ℓ values are −1, 0, and +1. −3 is not an allowed value when ℓ = 1.

Answer is c ---> −3 not possible with PQN 2 Notice that this method is comparing the n value to the m ℓ . The absolute value for m ℓ will always be lesser than n, never greater than or equal to n. In this question, the absolute value of the m ℓ value is 3, which is greater than the n value of 2. Look at this set: n = 2, ℓ = 1, m ℓ = −2, m s = − 1 ⁄ 2

(a) 2, 2, −1, + 1 ⁄ 2 (g) 2, 1, −1, + 1 ⁄ 2 (b) 0, 2, 1, + 1 ⁄ 2 (h) 1, 2, 0, + 1 ⁄ 2 (c) 2, 0, 0, − 1 ⁄ 2 (i) 1, 0, 0, ± 1 ⁄ 2 (d) 3, −2, −1, − 1 ⁄ 3 (j) 4, 3, 1, − 1 ⁄ 2 (e) 3, 2, 1, + 1 ⁄ 2 (k) 3.5, 3, 1, + 1 ⁄ 2 (f) 4, 3, −5, − 1 ⁄ 2 (o) 3, 2, 1, −1

We find that (b) and (k) have invalid n values. n starts at 1 [which eliminates (b)] and goes by positive integer values [which eliminates (k)]

We find that (d), (i), and (o) have invalid m s values. (d) is invalid because of the use of the 1 ⁄ 3 value. The m s numerical value is only 1 ⁄ 2 , either positive or negative. (i) is invalid because the m s is EITHER positive or negative, not both. You must use a single positive sign or a single negative sign, not both at the same time. (o) is invalid becase it has a value of 1. Only the numeral 1 ⁄ 2 is used for m s values.

(a) is not valid because ℓ always ends at n − 1. ℓ can never be equal to n. In addition to failing the definition of m s , (d) also fails with its use of a negative value for ℓ. ℓ is never negative. (h) has an ℓ which is greater than n. ℓ is always less than n. (h) is not a valid set of quantum numbers.

Here's the relationship between ℓ and m ℓ : m ℓ starts at negative ℓ, runs by whole numbers to zero and then goes by whole numbers to positive ℓ

(c) 2, 0, 0, − 1 ⁄ 2 (e) 3, 2, 1, + 1 ⁄ 2 (g) 2, 1, −1, + 1 ⁄ 2 (j) 4, 3, 1, − 1 ⁄ 2

(a) n = 2, ℓ = 1, m ℓ = 0, m s = + 1 ⁄ 2 (b) n = 3, ℓ = 3, m ℓ = −2, m s = − 1 ⁄ 2 (c) n = 4, ℓ = 3, m ℓ = −2, m s = + 1 ⁄ 2 (d) n = 3, ℓ = 2, m ℓ = 2, m s = + 1 ⁄ 3 (e) n = 2, ℓ = 1, m ℓ = −2, m s = − 1 ⁄ 2 (f) n = 3, ℓ = 2, m ℓ = −1, m s = − 1 ⁄ 2

m s can only take on values of + 1 ⁄ 2 and − 1 ⁄ 2 Now, you look for any that violate that rule, to find: (d) n = 3, ℓ = 2, m ℓ = 2, m s = + 1 ⁄ 3 We don't care what the other values in (d) are, the presence of the + 1 ⁄ 3 makes choice (d) disallowed. Now, scan the n values. Sometimes a value of 0 is used when the question is asking for incorrect sets. In this question, all the n values are allowed.

n = 1, 2, 3, . . . (all integer values) ℓ values range from zero to n − 1 (all integer values) The key is that ℓ must ALWAYS be smaller than n. See choice (b): (b) n = 3, ℓ = 3, m ℓ = −2, m s = − 1 ⁄ 2 When n equals 3, then the allowed ℓ values will be 0, 1, and end with 2. Three (3) would not be included.

m ℓ ranges from −ℓ to zero to +ℓ (all integer values) The point is that the absolute value of m ℓ cannot be greater than that of ℓ. Look at (e): (e) n = 2, ℓ = 1, m ℓ = −2, m s = − 1 ⁄ 2 When ℓ = 1, the allowed m ℓ values are −1, 0, +1. So m ℓ = −2 is disallowed.

(a) n = 1, ℓ = 0, m ℓ = 0, m s = +1 (b) n = 1, ℓ = 3, m ℓ = 3, m s = + 1 ⁄ 2 (c) n = 3, ℓ = 2, m ℓ = 3, m s = − 1 ⁄ 2 (d) n = 0, ℓ = 1, m ℓ = 0, m s = + 1 ⁄ 2 (e) n = 2, ℓ = 1, m ℓ = −1, m s = + 3 ⁄ 2 (f) n = 4, ℓ = 3, m ℓ = 5, m s = + 1 ⁄ 2

n = principal quantum number = major energy level Values are 1, 2, 3, . . . ℓ = azimuthal quantum number = energy sublevel Values are 0 to n − 1. m ℓ = magnetic quantum number = the orbital in the sublevel Values are −ℓ, . . . , 0, . . . , +ℓ m s = spin quantum number = electron in orbital Values are + 1 ⁄ 2 or − 1 ⁄ 2

The error is that m s = +1. The m s value can only be + 1 ⁄ 2 or − 1 ⁄ 2 .

Since n = 1 in (b), the value for ℓ MUST be 0. Remember, ℓ starts at 0 and goes up by integers to n − 1. So, we start at 0 and work our way up to 1 minus 1, which also equals 0. With n = 1, there is only one possible ℓ value and it is 0. This set also fails a comparison of n and m ℓ . The absolute value for m ℓ is always smaller than that for n.

Since ℓ = 2, the values for m ℓ will be −2, −1, 0, +1, +2. The error in (c) is that m ℓ is 3, when the maximum m ℓ (when ℓ = 2) can only be 2. Choice (c) also fails the n and m ℓ comparison. The absolute value for m ℓ is always smaller than n, never equal or larger.

n values start with 1, not 0.

m s is allowed to only be values of + 1 ⁄ 2 or − 1 ⁄ 2 . + 3 ⁄ 2 fails this rule.

−3, −2, −1, 0, +1, +2, +3 The m ℓ value of 5 is not permitted in this set of quantum numbers.

With n = 5, the possible ℓ values are 0, 1, 2, 3, 4. Since m ℓ ranges from −ℓ to +ℓ, we need all ℓ that generate a −2. When ℓ = 0, m ℓ can only equal 0. When ℓ = 1, the most negative m ℓ generated is −1. When ℓ = 0 or 1, an m ℓ value of −2 is not generated. ℓ values of 2, 3, and 4 will each include an m ℓ value equal to −2 Here's one example: When ℓ = 3, m ℓ values are 3, 2, 1, 0, −1, −2, −3.

(a) 3, 2, −1, +1 (b) 3, 3, −1, + 1 ⁄ 2 (c) 3, 1, −2, − 1 ⁄ 2 (d) 3, 1, 0, + 1 ⁄ 2

(a) 3, 2, −1, +1 s is either + 1 ⁄ 2 or − 1 ⁄ 2 , not +1 (b) 3, 3, −1, + 1 ⁄ 2 (c) 3, 1, −2, − 1 ⁄ 2 ℓ takes on values from −ℓ to +ℓ, by integers. When ℓ = 1, m ℓ takes on the values of 1, 0, and −1. A m ℓ value of −2 is not permitted when ℓ = 1 (d) 3, 1, 0, + 1 ⁄ 2 ℓ , then m s

All possible ℓ values range from 0 to n − 1 by integers, so: 0, 1, 2, 3

1) ℓ = 0 is associated with the s orbital and each principal quantum number has an ℓ = 0, so 4 occurences in K gives 2 in the first shell, 2 in the second shell, 2 in the third and 1 in the 4th. Total = 7. The quantum number sets are: 1, 0, 0, + 1 ⁄ 2 2, 0, 0, + 1 ⁄ 2 3, 0, 0, + 1 ⁄ 2 4, 0, 0, + 1 ⁄ 2 1, 0, 0, − 1 ⁄ 2 2, 0, 0, − 1 ⁄ 2 3, 0, 0, − 1 ⁄ 2

2, 1, 0, + 1 ⁄ 2 3, 1, 0, + 1 ⁄ 2 2, 1, 0, − 1 ⁄ 2 3, 1, 0, − 1 ⁄ 2 By the way, K does have an electron in the 4th shell, but it's not in a p orbital. It's in the s orbital of the 4th shell, discussed just above.

The other 8 electrons have ℓ = −1 (four of them, two in 2nd shell and 2 in 3rd shell) and ℓ = +1 (four of them, two in 2nd shell and 2 in 3rd shell). Writing out these eight quantum number sets is left as an exercise for the student.

n is an integer and can range from 1 to infinity ℓ is an integer and can range from 0 to n − 1 m ℓ is an integer and can range from −ℓ to +ℓ m s is either + 1 ⁄ 2 or − 1 ⁄ 2

ℓ = 0 Only an m ℓ of 0 can be generated. ℓ = 0 is not part of the answer.

Three m ℓ values are generated when ℓ = 1. They are −1, 0 , and 1. Since a 2 cannot be generated, ℓ = 1 is not part of the answer.

The m ℓ values generated are −2, −1, 0, 1, and 2. Since a 2 is generated, this will become part of the correct answer.

Since a 2 for the m ℓ is also generated here, this is the other part of the correct answer. The m ℓ values generated are −3, −2, −1, 0, 1, 2, and 3.

4, 2, 2 and 4, 3, 2 However, we are not yet done.

the maximum number of electrons that can have the quantum numbers n = 4 and m ℓ = 2 is four.

4, 2, 2, + 1 ⁄ 2 4, 2, 2, − 1 ⁄ 2 4, 3, 2, + 1 ⁄ 2 4, 3, 2, − 1 ⁄ 2

(a) n = 5, ℓ = 3, m ℓ = −1, m s = + 1 ⁄ 2 (b) n = 1, ℓ = 0, m ℓ = 0, m s = − 1 ⁄ 2 (c) n = 2, ℓ = 2, m ℓ = 2, m s = + 1 ⁄ 2 (d) n = 4, ℓ = 1, m ℓ = 0, m s = − 1 ⁄ 2 (e) n = 6, ℓ = 4, m ℓ = −3, m s = + 1 ⁄ 2

For n = 5, ℓ = 0, 1, 2, 3, 4. Thus, ℓ = 3 is allowed. For ℓ = 3, m ℓ = −3, −2, −1, 0, 1, 2, 3. Thus m ℓ = −1 is allowed. m s = + 1 ⁄ 2 is allowed.

For n = 1, ℓ = 0 only. Thus, ℓ = 0 is allowed. For ℓ = 0, m ℓ = 0 only. Thus m ℓ = 0 is allowed. m s = − 1 ⁄ 2 is allowed.

For n = 2, ℓ = 0, 1. Thus, ℓ = 2 is NOT allowed.

For n = 4, ℓ = 0, 1, 2, 3. Thus, ℓ = 1 is allowed. For ℓ = 1, m ℓ = −1, 0, 1. Thus m ℓ = 0 is allowed. m s = − 1 ⁄ 2 is allowed.

For n = 6, ℓ = 0, 1, 2, 3, 4, 5. Thus, ℓ = 4 is allowed. For ℓ = 4, m ℓ = −4, −3, −2, −1, 0, 1, 2, 3, 4. Thus m ℓ = −3 is allowed. m s = + 1 ⁄ 2 is allowed.

(a) n = 0, ℓ = 3, m ℓ = −3, m s = + 1 ⁄ 2 (e) n = 1, ℓ = −1, m ℓ = 1, m s = − 1 ⁄ 2 (b) n = 3, ℓ = −1, m ℓ = 0, m s = + 1 ⁄ 2 (f) n = 3, ℓ = 3, m ℓ = −2, m s = − 1 ⁄ 2 (c) n = 3, ℓ = 2, m ℓ = −3, m s = − 1 ⁄ 2 (g) n = 0, ℓ = −2, m ℓ = 1, m s = + 1 ⁄ 2 (d) n = 5, ℓ = 3, m ℓ = −2, m s = −1 (h) n = 3, ℓ = −2, m ℓ = 1, m s = + 4 ⁄ 3

(a) n cannot equal zero. It starts at 1 and goes up by integers from there. (b) n is correct, but ℓ is wrong. ℓ starts at zero and goes up by integers to n − 1. (c) m ℓ is incorrect. When ℓ = 2, m ℓ can only take on these five values: −2, −1, 0, 1, 2. (d) m s is incorrect. Values of + 1 ⁄ 2 and − 1 ⁄ 2 are the only two values allowed. (e) ℓ is incorrect. When n = 1, ℓ can only take on the value of 0. (f) ℓ is incorrect. When n = 3, ℓ can take on values only up to n minus 1, so ℓ = 3 is not allowed. (g) n is incorrect. The value of n starts at 1, zero is not allowed. (h) ℓ is incorrect. It cannot be a negative number. m s also is incorrect.

m ℓ starts at negative 'ℓ,' runs by whole numbers to zero and then goes by whole numbers to positive 'ℓ.'

−1, 0, +1

(a) ℓ = 0, 1, 2, 3 (b) ℓ = −2, −1, 0, 1, 2 (c) ℓ = −3, −2, −1, 0, 1, 2, 3 (d) ℓ = 0, 1, 2

(a) n = 1, ℓ = 0, m ℓ = 0, m s = + 1 ⁄ 2 (b) n = 1, ℓ = 3, m ℓ = 3, m s = + 1 ⁄ 2 (c) n = 3, ℓ = 2, m ℓ = −2, m s = − 1 ⁄ 2 (d) n = 2, ℓ = 1, m ℓ = −1, m s = + 3 ⁄ 2

The n value is allowed, but ℓ = 3 when n = 1 is not allowed. When n = 1, the only allowed ℓ value is zero. By the way, that is the case in (a). It has n = 1, so the only allowed ℓ value would be zero, which is what (a) has. When ℓ = 0, the only possible m ℓ value is also zero and that's what (a) has.

The disallowed value in (d) is m s equalling + 3 ⁄ 2 . For all possible sets of n, ℓ, and m ℓ the only choices for m s are + 1 ⁄ 2 and − 1 ⁄ 2 The n, ℓ, and m ℓ values for (d) follow the rules correctly, however, the m s value makes the set be not allowed.

(a) n = 4, ℓ = 2, m ℓ = −1, m s = − 1 ⁄ 2 (b) n = 6, ℓ = 2, m ℓ = 1, m s = + 1 ⁄ 2 (c) n = 1, ℓ = −1, m ℓ = −2, m s = + 1 ⁄ 2 (d) n = 6, ℓ = 0, m ℓ = 1, m s = + 1 ⁄ 2

Rationale #1: ℓ values start at zero and go by integers up to n − 1. When n = 1, the only possible ℓ value is zero. A value of −1 is not allowed in this example. Rationale #2: ℓ values cannot be negative. Ever.

The n, ℓ combination of 6, 0 is allowed. The problem comes with the m ℓ value. Remember, m ℓ values go from −ℓ to 0 to +ℓ. When ℓ = 0, the only possible m ℓ value is 0. Since m ℓ is incorrect, (d) is the set that is not allowed.

1, 0, 0, + 1 ⁄ 2 1, 0, 0, − 1 ⁄ 2 These two electrons are in the 1s orbital. s orbitals are always characterized by ℓ and m ℓ equalling zero.

The 2s orbital: 2, 0, 0, + 1 ⁄ 2 2, 0, 0, − 1 ⁄ 2

2, 1, −1, + 1 ⁄ 2 2, 1, −1, − 1 ⁄ 2 2, 1, 0, + 1 ⁄ 2 2, 1, 0, − 1 ⁄ 2 2, 1, 1, + 1 ⁄ 2 2, 1, 1, − 1 ⁄ 2

3, 0, 0, + 1 ⁄ 2 This is the answer. By convention, the positive m s is used first. This electron is in the 3s orbital.

The use of 5d provides the answer. The 5 in 5d is the value of n the question wants. Other examples: the value for n in the 4p orbital is 4. The value for n in the 2s orbital is 2.

We need to determine what ℓ value is associated with d orbitals. The answer to that is 2. (s orbitals have ℓ = 0, p orbitals have ℓ = 1, d has ℓ = 2 and f has an ℓ of 3.) We now apply the rule for m ℓ and determine the m ℓ values for the 5d orbital: −2, −1, 0, 1, 2

5, 2, −2 5, 2, −1 5, 2, 0 5, 2, 1 5, 2, 2 Sometimes, a teacher will insist that positive signs for the last two m ℓ be included. It's just a stylistic thing, so go along with it if it happens to you.

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## Assume that the rules for quantum numbers are different and that the spin quantum number ms can have any of three values, ms = -1/2, 0, +1/2, while all other rules remain the same. (a) Draw an orbital-filling diagram for the element with Z = 25, showing the individual electrons in the outer-most subshell as up arrows, down arrows, or 0. How many partially filled orbitals does the element have?

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