Electron Configuration

From MyMCAT

Contents 1 Introduction 2 Quantum Numbers 3 Pauli Exclusion Principle 4 The Aufbau Principle

Introduction

Electron configuration describes the arrangement of electrons in an atom, molecule, or other physical structure (e.g., a crystal). The very first model of an atom's electron configuration was proposed by Niels Bohr. His model was that of an electron in orbit around a nucleus just as a planet orbits a star. The model described quite well the observation that each atom can have multiple orbitals and that with either the absorption or release or energy, electrons could be moved between orbitals.

This model intuitively makes sense, electrons are negative and spin around a positive center, but it fails to explain many phenomena. Why don't the electrons just fall in to the nucleus? Why don't the absorption spectra of the elements correlate with orbital size? and why are the orbitals fixed in energy?

The reason for the failures in Bohr's model is because electrons do not orbit in the sense that we accustomed to. According to Heisenberg, at the microscopic scale, all elementary particles can exhibit both particle-like and wave-like properties. Thus, electrons can act like waves, and as such, electrons interact with each other in complex ways and the states of electrons are no longer viewed as simple particles in orbit, but instead by complex functions of interacting waves. These properties impose certain constraints on how electrons exist and also helps explain why the periodic table is organized the way it is.

The view of electron configuration today is now of electrons in a "blurry" cloud around the nucleus with unknown positions and velocities. An orbital is no longer a simple ring around the nucleus, but instead a region in space where there is a high probability that the electron will be located at any given time (but we have no guarantee of where the electron precisely is). The use of four quantum numbers helps us explain this new view.

Quantum Numbers

In the quantum mechanical view of electron configuration, electrons exist only at specific energy levels (they are quantized) and the shape of an orbital is a probabilistic interpretation of where the electrons are most likely to exist in space around the nucleus. Orbitals, instead of referring to a path the electron would take around the nucleus now refer to regions where there is a high probability the electron is present.

The state of each electron in an atom is given by four quantum numbers. Three of these properties describe where the electron is in space around the nucleus and the fourth is independent of the others describing instead how electrons in an orbital are behaving.

Name:	Denoted:	Value:	Description:
The Principal Quantum Number	n	integer, 1 or more	This number partially defines the overall energy of the orbital and the general distance the orbital is from the nucleus. (In the classical/Bohr view you could consider this analogous to how far away the orbit was from the nucleus)
The Azimuthal Quantum Number	l	integer, 0 to n-1	This number describes the orbital's angular momentum, or otherwise known as the orbital. By convention, letters have been assigned to the integer values (s=0, p=1, d=2, f=3, ...). It however can also be interpreted as the number of nodes in the density plot of the wave function which is describing the electron. Together the Principal and Azimuthal numbers describe the overall energy that an orbital has.
The magnetic Quantum Number	ml	integer, -l to +l, including zero.	This number indicates the spatial orientation of a given orbital type (previously described by the other two numbers). Each unique orientation will share the same energy level for a given n and l value.
spin quantum number	ms	+½ or -½ (sometimes called "up" and "down")	Spin is an intrinsic property of the electron and independent of the other numbers.

Putting it all together then, we can discuss any electron in terms of these four numbers. For example, we could say that hydrogen's one electron exists in a n=1, l=0, m=0, s=+½ state, or simply it is in a "1s" orbital (where the s implies l=0 and we are ignoring the spin as it often is not important to an electrons orbital state). Any combination of these numbers describes a valid orbital so long as the basic rules above are not broken. In the next two sections however, we will see that some additional rules are necessary to accurately describe multiple electrons and energy states of an atom.

1. How many different d orbitals exist at a given energy level?

	1
	3
	5
	7
→	When we are referring to the "d" state, we are implying that l=2 (l=0 is s, l=1 is p, and l=2 is d). For l=2, there are 5 possible ml values, -2, -1, 0, 1, 2. Each one of these values, represents a unique spacial orbital, thus the answer is 5.

2. In the electron configuration of an atom, it is found that there exists exactly 3 orbitals of equal energy, what energy state do these three orbitals most likely correspond to?

	1s
	2s
	2p
	3d
→	Three orbitals with exactly the same energy imply that they all have the same n and l values, therefore they must be representing different ml values. 1s and 2s only have ONE possible ml value, and 3d has 5, therefore 2p must be the answer, and as expected, it has 3 values for ml.

3. Which of the following electron configurations represents a valid quantum number <n,l,m_l,m_s>?

	3,2,0,½
	1,3,0,-½
→	if n=1, then the only possible values for l are 0 to n-1, which implies only l=0 would be valid, not l=3!
	3,1,2,½
→	if n=3, then l can be 0,1, or 2 so this part is correct. In the case of l=1 however, the only possible values for ml are -l to +l, or -1,0, and +1. 2 is NOT valid and violates this.
	3,2,1,0
→	While the first three quantum numbers are a valid configuration, the final spin orientation can only be +½ or -½, 0 is not a valid designation.

Your score is 0 / 0

Pauli Exclusion Principle

When we are working with multiple electrons (basically anything beyond neutral hydrogen) it becomes important to ensure there are no conflicts in electron configuration. Specifically, the Pauli Exclusion Principle states that every electron in an atom must have a unique set of quantum numbers. Therefore, if an element has 5 electrons, no two of these electrons can share the same four numbers. What does this mean? Well if one electron in an atom is in the 2s orbital with a spin of +½, then only one other electron can be in the 2s orbital with a spin of -½ and ALL the remaining electrons must be in different orbitals.

This Pauli Exclusion Principle also allows one to determine how many electrons can exist at a given energy level. For example, in the 3p state, n=3 and l=1. m_l thus, can have the values from -l to l, or -1,0, and 1 implying there are three different orbitals at this state. Additionally, m_s can be either +½ or -½. This means that the electrons can have anyone of 3 magnetic quantum numbers and 2 spins for a total of 6 different possibilities. Thus, the 3p orbitals can hold up to 6 electrons without violating the Pauli exclusion principle.

1. What is the maximum number of electrons a single p orbital can have?

	1
	2
	4
	6
→	For any given obital, there can be a maximum of 2 electrons, one with +½ spin and the other -½. This question was a bit of a trick as often it is asked how many electrons can exist at a given shell/subshell combination, which may include multiple orbitals.

2. How many electrons can exist in the 3p state?

	2
	4
	6
	8
→	Here, we must determine how many orbitals exist at the 3p state. "p" implies that the l value is 1, and therefore, the ml can be -1, 0, or +1. Thus we have a total of 3 orbitals, and each can hold two electrons, giving a total of 6.

Your score is 0 / 0

The Aufbau Principle

Now that we have described the quantum numbers and the requirement of each electron having a unique set of quantum numbers lets look at all the electrons in an atom. How do we decide which orbital each electron should go into? The theory is quite simple, but in practice can often be difficult to determine, but generally, electrons enter into each state in order of increasing energy; i.e., the first electron goes into the lowest-energy state, the second into the next lowest, and so on. The Aufbau principle describes this order and with a few exceptions is generally correct.

When determining the overall energy of an orbital both the principal (ie the shell) and azimuthal (ie the subshell) quantum numbers must be taken into account. Simply stated, the sum of "n" and "l" (the principle and azimuthal quantum numbers) provides a rough approximation of the energy, and thus in comparing two orbitals, the lower n+l orbital is filled first. If the n+l values are the same, the one with the lower principle quantum number is filled first. (This rule is often referred to as the n+l rule.)

For example, n = 3 and l = 0 (3s) has the same n+l as n = 2 and l = 1 (2p) both are n+l = 3 so the "rule" guides you to complete the 2p orbitals before filling the 3s.

The complete order of orbital filling can easily be derived for the whole set of possible orbitals by using the n+l rule: 1s, 2s, 2p, 3s, 3p, 4s, 3d... The order can also however be visually mapped out and described in the diagram to the right. Furthermore, one can determine the filling order by referring to the periodic table. Each row down the periodic table increases the n value, and each "block" is a different l value. The s block is the two leftmost columns, the p block is the 6 rightmost columns, the transition metals are the d block, etc.

This figure shows the orbitals organized by the aufbau principle. The superscript (top) numbers state how many electrons are in that orbital, while the subscript (bottom) numbers state how many electrons in total there from 1s to the current orbital. An atom in its ground state will always fill the maximum value possible for each orbital as it increases whereas atoms which are excited with extra energy often skip orbitals and contain electrons at higher than normal orbital states.

If a atom's electron configuration is to written out, often it is written as simply a noble gas configuration plus some additional orbitals. This reduces the number of orbitals to write and emphasizes only the valence shell electrons.

1. What neutral element has the following electron configuration? 1s² 2s² 2p⁶ 3s²3p⁶ 4s² 3d¹⁰ 4p⁶5s²

	Sr
	Ca
	Pb
	Zr
→	Firstly, if we know the number of electrons than because the element is neutral we also know the number of protons and thus the atomic number. In this case, there is a total of 38 electrons and so the element must be Sr. We can also observe that the configuration is that of a ground state and that the last orbital filled is the 5s, so one could identify that position on the periodic table to also answer with Sr with less counting.

2. What is the valence electron configuration of Arsenic?

	[Ar] 4s2 3d10 4p3
	[Ne] 3s2 3p3
	1s22s22p43s23p6
	[Ar] 3s2 3d5 3p3
→	The valence electron's of Arsenic are only those electrons which are in the same row as Arsenic. Thus we need only concern ourselves with the last few orbitals 4s, 3d, and 4p as the previous ones are not valence but instead internal. Furthermore, we must determine how many electrons Arsenic has (33) or more specifically, how many are in the valence shell (15) and fill the necessary orbitals to arrive at the answer. Only the first choice has the correct number of electrons AND use of the valence orbitals. Notice also that we must remember the Aufbau order to fill in the correct orbitals.

3. Order of the following orbitals from lowest to highest energy state: 4p, 3s, 2p, 3d, and 4s?

	4p, 3s, 2p, 3d, 4s.
	3s, 2p, 3d, 4s, 4p.
	2p, 3s, 4s, 3d, 4p.
	2p, 3s, 3d, 4s, 4p.
→	To answer this question one can either write out the Aufbau order and determine where each of these lies, or use the "n+l" to rank them.

4. What is the electron configuration of S^2-?

	[Ne] 3s2 3d4
	[Ne] 3s2
	[Ne] 3s2 3p6
	[Ne] 3s2 3p4
→	A charge of negative two implies that we have two extra electrons we need to fill. If we first determine the ground state of neutral S we would get [Ne] 3s23p4, then we need to add two additional electrons, which because the 3p isn't full yet would go into that orbital to make it [Ne] 3s2 3p6. Note that this is also the configuration of the next noble gas, Ar. This helps explain why S prefers a -2 charge as it implies its electron configuration has reached a full valence shell.

5. Which of the following configurations would best represent Mg in an excited state?

	1s2 2s2 2p6 3s2
	1s2 2s2 2p6 3s23p1
	1s2 2s2 2p6 3s1
	1s2 2s2 2p6 3s13p1
→	To answer this question we must recognize what an excited state really means. Excited states imply that electrons are not in their usual ground state of filling the lowest energy orbitals first, rather, one or more electrons may be existing in a higher energy state and thus some lower orbitals may be empty. The actual number of electrons however is still the same. Thus, for this question, we must find out how many electrons Mg normally has and then decide of those with the correct number, which is not in its ground state.

Your score is 0 / 0