So that k k E 5 = - E 2 = = x J = x J Therefore = E 5 - E 2 = x J Now so 631

So that k k E 5 = - E 2 = = x J = x J Therefore = E 5 - E 2 = x J Now so = x m 632

Definition: The ionization energy is the energy required to remove an electron from one mole of a substance in its ground state in the gas phase. 633

Definition: The ionization energy is the energy required to remove an electron from one mole of a substance in its ground state in the gas phase. For example, for substance X, X (g) X (g) + + e - 634

Problem: Calculate the ionization energy for the hydrogen atom. (k = x J and N A = x mol -1 ). 635

Problem: Calculate the ionization energy for the hydrogen atom. (k = x J and N A = x mol -1 ). The process is: H (g) H + (g) + e - 636

Problem: Calculate the ionization energy for the hydrogen atom. (k = x J and N A = x mol -1 ). The process is: H (g) H + (g) + e - The quantum numbers are n l = 1 and n u = (because the electron is completely removed from the atom). 637

Therefore = - E 1 = 0 - (-k) 638

Therefore = - E 1 = 0 - (-k) = k = x J 639

Therefore = - E 1 = 0 - (-k) = k = x J This corresponds to the energy to remove an electron from one atom. 640

Therefore = - E 1 = 0 - (-k) = k = x J This corresponds to the energy to remove an electron from one atom. To get the ionization potential, we need the energy expended for 1 mole of atoms. Hence 641

Therefore = - E 1 = 0 - (-k) = k = x J This corresponds to the energy to remove an electron from one atom. To get the ionization potential, we need the energy expended for 1 mole of atoms. Hence Ionization energy = x J ( x mol -1 ) = x 10 6 J mol

Definition: The electron affinity is the energy required to add an electron to one mole of a substance in its ground state in the gas phase. 643

Definition: The electron affinity is the energy required to add an electron to one mole of a substance in its ground state in the gas phase. For example, for substance Y, Y (g) + e - Y (g) - 644

Exercise: Calculate the electron affinity for the hydrogen positive ion. 645

Exercise: Calculate the electron affinity for the hydrogen positive ion. (k = x J and N A = x mol -1 ). 646

Exercise: Calculate the electron affinity for the hydrogen positive ion. (k = x J and N A = x mol -1 ). (By convention the electron affinity is the difference between the energy of the product formed and the energy of the starting species, in this case E(H) – E(H + ) ). 647

How does the Schrödinger eq. do for more complex systems, e.g. He atom, CO molecule, etcetera? 648

How does the Schrödinger eq. do for more complex systems, e.g. He atom, CO molecule, etcetera? It turns out that the Schrödinger equation is too complex to solve exactly for systems with more than one electron. However, approximate solutions have been obtained which match up with experimental results. 649

How does the Schrödinger eq. do for more complex systems, e.g. He atom, CO molecule, etcetera? It turns out that the Schrödinger equation is too complex to solve exactly for systems with more than one electron. However, approximate solutions have been obtained which match up with experimental results. For example, some of the energy levels of the He atom have been determined to around 10 significant figures! The ground state energy level for He is known to over 40 significant figures! 650

It is generally accepted that the Schrödinger equation provides a very good description of the ground and excited states of atomic and molecular systems. 651

It is generally accepted that the Schrödinger equation provides a very good description of the ground and excited states of atomic and molecular systems. The work of Schrödinger was one of the landmark breakthroughs in all of science. 652

The fact that the H atom can be solved exactly has turned out to be important for studying more complex atoms. 653

The fact that the H atom can be solved exactly has turned out to be important for studying more complex atoms. It is assumed the electronic behavior in many-electron atoms is not too different from that in the H atom. 654

The fact that the H atom can be solved exactly has turned out to be important for studying more complex atoms. It is assumed the electronic behavior in many-electron atoms is not too different from that in the H atom. So the results from the H atom can be used as a first approximation for describing the behavior of the electrons in more complex atoms. 655

The fact that the H atom can be solved exactly has turned out to be important for studying more complex atoms. It is assumed the electronic behavior in many-electron atoms is not too different from that in the H atom. So the results from the H atom can be used as a first approximation for describing the behavior of the electrons in more complex atoms. The justification for this approach is that it works! 656

Some final thoughts on the Bohr model 657

Some final thoughts on the Bohr model Obviously the result obtained by Bohr was very useful to Schrödinger. 658

Some final thoughts on the Bohr model Obviously the result obtained by Bohr was very useful to Schrödinger. Connection with Heisenberg Uncertainty Principle. 659

Some final thoughts on the Bohr model Obviously the result obtained by Bohr was very useful to Schrödinger. Connection with Heisenberg Uncertainty Principle. According to Bohr, an electron is always circling around the nucleus in a well specified orbit. 660

Some final thoughts on the Bohr model Obviously the result obtained by Bohr was very useful to Schrödinger. Connection with Heisenberg Uncertainty Principle. According to Bohr, an electron is always circling around the nucleus in a well specified orbit. Although the electron may switch from one orbit to another, as in an emission or absorption process, its position in an orbit at a specific distance from the nucleus is fixed once we know its energy state. 661

A simplified Heisenberg argument. 662

A simplified Heisenberg argument. Suppose the energy of the electron in the H atom is all kinetic energy (the argument could be modified to take account that there is a second energy component). 663

A simplified Heisenberg argument. Suppose the energy of the electron in the H atom is all kinetic energy (the argument could be modified to take account that there is a second energy component). m 2 v 2 p 2 E = ½ m v 2 = = 2m 2m 664

Now if E is known exactly, then so is p, hence the error in p is zero, so =

Now if E is known exactly, then so is p, hence the error in p is zero, so = 0. Now 666

Now if E is known exactly, then so is p, hence the error in p is zero, so = 0. Now so 667

Now if E is known exactly, then so is p, hence the error in p is zero, so = 0. Now so In other words, the electron cannot be in a well defined orbit! 668

Now if E is known exactly, then so is p, hence the error in p is zero, so = 0. Now so In other words, the electron cannot be in a well defined orbit! The electron could be anywhere!!!!! 669

Now if E is known exactly, then so is p, hence the error in p is zero, so = 0. Now so In other words, the electron cannot be in a well defined orbit! The electron could be anywhere!!!!! This is the death certificate of the Bohr model. 670

We learn from quantum theory that we should never think of the electron as being confined to a certain path. 671

We learn from quantum theory that we should never think of the electron as being confined to a certain path. It is more appropriate to speak of the probability of locating the electron in a certain region of space. 672

We learn from quantum theory that we should never think of the electron as being confined to a certain path. It is more appropriate to speak of the probability of locating the electron in a certain region of space. This probability is given by the square of the wave function. 673

Since the electron has no well-defined position in the atom, it is most convenient to use terms like: 674

Since the electron has no well-defined position in the atom, it is most convenient to use terms like: Electron density 675

Since the electron has no well-defined position in the atom, it is most convenient to use terms like: Electron density Electron Charge Cloud 676

Since the electron has no well-defined position in the atom, it is most convenient to use terms like: Electron density Electron Charge Cloud Charge Cloud to represent the probability concept. 677

To distinguish the quantum mechanical description from Bohr’s model, the word “orbit” is replaced with the term orbital or atomic orbital. 678

To distinguish the quantum mechanical description from Bohr’s model, the word “orbit” is replaced with the term orbital or atomic orbital. Atomic orbital means exactly the same as wave function describing one electron. 679

When we say that an electron is in a certain orbital, we mean that the distribution of the electron density or the probability of locating the electron in space is described by the square of the wave function associated with that energy state. 680

When we say that an electron is in a certain orbital, we mean that the distribution of the electron density or the probability of locating the electron in space is described by the square of the wave function associated with that energy state. For each atomic orbital, there is an associated energy as well as an associated electron density. 681

Quantum Numbers 682

Quantum Numbers From quantum mechanics it is found that four quantum numbers are necessary to describe the placement of electron(s) in the hydrogen atom or in any other atom. 683

Quantum Numbers From quantum mechanics it is found that four quantum numbers are necessary to describe the placement of electron(s) in the hydrogen atom or in any other atom. The quantum numbers are of significance if we wish to understand the sizes and shapes of orbitals and their associated energy levels. 684

These are important, because the size, shape, and energy of the electron cloud influence the behavior of atoms. 685

1. Principal quantum Number 686

1. Principal quantum Number Symbol n 687

1. Principal quantum Number Symbol n The principal quantum number determines the energy of an orbital (remember that E n ). 688

1. Principal quantum Number Symbol n The principal quantum number determines the energy of an orbital (remember that E n ). The principal quantum number also characterizes the “size” of the orbital. 689

1. Principal quantum Number Symbol n The principal quantum number determines the energy of an orbital (remember that E n ). The principal quantum number also characterizes the “size” of the orbital. The larger the value of n, the larger the orbital, and the farther on the average the electron is from the nucleus. 690

Roughly speaking, the “size” of an orbital is proportional to n 2. As n increases, the “size” differences among orbitals becomes very large. 691

Roughly speaking, the “size” of an orbital is proportional to n 2. As n increases, the “size” differences among orbitals becomes very large. Because the “sizes” of orbitals with different n values differ so significantly, the regions of space corresponding to particular values of n are referred to as shells around the nucleus. 692

Shell K L M N O … n … 693

2. The Angular Momentum Quantum Number Also called the azimuthal quantum number. 694

2. The Angular Momentum Quantum Number Also called the azimuthal quantum number. Symbol l 695

2. The Angular Momentum Quantum Number Also called the azimuthal quantum number. Symbol l The angular momentum quantum number determines the “shape” of the orbitals. 696

2. The Angular Momentum Quantum Number Also called the azimuthal quantum number. Symbol l The angular momentum quantum number determines the “shape” of the orbitals. The possible values of l depend on the value of the principal quantum number n. 697

For a given value of n, l takes values from 0 to n – 1 (in steps of 1). 698

For a given value of n, l takes values from 0 to n – 1 (in steps of 1). If n = 1, there is only one value of l, that is l = n – 1 = 1 – 1 = 0 699

For a given value of n, l takes values from 0 to n – 1 (in steps of 1). If n = 1, there is only one value of l, that is l = n – 1 = 1 – 1 = 0 If n = 2, there are two values of l, that is l = 0 and l = 1 700

For a given value of n, l takes values from 0 to n – 1 (in steps of 1). If n = 1, there is only one value of l, that is l = n – 1 = 1 – 1 = 0 If n = 2, there are two values of l, that is l = 0 and l = 1 If n = 5, there are five values of l, that is l = 0, l = 1, l = 2, l = 3, l = 4 701

Each value of l for a given value of n defines a subshell. 702

Each value of l for a given value of n defines a subshell. The following letters are used as symbols to designate the different values of l. 703

Each value of l for a given value of n defines a subshell. The following letters are used as symbols to designate the different values of l. l value … orbital designation s p d f g h …. 704

3. The Magnetic quantum Number 705

3. The Magnetic quantum Number Symbol m l 706

3. The Magnetic quantum Number Symbol m l This quantum number is used to explain the additional lines that appear in the spectra of atoms when they emit light while confined in a magnetic field. 707

The magnetic quantum number determines the orientation of the orbital in space. 708

The magnetic quantum number determines the orientation of the orbital in space. The value of m l depends on the value of l. For a given value of l there are 2 l + 1 integer values of m l ranging from - l to l. 709

The magnetic quantum number determines the orientation of the orbital in space. The value of m l depends on the value of l. For a given value of l there are 2 l + 1 integer values of m l ranging from - l to l. Examples: If l = 0, m l = 0 710

The magnetic quantum number determines the orientation of the orbital in space. The value of m l depends on the value of l. For a given value of l there are 2 l + 1 integer values of m l ranging from - l to l. Examples: If l = 0, m l = 0 If l = 1, m l = -1, or 0, or 1 711

The magnetic quantum number determines the orientation of the orbital in space. The value of m l depends on the value of l. For a given value of l there are 2 l + 1 integer values of m l ranging from - l to l. Examples: If l = 0, m l = 0 If l = 1, m l = -1, or 0, or 1 If l = 2, m l = -2, -1, 0, 1, 2 712

The number of different values that m l may take for a given subshell, indicates the number of individual orbitals. 713

The number of different values that m l may take for a given subshell, indicates the number of individual orbitals. Examples: If l = 0, there is one value for m l and only one orbital. 714

The number of different values that m l may take for a given subshell, indicates the number of individual orbitals. Examples: If l = 0, there is one value for m l and only one orbital. If l = 1, there are three values for m l and three orbitals. 715

4. The Electron Spin Quantum Number 716

4. The Electron Spin Quantum Number Symbol m s 717

4. The Electron Spin Quantum Number Symbol m s There are only two possible values for m s : m s = ½ or m s = - ½ 718

4. The Electron Spin Quantum Number Symbol m s There are only two possible values for m s : m s = ½ or m s = - ½ To explain certain spectral lines from atoms in the presence of a magnetic field, it was found to be necessary to assume that electrons act as tiny magnets. 719

720