Quantum NumbersTonya Patterson

Bohr’s Theory of the Hydrogen Atom

• Einstein’s work lead the way for the “mystery” of the emission spectra of atoms.

• In the 17th century Newton showed that sunlight is composed of various color components that can be recombined to produce white light.

• This lead to study of the characteristics of the emission spectra.

• The emission spectra is either continuous or line spectra of radiation emitted by a substance.

Emission Spectrum

• Can be seen by energizing a sample of material either with thermal energy or with some other form of energy (high-voltage electrical discharge.) • Example:• A “red-hot” or “white-hot” iron bar freshly removed form a high-temperature

source produces a characteristic glow. This glow is the portion of its emission spectrum that is sensed by the eye.

• The warmth of the iron bar represents another portion of its emission spectrum – the infrared region.

Sun and Heated Solids

• Common feature to the emission spectra of the sun and of a heated solid is that both are continuous; all wavelengths of visible light are represented in the spectra.

Atoms in the Gas Phase

• The emission spectra of atoms in the gas phase do not show a continuous spread of wavelength from read to violet, but produce bright lines in different parts of the visible spectrum. • Line spectra – are the light emissions only at specific wavelengths.

Elements

• Every element has a unique emission spectrum.

• The characteristic lines in atomic spectra can be used in chemical analysis to identify unknown atoms, much like a fingerprint are used to identify people.

• When the lines of the emission spectrum of a known element exactly match the lines of the emission spectrum of an unknown sample, the identity of the sample is established. >

Emission Spectrum of the Hydrogen Atom• After Planck and Einstein discoveries, Bohr proposed a theoretical explanation of the

emission spectrum of the hydrogen atom.

• Bohr’s proposal is complex and many parts are consider to be inaccurate. We will only focus of the parts that are relevant.

• Scientist thought the electrons whirled around the nucleus of an atom at high velocity. Supporting the planetary model of the atom, which scientist liked.

• It was believed that the electrostatic attraction between the proton and electron pulls the electron inward and that this force is balanced exactly by the outward acceleration due to the circular motion of the electron.

Emission Spectrum of the Hydrogen Atom• Based on the laws of classical physics, and electron moving in an orbit

of a hydrogen atom would experience an acceleration towards the nucleus by radiating away energy in the form electromagnetic waves.• This would lead to the electron spiraling into the nucleus and

annihilate itself with the proton.

Bohr’s Explanation

• To explain why this does not happen, Bohr postulated that the electron is allowed to occupy only certain orbits of specific energies.

• The energies of the electron are quantized.

• An electron in any of the allowed orbits will not spiral into the nucleus and therefore will not radiate energy.

• He attributed the emission of radiation by an energized hydrogen atom to the electron dropping from a higher-energy allowed orbit to a lower one and emitting a quantum of energy (a photon) in the form of light.

• Bohr showed that the energies that an electron in a hydrogen atom can occupy are given by:

• RH = Rydberg constant for the hydrogen atom (2.18 x 10-18J).• n = integer called the principal quantum number (n = 1, 2, 3 …..)• - sign (arbitrary) signifying that the energy of the electron in the atom is lower than the

energy of a free energy. • Free energy – electron far from the nucleus and assigned an arbitrary value of o. • As an electron gets closer to the nucleus En becomes larger in absolute value, but also more negative. • The most negative value is when n = 1, which corresponds to the most stable energy level. • A hydrogen electron for which n is greater than 1 is said to be in an excited state.

2

1

nRE Hn

• Bohr’s theory enables us to explain the line spectrum of the hydrogen atom. • Radiant energy absorbed by the atom causes the electron to move

from a lower-energy state (smaller n) to a higher-energy state (greater n).• Radiant energy is emitted (photon) when the electron moves from a

higher-energy state to a lower energy state. • The amount of energy needed to move an electron in the Bohr atom

depends on the difference in energy levels between the initial and final states.

• Lets suppose that an electron is initially in an excited state characterized by the principal quantum number ni. During emission, the electron drops to a lower energy state characterized by the principal quantum number nf.

• This lower energy state may be either a less excited state or ground state.

• The difference between the energies of the initial and final states is

∆E = Ef – Ei

Apply to previous formula for the energy of a hydrogen atom and since this transition results in the emission of a photon of frequency and energy we write:

22

11

fiH nnRhvE

Photon

• When a photon is emitted ni > nf (- ∆E showing energy is lost to the surroundings)

• When energy is absorbed ni < nf (+ ∆E showing energy is taken in from surroundings)

• Each spectral line in the emission spectrum corresponds to a particular transition in a hydrogen atom.

• When large amounts of hydrogen atoms, all possible transitions and hence the corresponding spectral lines. • The brightness of a spectral line depends on how many photons of

the same wavelength are emitted.

Transition Series

• Emission spectrum of hydrogen includes a wide range of wavelengths from the infrared to the ultraviolet. • Below is the list of transitions in the hydrogen spectrum. They are

named after their discoverers.

Series nf ni Spectrum Region

Lyman 1 2,3,4,…… UltravioletBalmer 2 3,4,5,…… Visible and ultravioletPaschen 3 4,5,6,….. InfraredBrackett 4 5,6,7,….. Infrared

• The Balmer series was particularly easy to study because a number of its lines fall in the visible range. • The graph shows a singe transition.

• Each horizontal line represents an allowed energy level for the electron in a hydrogen atom.

Quantum Numbers

• In quantum mechanics, 3 quantum numbers are required to describe the distribution of electrons in hydrogen and other atoms.

• These numbers are derived from the mathematical solution of the Schrodinger equation for the hydrogen atom.

• They are called principal quantum number, the angular momentum number and the magnetic quantum number.

• Quantum numbers will be used to describe atomic orbitals and to label electrons that reside in them.

• A fourth quantum number – the spin quantum number – describes the behavior of a specific electron and completes the description of electrons in atoms.

The Principal Quantum Number (n)

• Can have integral values 1, 2, 3 and so on.

• It corresponds to the quantum number in Equation for energy of hydrogen.

• In a hydrogen atom, the value of n determines the energy of an orbital.• This is not the case for a many-electron atom.

• The principal quantum number also relates to the average distance of the electron from the nucleus in a particular orbital.

• The larger the n is, the greater the average distance of an electron in the orbital from the nucleus and therefore the larger the orbital.

The Angular Momentum Quantum (l)

• Tells the “shape” of the orbitals • The value of l depends on the value of th eprincipal quantum number,

n.• For a given value of n, l has possible integral values from 0 to (n-1).• If n=1, there is only one possible value of l; l = n – 1= 1-1=0. If n=2

there are two values of l, given by 0 and 1. If n=3, there are three value of l, given by 0,1 and 2. • The value of l is generally designated by the letters s, p, d, …… as

follows: L0 0 1 2 3 4 5

Name of orbital

s p d f g h

The Angular Momentum Quantum Number • If l = 0, we have an s orbital • If l = 1, we have a p orbital and so forth.

• The unusual sequence of letters (s, p, d, ..) has a historical origin. • Physicist who studied atomic emission spectra tried to correlate the

observed spectral lines with the particular energy states involved in the transitions. • Some lines where sharp (s)• Some lines where more spread out, or diffused (d)• Some were very strong and hence referred to a s principal lines (p) • (f) for fundamental and then alphabetical after that

Subshells

• A collection of orbitals with the same n is frequently called a shell. • One or more orbitals with the same n and l wavlues are referred to as

a subshell. • n = 2 is composed of 2 subshells, l = 0 and 1 (the allowed value for n = 2). • These subshells are 2s and 2p subshells where 2 denotes the value of n, and s

and p denotes the values of l.

The Magnetic Quantum Number (ml)

• Describes the orientation of the orbital in space.

• Within a subshell, the value of ml depends on the value of the angular momentum quantum number, l.

• For a certain value of l, there are (2l + 1) integral values fo ml as follows: • If l = 0, then ml = 0. If l = 1, then there are [(2 x 1) + 1], or 3 values of ml namely, -1, 0,

and 1. If l = 2, there are [(2 x 2) + 1], or 5 values for ml namely -2, -1, 0, 1, and 2.

• The number of ml values indicates the number of ortitals in a subshell with a particular l value.

Example 3 Quantum Numbers

• If we have n=2 and l= 1, the values of indicate that we have a 2p subshell, and in this subshell we have three 2p orbitals (because there are three values of ml, given by -1, 0, 1).

The Electron Spin Quantum Number (ms)• Experiments on the emission spectra of hydrogen and sodium atoms

indicated that lines in the emission spectra could be split by the application of an external magnetic fields. • Only way to explain was to assume that electrons act like tiny

magnets. • If electrons are thought of as spinning on their own axes, as Earth

does, their magnetic properties can be accounted for. • According to the electromagnetic theory, a spinning charge generates

a magnetic field, and it is the motion that causes an electron to behave like a magnet.

The Electron Spin Quantum Number (ms)• Shows 2 possible spins of the

electron, one clockwise and one counterclockwise. • To take the electron spin into

account, we introduce the 4th quantum number, called the electron spin quantum number. • Has a value of +1/2 or -1/2.

Assigning Quantum Numbers for electrons• Oxygen • Electron Configuration is 1s22s22p4 and the electron box diagram is:

• For the electron circled: It is located in the 1s, so n=1. s has a l value = to 0, we number the orbitals form a -1 to a +1. So, the ml = 0 ( only value ml can have for a s subshell). Since the electron is spinning down, the ms is -1/2. So the set of quantum numbers are (1,0,0,-1/2)• For the electron in the rectangle the set of quantum numbers are (2, 1, 1, +1/2)

Another perspective

• Which element in the ground state, is the first one with an electron with the set of quantum numbers. Given the quantum number (3,2,0, -1/2).

Practice

• What is the quantum number of the last electron placed in the following:• Barium (Ba)• Arsenic (As)

• Which of the following quantum numbers are not valid and why?• (3,3,2,1)• (4,2,1,-1/2)• (2,1,2,-1/2)

• What is the first element to have an electron in the ground state with the following set of quantum numbers?• (2,1,1,-1/2)• (4,2,2,+1/2)