Chapter 7

7.7 Bohr’s hydrogen atom

We know that the nuclei of atoms are very tiny and dense and that they are surrounded by a cloud of negatively charged electrons, but how are these electrons organized? Why can only certain energies of light be absorbed by an atom and why do elements emit only specific wavelengths of light (the line spectra)? Why are the line spectra of different elements different?

Bohr began by hypothesizing that the electron orbited the nucleus of a hydrogen atom at discrete distances that corresponded to the quantized energy levels. He was wrong about the orbiting part but was right about the quantized energy levels. Using the line spectra, he derived his explanation of energy levels mathematically:

The wavelength of light emitted or absorbed corresponds to the change in energy between different electron energy levels. Thus:

E_photon = ΔE_electron = E_f - E_i

Bohr came up with the following equation for the energy of any particular energy level:

E_n = -B/n²

where B is a constant derived from Plank’s constant as well as the mass and charge of the electron. B = 2.179 x 10^-18 J.

n is the energy level and is an integer value.

Now Bohr decided to make E = 0 when the electron is infinitely far away from the nucleus and when n = infinity. Thus, when an electron drops in energy levels, it releases energy as a photon.

This should make sense. When we heat a solid, we are giving it energy. If we heat it until it glows, then stop heating it, it will continue to glow until it cools off. Thus, the release of light must be a process of giving off energy, not absorbing it.

So Bohr’s first equation becomes:

ΔE = -B/n_f² - -B/n_i² = B (1/n_i² - 1/n_f²)

if n_f > n_i then energy is absorbed by the electron

if n_f < n_i then energy is emitted by the electron

Ex 7.6A

Calculate the energy change that occurs when an electron is raised from the n_i = 2 energy level to the n_f = 4 level of a hydrogen atom.

ΔE = -B/n_f² - -B/n_i² = B (1/n_i² - 1/n_f²)

ΔE = 2.179 x 10^-18 J (1/2² - 1/4² ) = 2.179 x 10^-18 J (1/4 - 1/16) = 4.086x 10^-19 J

Since n_f > n_i a photon must be absorbed by the hydrogen atom. What is it’s wavelength and where does it fall in the electromagnetic spectrum?

E_photon= 4.086x 10^-19 J = hu

since c = ul then u = c/l so:

E_photon= 4.086x 10^-19 J = hu= hc/l = 6.626 x 10^-34 Js (3.00 x 10⁸ m/s)/l

l = 6.626 x 10^-34 Js (3.00 x 10⁸ m/s)/ 4.086x 10^-19 J = 4.86 x 10^-7 m = 486 x 10^-9 m

l = 486 nm

looking at figure 7.10, this photon falls into the blue section of the visible spectrum.

If we run through all the possible combinations of energy levels, we can generate the line spectrum of hydrogen:

figure 7.18.

Those that fall into the IR part of the spectrum are called the Paschen series, those in the visible are the Balmer series and those in the UV, the Lyman series.

Several terms we will be using in the future are:

the ground state - the lowest possible energy level for a particular electron or atom

the excited state - the state of an electron or atom when it is not in the ground state.

So, Bohr got the calculations right but the model wrong. Why?

Bohr believed that electrons were solid spheres of matter with negative charge. But when we get down to subatomic particle sizes, weird things being to happen…..

Matter isn’t really what we think it is on the atomic scale. Most of an atom is empty space -- there’s absolutely nothing there. Even though the table feels solid to us on the macroscopic level, on the nanoscale level, there’s not much there, but there are energy fields that attract and repel. A table feels solid because the energy fields of the atoms making up our hands repel the energy fields of the matter making up the table.

Matter at times acts like a wave, just as light waves at times act like particles.

If you go to the beach at high tide and watch ripples run through a tidal pool, you can see interference patterns form. This is when one set of ripples overlaps another. Where the wave peaks overlap, they add together to form a bigger wave. Where troughs overlap, they form deeper troughs. The same thing happens with electrons. When electrons are shot through very tiny spaces such as the spaces between atoms in a crystal, they form interference patterns which we can capture on photographic film as dark spots (where the peaks of waves add).

demo with laser pointer and diffraction grating

So at times, subatomic particles actually have wavelengths. De Broglie showed mathematically that these wavelengths were dependant on the particle's mass and velocity:

l = h/(mv) where v = velocity of the particle and m is the mass.

Ex. 7.9 A

Calculate the wavelength in nanometers of a proton moving at a speed of 3.79 x 10³ m/s. The mass of a proton is 1.67 x 10^-27 kg.

l = h/(mv) = 6.626 x 10^-34J s / (1.67 x 10^-27 kg x 3.79 x 10³ m s^-1)

now remember that J = kg m² s^-2 so

l = 6.626 x 10^-34 kg m² s^-2 s / (1.67 x 10^-27 kg x 3.79 x 10³ m s^-1)

l = 1.047 x 10^-10 m = 0.1047 x 10^-9 m = 0.1047 nm

this is in the x-ray region of the spectrum.

What would wavelength be if mass were 0.100 g?

mass = 0.100g x 1 kg/ 1000 g = 1 x 10^-4 kg

l = 6.626 x 10^-34 kg m² s^-2 s / (1.00 x 10 ^-4 kg x 3.79 x 10³ m s^-1)

l = 1.75 x 10^{-33 m} - this is so small that we can’t detect it.

Because of this wave/particle duality of subatomic particles, they are very hard to study. Heisenburg noted that for particles this small we can’t accurately determine both where the particle is and where it is heading. This is called the Heisenberg Uncertainty Principle: the more we know about where a particular particle is the less we know about where it’s going.

Schrodinger came up with the idea of a wave equation to describe electrons around the nucleus of an atom. These are a complex set of equations that predicted the properties of the electron in the hydrogen atom.

A solution to the wave equations is called a wave function (y) which denotes the energy state of the hydrogen atom.

The square of the wave function (y²) gives the probability of finding an electron at a particular position in the atom. From these wave functions comes the four principle quantum numbers that we used to describe the electron configuration of an atom.

1. the principle quantum number n - must be a positive integer and represents the electron energy level or shell.

2. orbital angular momentum quantum number l = describes the shape of the orbitals in which the electrons reside.

l 0 1 2 3

orbital s p d f

3. the magnetic quantum number m_l = describes the orientation of each suborbital in space.

Since y² is the probability of finding an electron at a particular point in an atom, we can map the shape of the orbitals the electrons occupy by marking where the electron is likely to be found 90% of the time. Thus we get the shapes of the different orbitals:

The s orbital (figure 7.24 first, then figure 7.23 and 7.25)

ends up being spherical, centered around the nucleus. Note that the s orbital also includes the nucleus. Thus we say that s electrons penetrate the nucleus.

There are 3 p orbitals (figure 7.26) which have dumb bell shaped lobes that we denote as p_x, p_y and p_z. Note that there is a zero point (a node) right where the nucleus is. p electrons are far less penetrating than s electrons since they can’t enter the nucleus.

There are five d orbitals (figure 7.27). These do not penetrate the nucleus either. Notice that for both p and d orbitals, when they are fit together around the nucleus, they completely surround the nucleus.

The last, fourth quantum number is the electron spin. Electrons have a magnetic field which is caused by their electric fields moving. Magnetic fields have two poles, so electrons can have two potential magnetic field orientations - we call these spin up and spin down.

figure 7.28