electron orbits

atomic spectra

the Bohr atom

No quiz this week!

“How wonderful that we have met with a paradox. Now we have some hope of making progress.”—Neils Bohr

Exam 1 Post-Mortem

Make sure you got what you deserved! Check for grading mistakes, addition mistakes, grade recording mistakes, grading inconsistencies. Spreadsheet is here.

Specific Comments

Please don't make your grader search through a pile of chicken scratchings for the solution to a problem. Remember, a happy grader is your friend!

If you use E=pc what are you telling me? The particle in question is a photon, speed=c.

Why did 7 students work LESS than 92 points worth of problems, even though it said in 3 separate places to select 92 points worth of problems?

I thought the test was “ easy normal” difficulty. No problems that I felt were “tricky” or especially difficult. Any comments?

Do you like having the choice of problems to choose, or does that waste your time?

Now, picking up the lecture where we left off last time…

So a “snapshot” of Rutherford’s atom at some instant in time looks like this. -

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-

-

-Of course, the electrons and nucleus are very small, but I had to draw them big enough to see.

Rutherford didn’t use the word “nucleus” in his 1911 paper describing this model. He used the term “charge concentration.”*

*See http://dbhs.wvusd.k12.ca.us/AtomicStructure/Rutherford-Model.html

Rutherford was confident he had the atom figured out. "The question of the stability of the atom proposed need not be considered at this stage, for this will obviously depend upon the minute structure of the atom, and on the motion of the constituent charged parts."*

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-

-

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4.2 Electron Orbits

Here’s our picture of Rutherford’s model of the atom again. In the model, electrons and nucleus are tiny and the electrons are separated by large distances from the positively charged nuclei. Let’s build such an atom. Any problems here? The electrons would be sucked into the positively charged nucleus (after all, unlike charges do attract, and the electrostatic force is a very strong one).

Otherwise, they must be far enough away to not feel any attraction, in which case they ought to wander off on their own. Have you ever known an electron to just sit around and do nothing?

What holds the positive charges in the nucleus together? After all, unlike charges repel.

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-

-

-

This is a problem we won't solve until later in the course. Forgetting about the nuclear problem for now, let’s play with our atom builder a bit more and see if we can gain some insight.

Does anybody see another big problem with this picture?

Dynamically stable orbits, like those of the planets around the sun, would allow electrons to remain "attached" to nuclei.

That’s what Rutherford meant by “motion of the constituent charged parts.” Let's calculate such an orbit.

-4+

-

-

-

As you learned in Physics 23, there must be a centripetal force holding anything in circular motion (remember, circular motion is accelerated motion): Fc = mv2/r.

In atoms, that force is the Coulomb attraction between electrons and nuclei:

We can solve for electron velocity:

22

C 20

mveF = = .4 ε r r

0

ev = .

4 ε mr

What atom are we talking about here?

Yes, hydrogen.

The electron's kinetic energy is K=mv2/2 and its potential energy is

2

0

eU = - .4 ε r

Its total energy E is just E=K+U. Adding kinetic and potential energies gives

The total energy is negative, meaning the electron is bound to the nucleus. As Beiser points out, this total energy is actually shared between electron and nucleus. Check: use the ionization energy of hydrogen to calculate the velocity and radius of the electron orbit. Do the results agree with experiment?

The ionization energy is 13.6 eV, so the binding energy is EB = -13.6 eV, or -2.2x10-18 Joules.

2

0

eE = - .8 ε r

Solve the preceding equation for r:

Using this r, we calculate the electron velocity to be 2.2x106 m/s. It is interesting that the electron velocity is far from relativistic. This is not the case in heavier atoms.

Does anybody see problems with this orbiting-electron model?

This is a classical derivation, based on Newton's and Coulomb's laws. It contradicts electromagnetic theory, which says that the accelerated electron must radiate (i.e. lose) energy.

-19-11

-12 -180 B

1.6×102er = - = - = 5.3×10 m

8 ε E 8 8.85×10 -2.2×10.

The explanation? We simply can't use classical physics to explain the atom. We are going to need to consider the wave nature of electrons.*

An accelerated electron continuously radiates energy and should spiral into the nucleus. Here is another visualization.Clearly, atoms do exist, so this doesn’t happen.

*Of course, that was not an option in 1911 because de Broglie was just a young undergraduate about to enter the army in World War I. De Broglie postulated matter waves in 1924.

4.3 Atomic Spectra

In 1911, a young Danish physicist named Neils Bohr had just earned his Ph.D. and went to study with “Plum Pudding” Thomsom. Unfortunately, Thomson was not happy to hear that Bohr had ideas that showed some of Thomson’s earlier work to be wrong. Thomson didn’t have much to do with Bohr.In December 1911, Bohr met “Alpha Particle” Rutherford, who he did get along with, and the next year went to work in his lab.

The next few years saw rapid progress in understanding the atom, and one key element was the study of atomic spectra.

Here is a brief history of spectroscopy.

The science of spectroscopy probably began with Newton, who showed that the spectrum of light produced by a prism was that of sunlight.

In the years around 1820, Joseph Fraunhofer thoroughly characterized the spectrum of sunlight.

Between 1846 and 1849, bright line emission spectra from a number of substances were identified and studied.

By 1871, Ångström had measured the wavelengths of the four visible lines of the hydrogen spectrum.

helium

mercury

hydrogen

increasing wavelength increasing frequency

In 1885, Balmer came up with an empirical formula (remember what we think of those?) for the wavelengths of the hydrogen lines.

2 2

1 1 1 = R - n = 3,4,5,...

λ 2 n

What the heck does this mean? At the time, nobody had a clue.

empirical parameters*

*fudge factors

n=3 n=4 n=5 n=6

why start at 3?

Other scientists (Lyman, Paschen, Brackett, Pfund) came up with similar formulas for infrared and ultraviolet hydrogen spectral lines.

For your reading pleasure, these delectable formulas are reproduced in your text on page 130.

2 2

1 1 1 = R - n = 3,4,5,... Balmer

λ 2 n

2 2

1 1 1 = R - n = 2,3,4,... Lyman

λ 1 n

2 2

1 1 1 = R - n = 4,5,6,... Paschen

λ 3 n

2 2

1 1 1 = R - n = 5,6,7,... Brackett

λ 4 n

2 2

1 1 1 = R - n = 6,7,8,... Pfund

λ 5 n

OK, so what’s the point?

There was a large collection of spectral data for hydrogen. Only a single line could be explained by Thomson’s plum pudding model. Rutherford’s “new and improved” model predicted a continuous spectrum.

Wouldn’t it be a good idea to find a theory which explains the data?

Absolutely! This section has been looking at the data. The next section looks at the Bohr model which “explains” the data.

One more thing… Beiser introduces the different kinds of atomic spectra in this section.

Emission spectra: an atomic gas is excited, e.g., by an electric current, and emits radiation of specific wavelengths. Absorption spectra: an atomic gas absorbs radiation of specific wavelengths.

Of course, molecules can have emission and absorption spectra too.

http://csep10.phys.utk.edu/astr162/lect/light/absorption.html

Summarizing section 4.3: if we put a high voltage current through hydrogen gas, we see light.

The light is not a continuous spectrum, but a series of discrete, monoenergetic lines.

Recall that the first series of lines was discovered by Balmer, and the wavelengths are given by

R is a constant. Any good theory should be able to predict the value of R, with no empirical parameters. This is an empirical equation, which we would like to explain with the laws of Physics. We can't explain it with classical physics.

2 2

1 1 1 = R - n = 3,4,5,...

λ 2 n

background “glow” due to white light leaking into spectrometer

I showed you other series of spectral lines. I am not interested in having you memorize their names or formulas. They all follow something like

The important idea is that we need to explain these experimental observations, including the formula and the spectra themselves. For those who enjoy sparkling lights, here (dead link?) and here are some pages of spectra.

i ff f2 2fi

1 1 1 = R - n = n +1, n +2, n +3, ...

λ n n

4.4 The Bohr Atom (1913)

The scenario that Beiser presents in describing Bohr’s model of hydrogen is not the approach Bohr took in his work, because de Broglie and his matter waves didn't come for another decade.

Nevertheless, let’s follow Beiser for a while, and at the end of this section I’ll briefly mention Bohr’s approach.

“…if only sufficiently many physicists are placed at sufficiently big machines, then everything will fall into place in the end.”—W. Heisenberg (no, Heisenberg didn’t actually believe that)

Note that this is a model for the hydrogen atom.

Why choose the hydrogen atom...

Consider the wavelength of an orbiting electron.

An electron in orbit around a hydrogen nucleus has wavelength = h / mv and a velocity as given in section 4.2:

0

ev = .

4 mr

You can solve these two equations for the wavelength:

04 ε rhλ = .

e m

04 rhλ = .

e m

You can plug in the electron mass, r = 5.3x10-11 m for the radius of an electron orbit in hydrogen, e =1.6x10-

19 C, and 0 = 8.85x10-12…

…and you get a wavelength = 33x10-11 m, which (coincidentally?) is the circumference of the electron orbit.

2-12 -11

-34 2

-19 -31

C4 8.85 10 5.3 10 m6.63 10 J s N mλ =

1.6 10 C 9.11 10 kg

So what?

We have taken an electron and calculated its wavelength. We find that its wavelength exactly corresponds to one orbit of the hydrogen atom. See Figure 4.12.

This gives us a clue as to how to construct electron orbits.

Or try this visualization. (Click on the little “ball” to start.)

Actually, instead of "destroying" itself, such an electron simply can't fit into an orbit. It must either gain or lose energy, so as to have a velocity which gives it a wavelength which fits into the orbit.

If an electron wave going around a nucleus "meets itself" out of phase after one revolution, destructive interference will take place.

http://hyperphysics.phy-astr.gsu.edu/hbase/ewav.html#c2

We thus arrive at the postulate that an electron can orbit a nucleus only if its orbit contains an integral number of de Broglie wavelengths.

Of course the idea of de Broglie waves was a decade in the future when Bohr worked out his model for the hydrogen atom, so he couldn't have made this postulate.

Expressed mathematically:

Combining this expression for with the one we obtained earlier in this section gives us an equation for rn, the orbital radii in the Bohr atom.

nn λ = 2 r n = 1, 2, 3, ...

2 20

n 2e

n h εr = n = 1, 2, 3, ...

m

Like our particle in a box, an electron in hydrogen can “fit” only if its wave can “fit.” The wavelength is quantized.

Energies!

Here’s a visualization of the Bohr model.

The orbital radii are quantized.

What else was quantized for our particle in box.

I wonder if that will happen here… (not a question, just wondering out loud)

If that visualization was supposed to mean something, there are a couple of things we have to explain...

The integer n in the equations for and r is called the quantum number of the orbit.

When n=1,

This value of r1 is the radius of the innermost orbit. It is called the Bohr radius, a0, of the hydrogen atom, and a0=5.292x10-11 m. The other radii are given by rn = n2

a0.

2-34 -122-110

2 21 -31 -19e

6.62×10 8.85×10h εr = = = 5.292×10 m

m 9.11×10 1.6×10.

If this wasn’t Bohr’s approach to his model of the atom, what was?

As a student, Bohr hadn’t thought spectra were worth much.However, when Bohr was trying to understand the hydrogen atom, a colleague happened to show him some spectra and Balmer’s formula.

2 2

1 1 1 = R - n = 3,4,5,...

λ 2 n

"As soon as I saw Balmer's formula, the whole thing was immediately clear to me." –N. Bohr

Quoting from the class notes* of Taylor (U. of Virginia)…

*http://www.phys.virginia.edu/classes/252/Bohr_Atom/Bohr_Atom.html

“What he saw was that the set of allowed frequencies (proportional to inverse wavelengths) emitted by the hydrogen atom could all be expressed as differences.

This immediately suggested to him a generalization of his idea of a "stationary state" lowest energy level, in which the electron did not radiate.

There must be a whole sequence of these stationary states, with radiation only taking place as the atom jumps from one to another of lower energy, emitting a single quantum of frequency f such that

h f = En – Em ,

the difference between the energies of the two states.

Evidently, from the Balmer formula and its extension to general integers m, n, these allowed non-radiating orbits, the stationary states, could be labeled 1, 2, 3, ... , n, ... and had energies -1, -1/4, -1/9, ..., -1/n2, ... in units of hcRH (using f = c and the Balmer equation above).”

The main thing is to realize that Bohr pictured a series of “stationary states”* in which the electron could exist without radiating, and that radiation takes place only when an electron jumps between different states.

*How can an electron be orbiting if it is in a “stationary” state?

Did you spot the example of the misleading Physicsspeak* on the previous slide?

Furthermore, if, as we showed above, the speed is quantized, then so is the electron’s angular momentum.

More specifically, it is not difficult to show that the angular momentum is quantized in units of h/2, which we call ħ.

L = n h / 2 = n ħ .

I guess Beiser didn’t want to wrestle with angular momentum in this chapter. I can’t say that I blame him.

Let’s think about Bohr’s model for a minute.

It contains classical physics. Remember, we used our “dynamically stable” orbit equations in the model.

0

ev = .

4 ε mr

It doesn’t explain how an electron can exist in a “stationary state” orbit without radiating. (Just giving it a name doesn’t explain it.)

It predicts Balmer’s (and all those others—see next section in text) formula for the hydrogen spectral lines.

But it doesn’t explain why the electron radiates when it jumps between orbits.

Bohr’s model for hydrogen is a curious hybrid of classical and quantum physics.

Some physics faculty refuse to teach it because it is wrong.

Several prominent physicists of Bohr’s day threatened to quit the profession if Bohr was correct. “The prevailing impression was one of scandal, or at least bewilderment, before the undeserved success of such high-handed disregard of the canons of formal logic.”*

*http://www.phys.virginia.edu/classes/252/Bohr_to_Waves/Bohr_to_Waves.html

"If you aren't confused by quantum physics, then you really haven't understood it." –N. Bohr

Bohr himself admitted his model didn’t explain anything. But it does tie together previously unexplained observations, and tells us the direction we might go in looking for the “true” model.

See here for another triumph* of Bohr’s model: singly ionized helium (like hydrogen but with an extra neutron). However, the model fails for helium. As we will see a few chapters from now, that is nothing to be ashamed of.

*"This is an enormous achievement. The theory of Bohr must then be right." –A. Einstein

There is a happy ending to this story: Bohr won the 1922 Nobel prize “for his services in the investigation of the structure of atoms and of the radiation emanating from them.”