QUANTUM PHYSICS AND ATOMIC MODELS

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QUANTUM PHYSICS AND ATOMIC

MODELS

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Table of Contents

· Electrons, X-rays and Radioactivity

· Blackbody Radiation, Quantized Energy and the Photoelectric Effect· Atomic Models· Waves and Particles· Quantum Mechanics

Click on the topic to go to that section

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Electrons, X-rays and Radioactivity

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Physics by the 19th Century

Newtonian Mechanics.

Maxwell's Equations - unification of electricity and magnetism.

Thermodynamics - heat is another form of energy.

Speed of Light measured - and no stationary reference frame (luminiferous aether) found - helped lead to Einstein's Special Theory of Relativity in 1905. Many physicists were comfortable that nearly everything had been discovered and explained.

But then, things started popping up. Like the electron.

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Cathode Rays

About 150 years ago, "cathode rays" were discovered.

No one knew what they were. Physicists working on the problem just knew what they could observe.

What they saw was that when a high voltage was connected to two plates inside a glass tube, making one plate positive and the other negative, then a shadow was cast on the glass behind the positive plate.

Furthermore, the glass surrounding the shadow behind the positive plate glowed. When more air was pumped out of the tube - approaching a vacuum - and the glass was treated with special fluorescent chemicals, the effect was enhanced.

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Cathode RaysThat seemed to mean that something was traveling in a straight line from the negative plate (the cathode) towards the positive plate (the anode) and the positive plate stopped some of it - creating a shadow surrounded by the glow.

Since the "rays" seemed to come from the cathode, they were called cathode rays. Click on the website below the diagram to see it in operation:

http://www.youtube.com/watch?v=Xt7ZWEDZ_GI&sns=em

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Cathode Rays

Light casts a shadow - so it could be light.

Or, it could be a stream of invisible particles.

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Cathode Rays

What two ways could you check to see if the rays were charged particles?

One rule - you can't open up the glass tube!

Discuss in your group.

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http://njc.tl/w4

Apply an Electric Field

One idea is to apply an electric field across the path of the rays. If they are charged particles, they will be deflected; if they are electromagnetic they will not be.

Also, the direction they are deflected will indicate whether they are negative or positive.

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1 In the below diagram, the blue line indicates the path of the cathode rays in the presence of an electric field. Are the rays:

A Waves

B Positive particles

C Negative particles

D Neutral particles

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Or, apply a Magnetic Field

Another idea is to apply a magnetic field (shown with the red X's) across the path of the rays. If they are charged particles, they will be deflected; if they are waves, they will not be.

Also, the direction they are deflected will indicate whether they are negative or positive.

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2 In the below diagram, the blue line indicates the path of the cathode rays in the presence of a magnetic field. Are the rays:

A Waves

B Positive particles

C Negative particles

D Neutral particles

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Negatively Charged ParticlesBoth experiments confirmed that cathode rays are streams of negatively charged particles. Also, when different materials were tried for the cathode, the same results occurred - they were independent of the material.

In 1897, J. J. Thomson used this experimental procedure of using magnetic and electric fields to learn more about these particles. He and his assistants adjusted the fields so they pushed the particles in opposite directions and canceled out - resulting in the particles going straight through.

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3 What would happen if the electric and magnetic fields were turned on at the same time?

A If the electric force was stronger, the particles would deflect up.

B If the magnetic force was stronger, the particles would deflect down.

C If the forces were equal, the particles would go straight.D All of the above.

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Learning about Negatively Charged ParticlesFrom measuring the magnitudes of both fields, they computed the velocity of the particles, as shown below:

ΣF= ma

FB - FE = 0

qvB = qE

vB = E

v = E/BSince they could measure E and B, they could find v.

FB

FE

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4 What is the velocity of a charged particle that passes undeflected through a magnetic field of 4.0 T and an electric field of 4.0 x 103 N/C?

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5 What is the velocity of a charged particle that passes undeflected through a magnetic field of 8.0 T and an electric field of 6.0 x 103 N/C?

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6 What perpendicular electric field will allow a particle whose velocity is 610 m/s to pass straight through a magnetic field of 2.0 T, and not deflect in either direction?

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7 What perpendicular magnetic field will allow a particle whose velocity is 920 m/s to pass straight through an electric field of 8.0 x 103 N/C, and not deflect in either direction?

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Discovery of the Electron

Now, let's see how J. J. Thomson and his team used all of these experiments to find an interesting property of these negatively charged particles.

They balanced the electric and magnetic fields so that the particles went straight through the vacuum tube. This enabled the team to calculate the velocity of the particles.

Then, they turned off the electric field. The particles were deflected downwards, in a semicircular orbit and struck the side of the tube.

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FB

v

r

They measured the radius of the semicircular orbit, by seeing where the particles struck the tube and made it fluoresce.

From these measurements they could determine the ratio of the particle's charge to its mass: q/m.

Learning more about charged particles

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Calculating Charge to Mass RatioThe particles had a velocity v=E/B before the electric field was turned off. The magnetic field now exerted a perpendicular force on the electrons, which caused the semicircular orbit.

Thus, the charge to mass ratio was determined.

Substituting in the set values for E and B, and the measured r:

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The Electron

Thomson found that this particle had a very low mass for its charge. In fact, its mass per charge was 1800 times less than the previous lowest amount measured for a particle. Before this work, physicists were speculating that the Hydrogen atom was the smallest fundamental particle.

This led Thomson to propose that this negatively charged particle was new - and he called them "corpuscles." The name "electron" was taken from George Johnstone Stoney's work in 1874, and proposed again by George F. Fitzgerald - and the name stuck.

Furthermore, since the electron was so much lighter than the hydrogen atom, it was concluded that it must be part of an atom.

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8 Which of the following is not true about cathode rays?

A They orginate from the negative electrode.

B They travel in straight lines in the absence of electric or magnetic fields.

C They are electrons.

D They depend on the material from which they are emitted.

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Thomson used a Crooke's Tube as shown above. Electrons were emitted from the cathode on the left, and were deflected by an electric field set up in the middle of the apparatus.

Here's a replica of the Crooke's Tube used by J.J. Thomson at Cambridge University.

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A replica of the third Mass Spectrometer used by J. J. Thomson (you may have seen that in the Magnetism chapter), that he used to continue to refine the charge to mass ratio of the electron. Electrons enter the Spectrometer from the right, into the velocity selector, then through the magnetic field which separates out the particles according to their mass.

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Finding the charge on the Electron

Now that Thomson measured the charge to mass ratio of the electron, if physicists could figure out either the charge or the mass, they would know both quantities.

Thomson and his team then made an attempt to measure the charge of the electron, using water droplets in an electric field (don't worry, this will be discussed in a few slides).

Using this method, Thomson assigned 1.03 x 10-19 C to the charge on the electron. As we showed earlier in the course, the charge on an electron is 1.6 x 10-19 C - so although Thomson had the right order of magnitude, the percentage error of his measurement was 54% - not acceptable.

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But, it was left until 1909 when a graduate student built on many years of experimentation by other physicists and actually refined the method in an afternoon!

Then, after months of tedious experimentation, a more accurate value for the charge on the electron was found.

The idea is pretty straight forward - but first, we'll talk about how Thomson and others tackled the problem.

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In fact, we did some problems like this when we studied electricity.

A charged object will fall with constant velocity if it is in an electric field E such that qE is equal to the object's weight (mg).

mg

FE = qE

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Charged Water Drops?

The falling object can't be electrons themselves since they're too small to see, and their mass was unknown.

Thomson and other physicists had been using water droplets and attempting to get them to float in an electric field.

That way they could estimate the mass of the drop and then see what charge was on each drop.

By measuring a lot of drops they could find the smallest charge, and figure that was the electron's charge.

But, the experiment didn't give an accurate measurement since the mass of the drop changes as it evaporated.

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9 What problem do you see in using charged water droplets and observing how they fall?

A Water will contaminate the experimental apparatus.

B Water evaporates so the mass is not constant.

C Water evaporates so the charge disappears.

D Water does not conduct electricity.

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Millikan Oil Drop Experiment

A graduate student of Robert Millikan, Harvey Fletcher, solved this in an afternoon.

He bought a perfume bottle and filled it with very fine oil used for watches. Unlike water, oil will not evaporate.

He then drilled holes in a couple sheets of metal and connected a battery so that he could change the voltage to the plates.

He used a microscope so he could see the drops fall. The method worked immediately!

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The idea was that as the oil was sprayed it would pick up some charge (just like the water droplets).

If charge was only carried by the electron, then when they measured the charge of the oil drop, it would have to be some multiple of the electron charge.

You just had to find the common denominator of the charge measured on all the oil drops.

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He adjusted the electric field so a drop fell with constant velocity.

He estimated the size of the drop, and then figured its mass, since he knew the density of the oil.

So, he knew the electric field, E, and the mass of the drop, m.

FE

mg


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Then he calculated the charge on the oil drop.

He did this for a large number of drops, using the below approach.

FE

mg

ΣF= ma

FE - mg = 0

qE = mg

q = mg/E


Many, many hours were spent by Harvey staring into the microscope watching the sparkling oil drops. The calculations also added in a correction for the viscosity of the oil drops (which provided an upward frictional force) and were laboriously performed without the help of computers (which didn't exist then).

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The experiment was repeated again and again, with increasing accuracy, by many other groups, and the accepted value of the elementary charge is now 1.602177 x 10-19 C.


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Here's a picture of the experimental setup that was used by Robert Millikan and Harvey Fletcher to determine the charge on the electron.

It all fits on one lab bench.

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10 For instance, if you measured the following as the charges of six oil drops, what would you conclude is the most likely charge of the electron?

A 3.2

B 4.8

C 6.4

D 1.6

E 4.8

F 3.2

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Properties of the Electron

The actual data were messier than the previous problem, and the units were in 10-19 C, but the conclusion was the same.

The electron was shown to have a charge of 1.6 x 10-19 C and a mass of 9.1 x 10-31 kg.

qe = 1.6 x 10-19 C

me = 9.1 x 10-31 kg

This charge is also called the elementary charge. It is the smallest charge found in nature.

So, the charge on any object is always a multiple of 1.6 x 10-19 C.

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r

You may wonder why it's not called the Fletcher Oil Drop experiment.

Fletcher was Millikan's graduate student and Millikan had made special arrangements for him to attend the University of Chicago without the established prerequisites and supported him throughout his tenure there.

Millikan had been working on the Electron charge problem, but it was Fletcher's idea that galvanized the research. Millikan wanted the main credit and as a consequence he became the first American to win the Noble Prize for Physics.


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r


Harvey Fletcher was able to author other papers on the topic and received this Ph.D. - and went on to do seminal work in stereo sound, acoustics, atomic physics and many other topics at Bell Laboratories and recorded albums with Leopold Stokowski.

An article was published in Physics Today in 1982, written by Harvey Fletcher, that he only wanted made public after his death.

In this article, he described his relationship with Millikan - and stated that he was happy with the way everything turned out and was very grateful to Millikan.

(Physics Today 35(6), June 1982)

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11 Which of these could be the charge on an object? (e = 1.6 x 10-19 C)

A 0.80 x 10-19 C

B 2.0 x 10-19 C

C 3.2 x 10-19 C

D 4.0 x 10-19 C

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12 Which of these could be the charge on an object? (e = 1.6 x 10-19 C)

A 2.0 mC

B 4.5 mC

C 3.2 C

D 2.5 μC

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13 The charge on the electron was determined in the:

A Cathode Ray tube by J. J. Thomson

B Millikan Oil Drop experiment

C Rutherford Gold Foil experiment

D Dalton atomic theory

E Atomic theory of matter

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14 The charge to mass ratio for the electron was determined by:

A Robert Millikan

B James Clerk Maxwell

C J. J. Thomson

D Harvey Fletcher

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X-rays

The Crooke's Tube that was used to study cathode rays, and resulted in the discovery of the electron, was also instrumental in the discovery of X-Rays.

William Roentgen was experimenting with various configurations of the vacuum tube in order to study the properties of the cathode rays. He shielded the tube with black cardboard so no light or cathode rays would emerge, energized the tube and turned off the lights to check the effectiveness of the shield.

He then noticed, that several meters away a screen coated with barium platinocyanide was glowing. He had planned on using this screen later in the experiment.

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X-rays

Roentgen knew that the glow could not be caused by the cathode rays, so he theorized that some other kind of ray was passing through the shielding and causing the screen to glow.

Because it was unknown, he coined the term "X-rays."

Very rapidly, he tested other objects to determine the penetration power of these X-rays.

This is another example in science where a great discovery was made by accident. You can search on the web for other discoveries made like this.

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X-rays

At that time, the dangers of excessive exposure to X-rays were not known, and many demonstrations occurred which would not be allowed today.

Only two weeks after their discovery, Roentgen took an X-ray photograph of the hand of his wife, Anna Bertha Ludwig, by placing a photographic plate behind her hand and directing X-rays at her hand.

The plate would show light where the X-rays touched it. If the X-rays didn't strike the plate, it would be dark.

Can you explain why the picture of Anna's hand looks like it does?

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X-rays

Denser objects, such as Anna's finger bones and her very prominent wedding band would stop the X-rays, and this would result in a shadow on the photographic plate.

Less dense objects, such as skin would allow the X-rays to pass mostly unhindered through and the plate would be lit up.

Why would the medical profession be immediately interested in this?

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X-rays

Physicians could now see bone damage without having to open up a patient. If a patient had been wounded by a bullet, X-rays could locate them quicker than a doctor poking around.

There is a problem though; X-rays have a great deal of energy and can knock out electrons from atoms (ionizing them) and can disrupt molecular bonds. If you're X-raying a box to see what's inside of it, that's not a problem.

But, if you're X-raying a person - you run the risk of damaging tissue and causing radiation sickness. That's why when you go to the dentist, the doctor is very careful to shield your body with a lead drape to minimize your exposure. X-ray usage is very carefully controlled to minimize its side effects.

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X-rays

Further experimentation showed that the intensity of the X-rays increased with denser target plates (anodes).

It was also discovered that the rays were not affected by electric or magnetic fields, as were the cathode rays.

What does this say about the charge on the X-rays?

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X-rays

If the path of the X-rays were not affected by magnetic and electric fields, then the charge on them must be neutral.

Later research showed that X-rays were not particles - but another kind of electromagnetic radiation.

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15 Compare and contrast cathode rays and X-rays.

A Both rays are made up of particles.

B Both rays are different types of electromagnetic radiation.

C Cathode rays are particles and X-rays are electromagnetic radiation.

D Cathode rays are electromagnetic radiation and X-rays are particles.

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16 What is the charge on an X-ray?

A Neutral

B Positive

C Negative

D It depends on its energy

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17 X-rays are very valuable for seeing inside of objects and people. Is there a danger of unlimited X-ray use on people?

A No, because the X-rays pass through tissue and don't impact it at all.

B No, X-rays do not have enough energy to harm humans.

C Yes, too much exposure to X-rays will cause ionization of tissue and break molecular bonds.

D Yes, and any use of X-rays is hazardous to humans, no matter how small.

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18 X-Rays were first observed when

A protons impacted upon a metal.

B neutrons scattered off of a crystal.

C electrons passed through a double slit.

D electrons struck a metal target.

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X-raysOther uses for, and understanding of X-rays came rapidly.

In 1899, Hermann Haga and Cornelius Wind demonstrated X-ray diffraction through a narrow slit, and estimated the wavelength of X-rays to be 1 x 10-9 m. In 1912, Max von Laue showed X-rays diffracting through crystals, which enabled the study of the structure of materials. Both of these indicated a wave nature for X-rays.

Then, in 1923, Arthur Compton showed that X-rays were acting as particles when low intensity X-rays were scattered by electrons.

Much of quantum physics focused on the different ways of looking at light and electrons - whether they are particles or waves. More on this later..........

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X-rays

Here's a graph of how many X-rays are produced (y axis - intensity) at different wavelengths (x axis) for a target anode made up of a specific element. This type of graph is called a spectrum.

What do you notice about this graph?

X-R

ay In

tens

ity

0 λmin Wavelength

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X-R

ay In

tens

ity

0 λmin Wavelength

peaks

continuous - "Bremsstrahlung"

X-rays

You can see a continuous curve, upon which are two very distinct peaks. The continuous part is called Bremsstrahlung and results from energy emitted by the target electrons striking the target, losing kinetic energy and releasing this energy in the form of light.

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X-rays

The two peaks on this curve relate to the internal structure of the atoms that make up the anode - and will be discussed later in the Bohr model. Would you expect all elements to have an identical curve?

X-R

ay In

tens

ity

0 λmin Wavelength

peaks


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X-rays

Atoms of different elements are unique and possess different internal structure. So, these spectra can be used to identify different types of materials - like fingerprints for people!

X-R

ay In

tens

ity

0 λmin Wavelength

peaks


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X-rays

Just to show the uniqueness of various spectra - here's the spectrum resulting from a Cu X-ray tube. It has more peaks, and a higher relative intensity of the Bremsstrahlung at lower wavelengths.

Emission spectrum of an old Cu X-ray tube

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19 How are graphs helpful to physicists?

A You can make extremely precise calculations from just looking at them.

B They enable comparisons to be made between different experiments.

C They present a lot of information in a small place and help you to spot trends.

D B and C only.

E A and B only.

F A, B, and C.

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20 Multi-Correct: What distinctive phenomena are illustrated in the X-ray spectra?

A Bremsstrahlung - deceleration of the electrons striking the anode.

B Acceleration of the electrons after they strike the anode.

C Internal atomic structure of the anode.

D Cathode ray penetration power.

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Radiation

A year after Roentgen discovered X-rays, Henri Becquerel wanted to see if phosphorescent salts (uranium) could be made to emit X-rays by exposing them to the sun (a source of energy, like the voltage in the Crooke's Tubes).

After much experimentation, Becquerel found out that the uranium salts would expose the photographic plates (they would be darkened) - just like the X-rays - but it didn't need the sun! It would work indoors, shielded from the light.

What conclusion could you draw from this?

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Radiation

Before you answer, here's a picture of a photographic plate that was exposed to uranium salts by Becquerel. Once again (like Roentgen), a Maltese Cross was used to shield the plate from the rays.You can see its shadow, where the cross blocked the rays.

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The Atom Can Be Divided!

More people started working on this problem.

Most famous of these were Marie Curie, and her husband, Pierre Curie.

Their conclusions were that: > these "rays" were coming from the atoms themselves > atoms were not indivisible - they were breaking up and

decaying into other atoms

This is the real life version of what alchemists were trying to do hundreds of years earlier - transforming elements into other elements via chemical means.

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Chemistry is all about sharing and trading electrons.

But the energies at which this sharing and trading happens is much less than the newly discovered rays, or radiation.

The radiation, discovered by Curie and the other researchers had very higher energies.

This indicated that the radiation was coming from something deeper in the atom (which was later found to be the nucleus).

The Atom Can Be Divided!

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Radiation

During their research, the Curies discovered new elements -Radium and Polonium - as they found different mineral blends emitted different ray energies.

These rays had a much higher energy than Roentgen's X-rays.

In fact, prolonged exposure to these substances without adequate protection led to Marie's early death from cancer.

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Radiation

Marie was the first woman to be awarded the Nobel Prize, which she shared with her husband Pierre and Henri Becquerel.

She was also the only person to be awarded the Nobel Prize in two technical disciplines (Physics and Chemistry)

And... she was the first woman faculty member at the Ecole Normale Superieure.

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Radiation

Subsequent research by Becquerel, M. Curie, P. Curie, Rutherford, Andrade, Royds, and many others, found that these rays were not all the same. By 1901, three new types of radiation were isolated:

Gamma Rays - a type of high energy electromagnetic radiation arising from energy changes in the nucleus.

Beta Particles - electrons from beta decay in the nucleus > This was the radiation that Becquerel found in Uranium salts

Alpha Particles - Helium nuclei.

This was an exciting time for science! Will return to this in the Nuclear Physics chapter.

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21 The radiation discovery by Becquerel, M. Curie and P. Curie was

A X-Rays can be observed by placing a substance in front of a photographic plate.

B Radiation occurs only when an electric potential is applied to a Crooke's Tube.

C Radiation is spontaneously emitted from certain substances.

D All substances emit radiation.

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22 What was Marie Curie's conclusion when she found radiation being emitted from elements spontaneously (without an external energy source)?

A Atoms were indivisible.

B Atoms had no internal structure.

C Only electrons were being emitted spontaneously.

D Atoms were decaying into different atoms.

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23 What types of radiation were discovered by physicists working around Marie Curie's time?

A Alpha particles

B Beta particles

C Gamma rays

D All of the Above

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Blackbody Radiation, Quantized Energy and the Photoelectric Effect


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Classical Physics continues to evolve into Modern

Physics

At the start of the 20th century, our understanding of the physical world was changing rapidly.

The electron was discovered. There was evidence that the atom, which had been considered to be the smallest particle of matter, had an internal structure.

Cathode rays were found out to be particles (electrons), and the distinction of ray like properties (light) and solid matter (particles) blurred.

Next, the very nature of light was to be challenged.

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Blackbody Radiation

All objects emit electromagnetic radiation which depends on their temperature: thermal radiation. A blackbody absorbs all electromagnetic radiation (light) that falls on it, and as they do they heat up. Because no light is reflected or transmitted, the object appears black when it is cold. However, as blackbodies heat up, they emit a temperature-dependent spectrum, called blackbody radiation.

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Blackbody Radiation

At normal temperatures, we are not aware of this radiation.

But as objects become hotter, we can feel the infrared radiation or heat.

As objects continue to get hotter they glow red.

Then at even higher temperatures, they glow white, like the filament in a light bulb.

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24 Which is a property of a black body radiator?

A It absorbs all incident radiation and then re-emits it at a different frequency.

B It is always black, even when heated to high temperatures.

C It absorbs all incident radiation and does not re-emit any radiation.

D As it is heated, it changes color from black to white to red.

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25 Which of the following colors indicates the hottest temperature of an object?

A Black.

B Red.

C Yellow.

D Blue.

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26 Four black bodies emit light with one of the colors listed below. Which of the black bodies has the lowest temperature?

A Red.

B Orange.

C Yellow.

D Blue.

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27 Using Wien's Displacement Law, determine the wavelength that contributes most to the color of a blackbody at T = 4000 K?

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This figure shows blackbody radiation data for a black body at three different temperatures and the classical physics prediction of the intensity of its light at 5000 K.· The x axis shows the wavelength of light emitted by the black body. · The y axis shows how intense (strong) the light is.

Blackbody Radiation

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Compare the plots for 3000K, 4000K and 5000K.As the temperature rises, what happens to the curves?

Blackbody Radiation

The wavelength at peak intensity shifts to shorter wavelengths as the temperature rises. It starts as red light then shifts to yellow and then white when all the colors are present.

?

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How well does the classical theory (black line) predict the data (the blue, green and red lines)?

Notice that the black line just keeps increasing as the wavelength approaches zero.

Blackbody Radiation

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Classical theory indicates that the intensity of the emitted radiation should become infinite as the wavelength gets shorter. The shorter wavelengths predicted that blackbody should radiate strongly in the ultraviolet (X-ray...) end of the spectrum. Hence, the prediction was dramatically named the "Ultraviolet Catastrophe."

Data shows that this (fortunately) does not happen.

Blackbody Radiation

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28 How does using a graph of observed data from a black body radiator versus the theoretical prediction of the data help illuminate problems with the theory?

Students type their answers here

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29 As the temperature of the black body increases, what happens to the color of the emitted light?

A It goes from white to black.

B It moves from red to yellow to white.

C It moves from white to red to yellow.

D It moves from visible light to infrared.

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30 What is the problem with the theoretical prediction of the black body's behavior as its temperature increases?

A The theory predicts that the intensity of the emitted light does not change as the wavelength decreases.

B The theory predicts that the intensity of the emitted light approaches zero as the wavelength increases.

C The theory predicts that the intensity of the emitted light approaches zero as the wavelength decreases.

D The theory predicts that the intensity of the emitted light becomes infinite as the wavelength decreases.

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Because of this tremendous discrepancy between existing physics theory and reality, a revolutionary, and not easily understood new idea was needed to match the observed data.

The wave theory of light could not explain the way the black body glows as a function its temperature (its spectrum).

So, in 1900, Max Planck made a brilliant assumption that atoms do not emit electromagnetic radiation (light) with any continuous value - they could only emit in multiples of small, distinct amounts of energy - called quanta.

When he applied this assumption to the Blackbody Problem, the Ultraviolet Catastrophe was gone!

Planck's Quantum Hypothesis

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These quantum amounts are given by the following formula:

E is the energy of the light, f is the frequency of the light, and h is Planck's Constant:


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Planck didn't believe this was real...it just worked. It was like working from the answers in the book...getting something that works, but having no idea why.

It didn't make sense that atoms could only have steps of energy. Why couldn't they have any energy?

Planck thought a "real" solution would eventually be found...but this one worked for some reason.

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Planck continued his analysis to try and make h = 0. If he could show that h was equal to 0, he could get rid of this notion of quantized energy levels.

However, he didn't succeed because it turns out that "h" is a fundamental constant of the universe and numerous measurements made by other physicists confirmed it.

This now leads to another mystery of the physical world that would be worked on by Albert Einstein, and would use this quantum theory. It is the Photoelectric Effect.


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31 What discrepancy between experiment and theory helped lead Max Planck to his quantum theory?

A The discovery of Cathode Rays.

B The discovery of X-Rays.

C The Ultraviolet Catastrophe.

D The discovery of electrons.

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Photoelectric Effect

Between 1888 and 1889, Heinrich Hertz, Wilhelm Hallwachs and Philipp Lenard performed experiments that showed light of specific frequencies incident upon certain polished metals would emit electrons.

Light Electrons

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Below a certain cutoff frequency of the incident light, there were no electrons emitted.

When electrons were emitted, their number were directly proportional to the intensity of the light, but their energy was independent of the light intensity.

The electrons appeared instantly when illuminated by the proper frequency of the light.

These properties could not be explained with the classical wave theory of light!

Light Electrons

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Nobody could explain why this happened - if light were a wave, then electrons should be emitted no matter what the light's wavelength was.

And, if the light was more intense, than more electrons should be emitted. But that didn't happen either.

So, as Einstein looked at the problem, he made an assumption that had been made by Sir Isaac Newton.

Remember?

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Albert Einstein theorized that Light acted as a particle!

And that the observed energies can be explained by Planck's Law:

Since the frequency, wavelength and speed of light are related by c = fλ:

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Actually, Planck first applied his quantum theory to light trapped within a box, stating that it acted as a particle - but he still theorized that light was a wave outside the box.

Einstein expanded on this and said light also acts like a particle when it's free.

He stated that light energy came in discrete packets of energy, that were later named photons, again, with the energy described by Planck. Thus, light consists of photons - which act like particles.

This was another example of how physicists continue to build on previous research in coming up with new explanations.

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32 What is the energy of a photon with a frequency of 5.00 x 1022 Hz?

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33 What is the wavelength of a photon of energy of 5.00 x 10-11 J?

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34 If the wavelength of a photon is cut in half, what is the new value of its energy in terms of its original energy?

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35 The ratio of energy to frequency for a photon is

A its amplitude.

B its velocity.

C Planck's constant.

D its wavelength.

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36 What is a photon?

A An electron in an excited state.

B A small packet of electromagnetic energy that acts as a particle.

C One form of a nucleon - one of the particles that makes up the nucleus.

D An electron that has been made electrically neutral.

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37 The energy of a photon depends on

A its amplitude.

B its velocity.

C its frequency.

D All of the above.

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38 The Photoelectric Effect was explained by treating light as a

A particle.

B wave.

C a wave and a particle.

D None of the above.

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Photon Theory of Light

Examine this plot of the kinetic energy of escaping electrons vs. the frequency of incident light.

This graph cannot be explained by the wave theory of light.

This shows that light is made of particles, photons; light is not a wave.

What do you notice about this graph for low frequencies?

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Photon Theory of Light

· Light with a frequency less than a cutoff frequency, f0, will not emit any electrons.

· Light with a frequency greater than f0 will eject electrons up to a maximum kinetic energy dependent on the light frequency.

Why?

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Not, all of the photon's energy goes into the electron's kinetic energy. Electrons are bound to their atoms, so a minimum amount of energy, called the work function (W), is required to free them. The equation for the maximum kinetic energy of the emitted electron is:

So, " –W " is the y intercept of the previous graph.

All of the observed properties of the Photoelectric Effect were explained by Einstein's work, and for that he received his only Nobel Prize.

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We now have Einstein reintroducing Newton's idea that light wascomposed of particles and his explanation exactly matched the observable data.

Which is great - for the Photoelectric Effect and Einstein's career.

But now we have a variety of phenomena some explained by assuming that light is a particle and some by assuming light is a wave.

What is Light really?

We'll come back to that in the Waves and Particles section of this chapter. But, for now - we're moving on to Atomic Models - which will led to more surprises.

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39 A photoelectric cell has a work function of 5.0 x 10-19 J. What is the minimum frequency of the incident light that will eject electrons from its surface?

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40 Light, with a frequency of 9.5 x 1014 Hz, is incident upon a photoelectric cell with a work function of 5.0 x 10-19 J. What is the maximum kinetic energy of the ejected electrons?

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Atomic Models


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Evolution of Atomic Models

The atom was originally viewed around 400 B.C. by the Greek philosophers as an indivisible object whose shape depended on its macro physical properties. e.g., water "atoms" would be slippery, and iron "atoms" would be very solid and hard.

It wasn't until 1802, when John Dalton, based on experimental research, proposed that elements were made up of specific atoms that would combine with each other - and were indivisible, and could not be altered by chemical means.

Then models evolved very rapidly, starting with Thomson's discovery of the electron.

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Thomson's Plum Pudding Model

Thomson's discovery of the electron was the first evidence that the atom had an underlying structure.

In 1904, he proposed that the electrons moved about within a mass of positive charge - he never really said what the positive charges were.

A plum pudding is an English dessert where raisins (used to be called plums in England) are embedded in an egg, suet and molasses mixture, hence the nickname.

But, the model could not explain the different wavelengths of light emitted by excited atoms.

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Rutherford Model

The Plum Pudding Model only lasted five years, when work done by one of Thomson's former students, Ernest Rutherford with Hans Geiger and Ernest Marsden established that the positive charge was concentrated in a tiny nucleus of diameter 10-15 m.

They aimed a beam of alpha particles at gold foil. If the atom were a plum pudding, they expected that the alpha particles would just plow through as shown in the top right.

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Rutherford Model

But, a very small amount of alpha particles were deflected; some were bounced right back to the alpha source.

As Rutherford said, "It was quite the most incredible event that has happened in my life. It was almost as if you fired a 15-inch shell at a piece of tissue paper, and it came back and hit you."

Hence, the conclusion that the positive charge was all located in a tiny region of the atom as shown in the lower right.

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Rutherford Model

Since all the positive charge of the gold atoms were concentrated in a small nucleus, and most of the atom was just empty space, most of the positively charged alpha particles would not be affected by the positively charged nucleus.

However, for the small percentage of alpha particles that actually struck the nucleus, they would be deflected from their original paths, and in the most extreme case, just bounce right back. Rutherford proposed that the small nucleus was surrounded by electrons in planetary orbits.

Why do you think Rutherford proposed planetary orbits?

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Rutherford Model

The answer was probably because scientists were well versed in planetary orbits since Galileo, Copernicus, Brahe and Kepler.

This was also prompted by the similarities in the force equations for Gravity and the Electric Force.

Since the equations are similar, and gravity causes planets to move in orbit about the Sun - why wouldn't electrons do the same thing about the nucleus?

Rutherford did not propose a structure for the orbiting electrons - that would be left to Neils Bohr.

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Rutherford Model

To emphasize how much of the atom is just empty space - a void - if the nucleus was magnified so that it was the size of a baseball, then the atom would have a radius of 4 km.

The electrons would be the size of the period at the end of this sentence.

The below website presents a great simulation of Rutherford's gold foil experiment.

But, Rutherford could not explain the different wavelengths of light emitted by excited atoms. That also would be left for Niels Bohr.

http://phet.colorado.edu/en/simulation/rutherford-scattering

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http://phet.colorado.edu/en/simulation/rutherford-scattering

41 The Plum Pudding model of the atom was important because

A it was correct in every manner and lasted for many years without any corrections.

B it confirmed that electrons had a negative charge.

C it included protons.

D it was the first model that illustrated the underlying structure of the atom.

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42 The gold foil experiment performed under Ernest Rutherford's direction

A confirmed the plum pudding model of the atom.

B led to the discovery of the atomic nucleus.

C was the basis for Thomson's model of the atom.

D utilized the deflection of beta particles by gold foil.

E proved the law of multiple proportions.

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43 In the Rutherford Nuclear atom model,

A the heavy part of the atom is very small and is surrounded by electrons.

B the positive charge in the atom is uniformly spread throughout.

C the positive charges surround the electrons which are concentrated in the center.

D the nucleus is physically large compared to the rest of the atom.

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Bohr Model

Neils Bohr applied Einstein's concept of light as quantized photons to the Rutherford model to explain the optical spectrum for hydrogen like atoms. These are atoms with only one electron such as hydrogen, helium missing one electron, lithium missing two electrons, etc - and obtained good agreement with experimental data.

Despite only having one electron, Hydrogen emits many wavelengths of light when it is excited - and this is what Bohr explained with his model introduced in 1913.

Let's start with these optical spectra, and see how Bohr applied structure to Rutherford's orbiting electrons.

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Optical Spectra

When light is passed through a diffraction grating, interference causes different colors to show up on the detection screen. Here is a picture from the Electromagnetic Waves unit of this course:

With Xenon gas is in a tube, and an electric potential is applied to either end of the tube (like the Crooke's tube), the Xenon gas emits light with frequencies that are unique to Xenon. When this light is passed through a diffraction grating, it results in the above optical spectra.

The Rutherford model had no explanation for this.

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Hydrogen Spectra

Here's the spectrum from ionized Hydrogen gas. This is important because it has been well studied and equations were derived experimentally that would fit this spectra. They will be shown on the next page. What's very interesting is that these experimentally derived equations would match Bohr's theory exactly.

InfraredUltraviolet

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Hydrogen Spectra

An equation that determined where the visible spectral lines from Hydrogen would appear was devised back in 1885 by Johann Balmer.

In 1888, Johannes Rydberg expanded the equation to include the ultraviolet and infrared lines from Hydrogen and other Hydrogen like atoms.

Note - there was no theory for these lines - it is a fantastic case of matching mathematics to reality - and this is before any concept of Quantum Theory.

The constant in these equations that made them work was later named after Rydberg:

the "Rydberg Constant"

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Hydrogen Spectra

These equations match exactly where the Hydrogen spectra lines were found by experiment.

Balmer series

Lyman series

Paschen series

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Bohr and the Hydrogen Spectrum

Neils Bohr looked at these spectra and, in 1913, proposed that: > The electrons orbited the nucleus in specific energy levels. > When the electrons absorbed energy, they would jump to a

higher energy orbit. > When electrons dropped back into an orbit, they would

release energy in the form of light.

Remember E=hf; so the energy was related to the frequency of the spectra measured.These orbits were quantized - so only unique light frequencies could be emitted. Not just any light frequency could be emitted - it had to equal the energy difference between the orbits.The orbits were unique to each element.

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44 Albert Einstein used the concept of light as a particle to explain why the Photoelectric Effect only emitted electrons with specific, quantized energies. What led Bohr to use a similar explanation for the Hydrogen spectra?

A The particle theory of light is the only way to explain all of light's properties.

B Excited Hydrogen atoms emit light of specific frequencies.

C Excited Hydrogen atoms emit light at all frequencies.

D Bohr believed, with no data, that light was a particle.

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45 The Hydrogen spectra were first explained by:

A Investigation of Cathode rays.

B The Gold Foil Experiment.

C Fitting equations to experimental data.

D Bohr's theory.

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Planetary Orbit Problem

In addition to the Hydrogen spectrum, Bohr needed to address the "planetary orbits" of the electrons.

The electrons in the Rutherford model were kept in orbit by the Coulomb Force providing the centripetal force between the electrons and the positively charged nucleus.

That was good. But, as shown earlier in the 19th century, an accelerating charge would emit electromagnetic radiation, and since the electrons were constantly accelerating (centripetal acceleration), they would lose energy. This loss of energy would cause the electrons to slow down, fall out of orbit and collapse into the nucleus within 10-11 s. This doesn't happen, as evidenced by the fact that we're all still here!

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The Bohr Atom

An electron is held in orbit by the Coulomb force:

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Bohr's Solution

Neils Bohr made the following assumptions to address the shortcomings of the Rutherford model. It is called a semi classical model since Bohr combined classical physics with aspects of the emerging quantum world.

1. Electrons could only revolve around the nucleus in certain circular orbits with fixed angular momentum and energy. The further away from the nucleus, the greater momentum and energy.

2. When in these orbits, the electrons would not radiate electromagnetic energy.

3. Electromagnetic energy is emitted from or absorbed by the atom as electrons move between orbits. The energy of the emitted or absorbed photons would be the energy difference between the allowed orbits.

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Bohr's Solution

Model of Electron Orbits

In this diagram there are three orbits (n=1, 2 and 3) that the electrons can be in.

When an electron falls from a higher energy level (larger n) to a lower energy level (smaller n), it emits a photon with frequency f proportional to the energy difference between the levels.

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Bohr's Solution

Here's a quick tour through Bohr's mathematics.

The quantized angular momentum of each level is expressed as (where n is the number of the energy level):

The Coulomb attractive force between the nucleus of charge Ze and the single electron that provides the centripetal force (remember, Bohr was only calculating orbits for atoms with a single electron):

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Bohr's Solution

Combining these two equations, doing a little algebra, and solving for the radius of each quantum level, we get:

By using classical equations for the kinetic energy, potential energy, angular momentum, Coulomb's Law, Newton's Second Law, and Bohr's quantum assumption, the energy of the quantum levels in hydrogen like atoms is determined to be:

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Bohr's Solution

Using the radius equation from the previous slide, the Bohr radius (a0) is defined as the electron orbital radius for the ground state (n=1) of the Hydrogen atom (Z=1):

Radii of other Hydrogen like atoms are expressed as:

Notice how the orbits grow in size as the square of n, so they get much larger as n increases.

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Bohr's Solution

And, by substituting in the values of the constants in the Energy equation, it is found that:

Notice how the energy levels are all negative, which indicates that the electrons are bound in the atom.

The levels get closer together, and closer to zero as n increases.

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46 What is the energy of the second excited state (n=3) of the Hydrogen atom?

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47 What is energy of the fifth excited state (n=6) of the Hydrogen atom?

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The Electron Volt (eV)

In atomic physics, energies are so low that it's awkward to use Joules (J).

A smaller unit of energy is the electron volt (eV).

Its value equals the potential energy of an electron in a region of space where the voltage (V) is 1.0 volt.

UE = qV

UE = (1.6 x 10-19 C)(1.0V)

UE = 1.6 x 10-19 J # 1.0 eV

1 eV = 1.6 x 10-19J

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The Electron Volt (eV)

It's also convenient to convert Plank's constant to units of eV-s rather than J-s.

h = 4.14 x 10-15 eV-s

h = 6.63 x 10-34 J-s

h = (6.63 x 10-34 J-s)

h = 4.14 x 10-15 eV-s

1.6 x 10-19J 1.0 eV )(

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The Bohr Atom

The lowest energy level is called the ground state; the others are excited states.

The first excited state is n = 2.

Energies for emitted and absorbed photons are calculated using:

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48 What is the frequency of the photon emitted when an electron falls from the n=6 level to the n=3 level in Hydrogen? (h = 4.14 x 10-15 eV-s)

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49 What is the frequency of the photon required to excite the electron in Hydrogen from the n=2 level to the n=4 level? (h = 4.14 x 10-15 eV-s)

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Bohr's Solution

These calculations from Bohr are in excellent agreement with the optical spectra showed earlier.

The next slide shows the various transitions between quantum levels - and have been named for the physicists who first observed them.

The Balmer series of orbital transitions are the only ones that are in the visible spectrum.

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Bohr's Solution

Bohr's model exactly matched the observed spectra and theexperimentally determined equations shown earlier.

The lowest energy level (n=1) is the ground state; the others are excited states.

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50 Which series of Hydrogen spectral lines are in the visible section of the electromagnetic spectrum?

A Balmer Series

B Paschen Series

C Lyman Series

D Rydberg Series

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Bohr's Solution

The Bohr Model was a critical step in transitioning from the classical view of the world into the new Quantum world - in that he used classical equations to explain what was happening at the Quantum level.

Despite the accuracy of his theory in explaining the Hydrogen spectra, he had to make an incredible assumption that violated a well understood phenomenon of accelerating electric charges.

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51 What assumption did Bohr make about accelerating charges that violated previous work in electromagnetic theory?

A Electrons in an atom never radiate electromagnetic energy.

B When an electron was not in an allowed orbit, it would not radiate electromagnetic enegy.

C When an electron was in an allowed orbit, it would radiate electromagnetic enegy.

D When an electron was in an allowed orbit, it would not radiate electromagnetic enegy.

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Limitations of the Bohr Model

1. The model is applicable only to hydrogen like atoms (single electron).

2. The model is based on an assumption that accelerating charges do not emit electromagnetic radiation if they are in specific orbits.

3. The model predicts the frequencies of the photons that are emitted when an electron changes orbits, but does not predict their intensities.

More work would now be done by Bohr and many other physicists to transition the explanation of atomic phenomena into a more complete Quantum world.

Let's look at electrons and light together next.

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Compton EffectIn 1923, Arthur Compton scattered X-rays off of a graphite surface and observed that the wavelength of the scattered X-rays increased. This was called the Compton Shift.

Compton postulated that the photon, acting as a particle with an energy E=hf, collided elastically with the electron. Since momentum and kinetic energy are conserved in an elastic collision, the scattered photon would have less energy after the collision; some of its energy would be transferred to the electron. This would be observed as the scattered photon having a longer wavelength than the incident photon.

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Compton Effect

To derive the equation for how much the photon's wavelength is shifted, Einstein's equation for the relativistic energy of a particle is used so that a momentum can be assigned to the photon:

Since the mass of a photon is zero:

This equation also shows that a massless particle (a photon) still has momentum.

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Compton Effect

After solving the conservation of energy and momentum equations (using the relativistic versions as photons move at the speed of light), the following angular dependent equation was derived.

It shows how much the photon's wavelength is shifted when it collides with an electron.

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52 The Compton Effect showed that photons act as

A light rays.

B particles.

C both light rays and particles.

D None of the above.

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53 What is the wavelength for a photon of wavelength 3.0000 x 10-9 m (X-ray) that hits an electron, undergoes a Compton shift and scatters at an angle of 450?

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The Neutron

With all these incredible theories and experiments in the first twenty five years of the 20th century, it's interesting that the neutron's existence was not proven until James Chadwick demonstrated its properties in 1932.

Ernest Rutherford, in 1920, proposed that a neutral particle with a mass on the order of the proton existed to account for the atomic mass measurements of atoms.

Experiments were performed before Chadwick that actually found neutrons, but they were mistakenly believed to be gamma rays or even protons.

Why do you think it took so long to discover neutrons?

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Neutron

Because of the absence of charge on a neutron, it was impossible to detect with electric and magnetic fields in the way that made the discovery of the electron possible.

Chadwick placed a small quantity of radioactive Polonium in an evacuated chamber, where it decayed by emitting alpha rays (Helium nuclei) that then struck a beryllium target.

Neutrons were emitted from this reaction and were found to have the correct mass and a neutral charge.

Neutrons are essential to having more than one proton in a nucleus. More about that in the Nuclear Physics chapter.

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54 Neutrons have a

A neutral charge and a mass approximately equal to an electron.

B neutral charge and a mass approximately equal to a proton.

C negative charge and a mass approximately equal to a proton.

D positive charge and a mass approximately equal to an electron.

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Waves and Particles


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Light described as a Particle and as a Wave

Newton first described light as a group of corpuscles (particles) and was able to explain reflection, refraction and dispersion.

Thomas Young and others found a wave like description was the only way to explain diffraction phenomena.

James Clerk Maxwell described light as perpendicular magnetic and electric fields moving as a wave.

Einstein brought back the particle model to describe the Photoelectric Effect.

Arthur Compton used the particle model to explain the frequency shift of scattered photons off of electrons.

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Light described as a Particle and as a Wave

Physicists love symmetry.

Light has been described as being both a wave and a particle.

So, since electrons were discovered, and well understood as a particle.....what could physicists be thinking about?

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What about Matter - can it be described as a wave?

Since light had been described both as waves and particles, Louis de Broglie, in 1924, proposed that particles could be described as waves, with a wavelength equal to:

This is the same equation that Compton used to determine the momentum of a photon.

Thus, all particles, and by extension, objects like baseballs and bowling balls have a de Broglie wavelength associated with their motion! But, since h is so very small, the wavelength associated with large objects is vanishingly small.

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Electrons described as a Wave

However, for small particles, like electrons, their wavelengths are often 10-10 m, which are about the size of the atom, so one might expect that they could experience wavelike effects like diffraction and interference.

de Broglie proposed that an interference experiment be setup with crystals and electrons to validate his theory. But no one tried until decades later.

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Electrons described as a Wave

In 1927, Clinton Davisson and Lester Germer were studying how electrons bounce off a nickel surface. During the experiment, an accident occurred that resulted in the scarring of the nickel surface, making it act like a three dimensional diffraction grating.

They were startled to find an interference pattern in the scattering! Electrons were diffracting as if they were waves.

This was the first experimental proof of de Broglie's hypothesis.

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Wave Nature of Matter

Later on, the two-slit experiment that showed light was a wave as performed by Thomas Young in 1803, was replicated with electrons.

In this experiment the de Broglie wavelength of the electrons was on the same order as the distance between and the width of the two slits.

Electrons fired one at a time towards two slits show the same interference pattern on the screen as photons had in Young's experiment.

This indicated that something like an "electron wave" must go through both slits at the same time - which is something we can't imagine a single particle doing. This result has been verified over and over.

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These photos show electrons being fired one at a time through two slits. Each exposure was made after a slightly longer time.

It appears that individual electrons behave like a wave and pass through both slits.

But each electron must be a particle when it strikes the film, or it wouldn't make one dot on the film, it would be spread out.

This one picture shows that matter appears to act as a particle and a wave at the electron level!

Electrons acting like a wave and a particle

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55 What assumption did de Broglie make when he was studying the electron?

A Since light can behave like a particle and a wave, electrons could also behave like a wave.

B Electrons are not particles, they are waves.

C Electrons only exhibit particle like behavior.

D Electrons do not interract with light.

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56 If particles act like waves, why do we not observe the wave behavior of a thrown baseball?

A Wave behavior only applies to electrons.

B The de Broglie wavelength of a thrown baseball is too small to be observed.

C The de Broglie wavelength of a thrown baseball is too large to be observed.

D You can observe the wave behavior of a thrown baseball.

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57 What is the de Broglie wavelength of a 0.145 kg baseball moving at 40.0 m/s?

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58 What is the de Broglie wavelength of an electron moving at 2.50 x 107 m/s?

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de Broglie’s Hypothesis Applied to Atoms

A weakness of the Bohr Model is that Bohr just stated, without explanation, that an electron in an allowed orbit did not radiate.

Physicists (and people in general) like to have explanations!

But de Broglie's wave theory of matter explained it well.

As long as the circumference of an orbit was an integral multiple of the de Broglie wavelength of an electron in that orbit, it would not radiate.

This approach yields the same relation that Bohr had proposed. In addition, it makes more reasonable the fact that the electrons do not radiate, as one would otherwise expect from an accelerating charge.

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These are circular standing waves for n = 2, 3, and 5.

As long as the circumference of the orbit matched an integral number of the electron's de Broglie wavelength, it would not radiate.

This also explained why the electron's angular momentum is also quantized.

de Broglie’s Hypothesis Applied to Atoms

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Quantum Mechanics


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Heisenberg Uncertainty Principle

The pieces were now in place for a way to rationalize all these seemingly contradictory wave and particle behaviors of light and matter. But first - another weird observation.

A driving force behind physics has been the criticality of making accurate measurements of physical phenomena, and making predictions based on these measurements.

This still holds for Quantum Physics, but with a twist - there are times when the more accurate a measurement that you make of one quantity, the less accurate a measurement you can make of a complementary quantity.

This is expressed in the Uncertainty Principle, published by Werner Heisenberg in 1927.

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The first two quantities that cannot both be measured precisely at the same time are momentum and position - the more accurately that momentum can be observed, the less sure you are of where the particle (photon, electron, neutron, etc.) is located.

This imprecision, or uncertainty, is represented by Δpx for momentum in the x direction, and Δx for the position in the x direction. Here's the equation:

Δpx Δx ≥ h

Planck's constant sets the limit for how accurate the measurements of momentum and position can be at the same time.

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This equation also works at our level - where we measure, for example, the positions of cars and their momenta.

Δpx Δx ≥ h

If that's the case (and it is), why do we feel we can very accurately measure everything we need to know about a car's size and its momentum?

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Because h is so very small. Recall that h = 6.63 x 10-34 J-s.

Δpx Δx ≥ h

So, if we measure a car's position with a precision, Δx = 1 mm (10-3 m),we'll be able to find its momentum to a precision of:

We don't have any instruments that can measure a car's momentum that precisely, so we don't feel any restriction from the Uncertainty Principle!

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Δpx Δx ≥ h

This expression helps explain why electrons remain in their orbits and do not spiral down to collapse into the nucleus. Neils Bohr theorized that if an electron is in a quantized orbit, it does not radiate EM waves, so it does not lose energy.

Heisenberg's Principle predicts that if you try and move the electron very close to the nucleus, then Δr (there's nothing special about x, y, z or r here - it just represents a distance, and if we're talking about orbits, the radius is what we're interested in) gets very small.

This implies that Δpr gets very large, and the electron will have a larger momentum which will prevent it from getting closer to the nucleus.

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ΔE Δt ≥ h

There's another version of the Uncertainty Principle relating Energy and time as shown above.

Fortunately, for physicists making measurements at levels that we can all see, this principle does not affect these measurements due to the very small value of Planck's constant.

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59 If you measure the diameter of a baseball to be 74.2 mm with a precision (uncertainty) of ±0.5 mm, how precise can you measure its momentum, at the same time, according to Heisenberg's Uncertainty Principle?

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60 An electron is measured to have a momentum of 2.5 x 10-28 kg m/s with an uncertainty of 2.5 x 10-32 kg m/s. How precisely can we determine where it is?

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Schrodinger's Wave Equation

To explain the seeming contradictions between particle and wave behavior and the impossibility of measuring certain quantities simultaneously with perfect accuracy, Erwin Schrodinger developed the wave function, Ψ, in 1926.

Significant work was also done by Paul Dirac, and the two gentlemen shared the Nobel Prize for Physics in 1933.

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In classical physics, if all the forces on an object are known, all other quantities, such as position, momentum, acceleration, etc., can be found.

And in classical physics, we've always assumed we can measure things to an incredible precision, and then predict exactly what is going to happen.

Things act very differently at the very small level of atoms.

We've got light being described as a wave and a particle, we have electrons being described as a wave and a particle, and we have Heisenberg telling us there is a limit to how precisely we can measure position, momentum, energy and time.

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Schrodinger conceived the idea of the wave function, where electrons and light are assigned specific wave functions. These wave functions perform a similar role to that of Force in classical physics.

But - the major difference is you can only find the probabilities of a particle's position, energy, momentum, etc.

The key word is PROBABILITY, not EXACT.

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An example of the use of ψ is in the double slit experiment for electrons. Young's experiment using monochromatic light incident upon two slits showed a unique interference pattern.

Electrons showed the same pattern! This leads to a key property of ψ - it is used to represent all the knowledge we have about an electron in this case. And, if a stream of electrons are incident on the double slits, the wave functions of each particle will interfere with the other particles' wave functions.

Thus, the interference patterns are explained by analyzing the wave functions and their interactions.

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To find the probability of where a particle is, the wave function, Ψ, is squared. This was first suggested by Max Born.

Just to show the beauty of the Wave Equation, here it is for one specific problem:

Solving this equation for Ψ, will enable you to understand every thing there is to know about the particle. But, this is College level physics - even beyond AP.

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The solutions to the Wave Equation show how the periodic table, electron orbitals, atoms, molecules, chemical bonds work. It describes the world at the atomic level - which is so different and counter intuitive to what we see at our level.

The Uncertainty and Probability concepts contradict what can be seen at the macroscopic level where we know, pretty exactly, where things are and how they're moving.

But - that's how nature works.

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61 The Schrodinger Wave Equation is used to

A describe exactly where a particle is at any specific time.

B calculate the exact momentum of a particle at any time.

C calculate the probability of a particle being at a certain point at a certain time.

D calculate only the probability of properties belonging to electrons.

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62 How is the probablility of a particle being in a certain position calculated from the wave function?

A The wave function is the probability of where a particle is located.

B Divide the wave function by 2.

C Triple the wave function.

D Square the wave function.

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63 What do the solutions of the Schrodinger's wave equation allow us to understand?

A Atoms

B The Periodic Table

C Chemical Bonds

D All of the above

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Quantum Mechanics

Quantum Mechanics is very different from classical physics - for example, you can predict what a lot of electrons will do on average, but you have no idea what any individual electron will do.

It's also different in that there is a limit to how precisely we can measure phenomena. But - it accurately matches experimental observations, and it predicts other effects which are then verified by experimentation.

In Chemistry, you'll be using the solutions to Schrodinger's Equation, and the physics you've learned this year, to explore the nature of matter.

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QUANTUM PHYSICS AND ATOMIC MODELS