**Introduction**

In the beginning of the 19th century it was taken for granted that classical mechanics, which assumed that energy values were continuous and not discrete, sufficiently explained all physical phenomena. However, it was discovered through problems with classical models in areas such as blackbody radiation and the photoelectric effect that the classical model did not adequately explain certain observations - because of this, a new perspective based on the assumption that energy occurs in discrete quantities was developed, and scientists such as Maxwell Planck, Albert Einstein, and Niels Bohr began to create approximate models for a fundamentally changed understanding of the physical world.

**Blackbody Radiation**

One of the last remaining problems with classical mechanics at the end of the 19th century involved solid objects that radiated energy. When an object is heated, it emits electromagnetic radiation, and as it is heated the wavelength of the emitted radiation becomes shorter and shorter. This is a concept termed **blackbody radiation**.

A **blackbody **is a theoretical object, considered an ideal emitter, capable of absorbing and emitting all radiation - because it absorbs all incoming energy there is zero reflection, so all radiation measured off of this blackbody is from energy that has been absorbed and then re-emitted. An approximation of a blackbody used in lab settings is an empty container with a pinhole at a constant temperature. The radiation is released from the walls due to thermal energy from a heater, and this radiation is re-absorbed and re-emitted many times before it leaks out of the pinhole at the end - by this point the radiation is at equilibrium with the walls^{1}.

Figure 0. An image of an object designed to emulate a blackbody. Note that the radiation from the chamber is emitted and reabsorbed by the walls multiple times before it escapes through the pinhole.

In classical mechanics, the behavior of a blackbody was modeled using an equation, derived from other classical equations, termed the **Rayleigh-Jeans Law**:

where *ρ*_{T}(v) is the intensity of the radiation, *v* is the frequency, *k*_{B} is a constant called the Boltzmann Constant, *T* is the temperature in Kelvin, and *c* is the speed of light^{1}. This equation was conceptually based on a couple things (accepted as true by classical mechanics); firstly, that blackbody radiation comes from oscillating electrical charges on the surface of the object, and secondly, that these oscillators can gain or lose any amount of energy in a continuous manner. However, the Rayleigh-Jeans Law had a significant problem - it predicted infinite intensity at very short wavelengths of light, which was in direct contradiction of collected experimental data which stated that radiation intensity would fall to zero at extremely high frequencies^{3}.

This was termed **the ultraviolet catastrophe** - intuitively, intensity could not infinitely increase because then any object sufficiently heated would return an enormous amount of energy - however, the Rayleigh-Jeans Law was grounded in demonstrated classical principles and was mathematically confirmed by multiple scientists. Therefore, a new predictive model was required, and it was discovered by Maxwell Planck.

Figure 1. This graph shows the intensity predictions that each law made. As you can see the Rayleigh-Jeans Law showed the erroneous belief that intensity could be infinite whereas Planck’s Radiation formula is consistent with the observation of increasing intensity with decreasing wavelength^{2}.

Max Planck noticed that the Rayleigh-Jeans Law produced results inconsistent with experimental results, particularly at low wavelengths or high frequencies. In order to create a law consistent with all results, Planck reasoned that high-frequency oscillators cannot be excited as much as low-frequency oscillators. This was the first step towards quantum mechanics in that Planck was stating that the energy (E) of the oscillator had restrictions and did not allow for continuous values of E. His hypothesis therefore had two parts. The first was that the total energy of the oscillator is discrete and *E*_{osc}can only have a value of *nhv *where n= 1,2,3…. The second was that oscillators gain or lose energy in “packets” of energy that he called quanta; these quanta have the magnitude of *hv *where *h *is some constant. He went on to prove that h=6.626 * 10^{-34} J*s, now known as Planck’s constant.

The next component to Planck’s Law was that the excitement of an oscillator occurs only after a threshold frequency. Planck’s idea was that since the oscillator has only discrete possible energy values, the only way the oscillator could reach a higher energy is if the amount of energy absorbed or emitted corresponds to the difference between two levels (creating a threshold to be reached prior to a new energy level). His law to explain why the intensity of the radiation decreases with frequency was based on his initial idea that high-frequency oscillators are less likely to be excited as as low-frequency oscillators. The equation that Planck used to express his laws uses the rightmost fraction to factor in the the probability that an oscillator with a given frequency *v* is activated at temperature T.

Difficult Concepts

Concept 1 - The definition of blackbody radiation

When a solid object is heated, it emits electromagnetic energy. As the object is heated further the intensity over all wavelengths increases and the maximum intensity moves to a lower wavelength and higher frequency. Blackbody radiation is the radiation that comes from a theoretical concept called a “blackbody” which is an object that uniformly absorbs and emits all frequencies of radiation. Although blackbodies are an idealized entity, as no object uniformly absorbs and emits all radiation, they are the entity modeled by equations like the Rayleigh-Jeans Law and Planck’s Law and these equations modeling blackbody radiation are used to predict the emissions of other solid objects..

Question 1: True or false: Blackbody radiation is electromagnetic energy emitted from a heated solid object.

A.True- the blackbody radiation models the electromagnetic energy emitted from all heated solid objects, because an object that emits electromagnetic energy when heated is called a blackbody.

B. True - all ordinary solid objects always act like perfect blackbodies

C. False - Blackbody radiation is energy reflected by a solid object

D. False - Blackbody radiation only comes from perfect blackbodies - it only approximates the behavior of real world objects

Concept 2 - The difference between classical and quantum interpretations of blackbody radiation

The classical understanding of blackbody radiation was that there were oscillating charges in the solid which, when bombarded with some sort of radiation, absorb all of the incoming energy indiscriminately. The quantum interpretation states that these oscillators cannot absorb any frequency of energy - it has to be a high enough frequency to activate the specific oscillator, and when the oscillator absorbs the energy it reaches a discrete energy level.

Question 2: Which of the following is an assumption made by Planck’s Law:

A. Energy is absorbed by oscillators only above a threshold frequency

B. Energy is absorbed by oscillators only in discrete quanta

C. Oscillators can only take discrete energy levels

D. The energy of an oscillator is proportional to the frequency

E. All of the above

*Scroll down to see answers*

**Photoelectric Effect**

In addition to the conceptual problems with blackbody radiation, the observation of the photoelectric effect arose additional conflict in the development of modern quantum mechanics. The photoelectric effect was introduced from the observation of light causing electron to eject from a metal surface producing a flow of an electric current. The amount of energy to eject the electrons depended on the frequency and intensity of the electric current. It was shown that the electric current does not begin to flow until it exceeds a specific threshold value v_{0}, which is different for each metal. A light with a low frequency which produces a longer wavelength cannot give enough energy to eject electrons from the metal, regardless of how intense it is. In contrast if a lights frequency were to increase and pass the threshold value v_{0} then the electrons are emitted from the metal which leads to the electric current being directly proportional to the light intensity. This led to the notion that the key to having enough energy for the electrons to be emitted from a metal is for the light frequency to be above the threshold value v_{0}. (see Figure 2) However, classical physics could not explain these results due to the classical electromagnetic theory. The classical electromagnetic theory stated that the energy correlated to the electromagnetic radiation only depends on the intensity and not the frequency of the radiation.

** **

Figure 2

**Question:** Then, why can electrons eject form a metal like sodium when a low- intensity beam of blue light with high frequency hit its but can eject when a high intensity beam of red light with low frequency hits it?

The key to that answer lies in the connection between the energy transmitted by light at the treshold value, v_{0}, and the maximum energy, E_{max} that the electrons are emitted from the atoms in the metal. In the early 1900’s, Albert Einstein used Planck’s quantum hypothesis to explain the photoelectric effect. Einstein first proposed that a light wave of frequency (v) consisted of photons that carry energy found through E_{photon}= hv, where h is also knowns as Plancks constant. He then believed that an electron gains energy to escape from a metal from absorbing a photon of light. An electron emitted from below the surface of a metal will lose energy E from colliding with other atoms in the metal and will lose in leaving the surface of the metal with kinetic energy. With this conservation of energy Einstein predicted that “the maximum kinetic energy of ejected electrons varies linearly with the frequency of light used.”^{3} This theory is also given by the following equation; .

To test this theory experiments were conducted to illustrate that the relationship between E_{max} and frequency is linear. (see Figure 3)

Figure 3.

The results of the experiment showed that the slope of the data is identical to the value,h that Planck discovered in his experimental data for his “theoretical blackbody radiation intensity distribution.”^{3} Through Einstein’s experiment he was also able to find a way to obtain the value which is called the work function. The work function represents the “energy barrier”^{3} that electrons have to overcome to leave the metals surface after absorbing a photon inside the metal. The fact that both Planck and Einstein’s experimental data led to the same value h gave great confidence that their hypotheses are valid.The photoelectric effect has played a important part to many inventions that we use today. For example night vision devices and image sensors both apply the use of the photoelectric effect.

Playing with the Photoelectric Effect

If you would like a better grasp on the concept the following is a simulation website in which you can look at and manipulate the different intensities of the photoelectric effect; http://phet.colorado.edu/en/simulation/photoelectric .

**Bohr Model**

The Bohr model is a combination of classical and quantum models. Being the first atomic model to incorporate the famous quantum theory, it might be challenging to comprehend at the beginning. Originally, the way electrons were predicted to move in the classic model is governed by Newton’s Laws. The Bohr model originated from the Rutherford (classic) model, in which electrons move around the nucleus in given orbitals; while in Bohr model, the electrons move in discrete energy orbitals.

However, this classical model disobeys the Maxwell’s theory, which predicts that when the electron is revolving around the protons, it will radiate energy and thus lose energy bit by bit, spiraling into the nucleus (as shown to the left). If Maxwell and the classic model were both right, the atoms would not be as stable as we observe.

Around the same time, Balmer and Rydberg developed their formula (on the right) from atomic (Hydrogen) spectra, which suggests that there are discrete energy levels within atoms.

Combining all these clues, Bohr finally comes up with the idea that electrons should be revolving in discrete orbitals and applies it to his new model (shown below).

To summarize, the Bohr model is based on the classic model, where the electrons are revolving around the nucleus like the planets revolving around the sun. However, the electrons have distinct orbitals, which is the application of quantum theory. Also, Bohr model cannot predict accurately besides Hydrogen, which is one of the reasons that it's not used today. Despite the imperfectness, we still study it today because it was the first model to incorporate quantum theory into our understanding of the atom, although the orbitals are not correctly predicted.

Difficult Concepts:

The exploration might be confusing in how Rydberg’s formula influences the Bohr model. The Rydberg formula is a mathematical formula used to predict the wavelength of light resulting from an electron moving between energy levels of an atom, which is to say, from the ground state to the excited state or vice versa. Below are the four visible lines of the hydrogen emission spectrum. The spectra of different atoms are distinct because the energy levels require for the excitation are different in each cases.

When an electron changes from one atomic orbital to another, the electron's energy changes. When the electron changes from an orbital with high energy to a lower energy state, a photon of light is created. When the electron moves from low energy to a higher energy state, a photon of light is absorbed by the atom. Therefore we usually say, when an electron absorb energy, it would get excited and jumps from a orbital to a higher orbital. The fact it is “jumping” instead of “moving” comes from the experimental result that the spectrum only have discrete lines. Based on the result that electrons have to “jump” when they are excited, Bohr makes the electrons moving in different orbitals that never overlap in his own model.

Question 3: Based on the exploration which of the following statement is false:

A. Bohr model can be used in both classic mechanics and quantum mechanics

B. The classic model failed because if it were true then electrons would spiral into the nucleus

C. Bohr got the idea of “discrete energy level” idea directly from the emission spectrum of hydrogen atom

D. The classic model stems from the Rutherford model, where the electron orbitals could overlap

E. Both A and C

Question 4: Based on the second part, which of the following is false:

A. Rydberg’s formula is developed based on the fact that there are discrete lines in the hydrogen emission spectrum

B. Electrons jumps from the ground state to the excited state, causing some lines missing in the spectrum

C. A photon is emitted when an electron jumps from lower energy state to a higher energy state

D. The spectra of different atoms are distinct

E. Both B and C

*Scroll down to see answers*

**Quantum vs Classical Mechanics Summary**

**Answer Key**

1. D 2.E 3.E 4.C

**Works Cited**

(1) Atkins, P. W., and Julio De Paula. *Atkins' Physical Chemistry*. Oxford: Oxford UP, 2006. Print.

(2) "Blackbody Radiation." *Blackbody Radiation*. Hyperphysics, n.d. Web. 20 Sept. 2014.

(3) Oxtoby, David W., H. P. Gillis, and Alan Campion. *Principles of Modern Chemistry*. 7th ed.

Cengage Learning, 2011. Print.

(4) About Education. What Is the Rydberg Formula? Definition and Examples. http://chemistry.about.com/od/chemistryfaqs/f/What-Is-The-Rydberg-Formula.htm (accessed Sep 15, 2014)

(5) Annenberg Learner - Teacher Professional Development. Visuals: Bohr's Model of the Atom. http://www.learner.org/courses/physics/visual/visual.html?shortname=bohr_model (accessed Sep 15, 2014)

(6) ChemWiki: The Dynamic Chemistry E-textbook. Bohr's Hydrogen Atom. http://chemwiki.ucdavis.edu/Physical_Chemistry/Quantum_Mechanics/Quantum_Theory/Bohr's_Hydrogen_Atom (accessed Sep 15, 2014)

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