Two Equations Governing Light's Behavior: Part Two
E = hν

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There are two equations concerning light that are usually taught in high school. Typically, both are taught without any derivation as to why they are the way they are. That is what I will do in the following.

Equation Number Two: E = hν

Brief historical note: It is well-known who first wrote this equation and when it happened. Max Planck is credited with the discovery of the "quantum," the discovery of which took place in December 1900. It was he who first wrote the equation above in his announcement of the discovery of the quantum.

1) E is the energy of the particular quantum of energy under study. When discussing electromagnetic quanta (of which light is only one example, x-rays and radio waves being two other examples), the word photon is used. A photon (the word is due to Albert Einstein) is a quantum of electromagnetic energy. The word quantum (quanta is the plural) is usually used in a more general sense, to describe various ideas of quantum theory or even, as I just did, to describe the entire theory itself.

2) h stands for a fundamental constant of nature now known as Planck's Constant. By the way, the discovery of the quantum had, and continues to have, many profound effects. Enough so that all of science (especially physics) before 1900 is refered to as "classical" and the science since 1900 is called "modern."

The value for Planck's Constant is 6.6260755 x 10¯34 Joule second. Please note that the unit is Joule MULTIPLIED BY second. It is not a division, both Joule and second are in the numerator.

3) ν is the frequency of the particular photon being studied. The discussion about frequency above applies here.

Before going on, I want to discuss one little issue (heh, heh, heh). Frequency is a wave-based idea. What is it doing in a particle-based idea like the quantum? Good question. So much so that the term "wave packet" is often used in discussing these ideas. Indeed, modern science now speaks of "wave-particle duality" rather than "Light is a wave" or "Light is a particle." This whole area is profound and can lead to years of probing discussion. Albert Einstein and Niels Bohr (who were great friends) discussed these issues (and more) often over a period of many years, especially in the late 1920's and early 1930's. Their discussions are still important enough to merit historical study today. Every year, several books are published which delve into one or more of the implications of "wave-particle duality."


Problem #1:

(a) Identify λν = c as either a direct or inverse mathematical relationship.
(b) Do the same for E = hν.
(c) Write a mathematical equation for the relationship between energy and wavelength.
(d) Identify the equation for (c) as either direct or inverse.

Solution:

(a) λν = c is an inverse relationship. As one value (say the wavelength) goes up, the other value (the frequency) must go down. Why? Because the product of the two must always equal the same value, c, which is a constant.

(b) E = hν is a direct relationship. As the frequency increases, so does E. Why? Because h remains constant. Notice that in this equation the two quantities which can change (ν and E) are on DIFFERENT sides of the equation whereas in the inverse relationship in (a), both λ and ν are on the same side, with the constant on the opposite side.

(c) Rearrange the light equation thusly:

ν = c / λ

Substitute for ν in E = hν:

E = hc / λ

Rearrange to group the variables and the constants:

Eλ = hc

(d) The mathematical relationship is an inverse one. Note the equation's similarity to λν = c, with two values that can vary on the left side and a constant (h times c) on the right.


Problem #2: How many Joules of energy are contained in a photon with λ = 550 nm? How many kJ/mol of energy is this?

Solution:

I will first do the problem step by step, then in a more combined way.

Use ν = c / λ to get the frequency:

x = (3.00 x 108 m s¯1) / (550 x 10¯9 m)

This equals 5.4508 x 10141. I left a couple extra digits in the answer and notice that the wavelength is not in scientific notation. Why? Think about how I got from nm to m for the value used. (I also used 299,792,458 in the calculation, not 3.00 x 108. Sorry!! Using 3.00 x 108 gives 5.454 x 10141.)

Now use E = hν to get the energy:

x = (6.6260755 x 10¯34 J s) (5.4508 x 10141)

This equals 3.612 x 10¯19 J. Most emphatically, this is not the final answer. This is the energy for one photon!!

The last step is to find the kilojoules for one mole and for this we use Avogadro's Number:

x = (3.612 x 10¯19 J/photon) (6.022 x 1023 photon mol¯1)

Dividing the answer by 1000 to make the change to kilojoules, we get 217.5 kJ/mol. Yes, this is my final answer!

A more condensed way would be to use Eλ = hc and solve for E, then multiply the answer times Avogadro's Number.


Problem #3: How many kJ/mol (of photons) of energy is contained in light with a wavelength of 496.36 nm?

The answer is 241.00 kJ/mol.


Problem #4: When it decays by beta decay, cobalt-60 also produces two gamma rays with energies of 1.17 MeV and 1.33 MeV. For the 1.17 MeV photon, determine (a) the energy in Joules of one photon as well as (b) the energy produced in units of kJ per mole.

Solution:

To solve this problem, you must know the relationship between electron-volts (eV) and Joules. This value can be looked up. I will use 1 eV = 1.6022 x 10¯19 J for this problem.

(a) convert MeV to Joules:

1.17 MeV = 1.17 x 106 eV

(1.17 x 106 eV) x (1.6022 x 10¯19 J/eV) = 1.87 x 10¯13 J (to three sig. figs.)

(b) convert J of one photon to kJ/mol of photons:

(1.874574 x 10¯13 J) x (6.022 x 1023 mol¯1) = 1.129 x 1011 J/mol = 1.129 x 108 kJ/mol

Note the use of the unrounded off value for the energy of one photon.


Problem #5: What is the energy of a photon of green light with a frequency of 5.76 x 10141.

Problem #6: A particular x-ray has a wavelength of 1.2 Å. Calculate the energy of one mole of photons with this wavelength.

Problem #7: When excited, some atoms produce an emssion with a frequency of 7.25 x 1012 Hz.

(a) calculate the energy, in Joules, for one photon with this frequency.
(b) calculate the energy, in kJ/mol.
(c) Is this light visible? Why or why not?

Go to the answers for 5 and 6 plus some advice on the solution technique for problem 7.


Go to Part One of Light Equations

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