Collaborative Research: Learning Discrete Mathematics and Computer ...

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Collaborative Research: Learning Discrete Mathematics and Computer Science via Primary Historical Sources Janet Barnett, Guram Bezhanishvili, Hing Leung, Jerry Lodder, David Pengelley, Inna Pivkina, Desh Ranjan Jan. 2007 1 Our List of Projects • Summation of Numerical Powers. The discovery of closed formulas for discrete sums of numerical powers, motivated by application to approximations for solving area and volume problems in calculus, is probably the most extensive thread in the development of discrete mathematics, spanning the pe- riod from ancient Pythagorean interest in patterns of dots to the work of Euler on a general formula for discrete summations.

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• Publié par: vosom
• Nombre de lectures: 12
• Langue: English

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Self-Study Assignment
You will have a QUIZ on the attached pages on _____________________ .
Your assignment is: READ the pages attached. WORK the examples in the lesson.
Complete the pages as homework.
To work the examples,
• use a sheet of paper to cover below the * * * * * line,
• try the problem on your paper,
• then check your answer below the * * * * * line.
Start early. This assignment will require 4-6 hours of work outside of class.





Introduction to
Chemistry
Calculations
* * * * *
Module 19 – Light and Spectra ................................................................................... 381
Lesson 19A: Waves ................................................. 381
Lesson 19B: Wave Calculations and Consistent Units ....................... 386
Lesson 19B: Planck's Constant ............................. 391
Lesson 19C: The Hydrogen Atom Spectrum ....................................................................... 395
Lesson 19D: The Wave Equation Model .............. 402 Module 19 — Light and Spectra

Module 19 — Light and Spectra
* * * * *
Lesson 19A: Waves
Waves and Chemistry
Electromagnetic energy includes gamma rays, x-rays, ultraviolet, visible, and infrared
light, microwaves, and radio waves. Each of these types of energy occupies a different
region of the electromagnetic spectrum.
Chemical particles can both absorb and release electromagnetic energy. This absorption
and release of energy can be a powerful tool in identifying chemical particles. Exposure to
certain types of electromagnetic energy can also cause chemical particles to change and
react.
In some cases, the behavior of electromagnetic energy is best predicted by assuming that
the energy is a particle, but in other cases, energy is best understood as a wave. Let us
begin by investigating the properties of waves.
Wave Terminology
Crest1 Wavelength ( λ)
0 90 180 270 360 450 540 630 720 810 900 990
Trough

The following are some of the components of a wave that are important in chemistry.
1. Wavelength is the distance between the crests of a wave, which is equal to the distance
between the troughs of a wave.
a. The symbol for wavelength is λ (the lower-case Greek letter lambda).
b. Since a wave length is a distance, the units of wavelength are distance units, such as
meters, centimeters, or nanometers.
Page 381
Module 19 — Light and Spectra

2. Frequency is a number of events per unit of time. The unit for frequency is 1/time.
For waves, frequency is the number of wave crests that pass a point per unit of time.
a. In wave equations, the symbol for frequency is υ (the lower-case Greek letter nu).
b. The SI unit for time, a fundamental quantity, is seconds. Because frequency is a
―1derived quantity that is 1/time, the SI unit must be 1/seconds (s ). The unit
―1 second is also called a hertz (Hz). During calculations, it is best to write hertz as
―1 ―1 s . Hertz and s are equivalent and can cancel.
c. When wave frequency is expressed as “cycles per second,” wave cycles are the
entity being measured, and 1/seconds is the unit. When writing wave units, the
term “wave cycle” or “cycle” is often included as a helpful label in conversion
calculations, but is usually omitted as understood in equation calculations.
3. The speed of a wave is equal to its frequency times its wavelength.
wave speed = λ υ = (lambda)(nu).
Memorize the equation for wave speed in words, symbols, and names for the symbols.
Wave Calculations
Because wave relationships are often defined by multi-term equations, wave calculations
are generally solved using equations rather than conversions. We will start with a simple
problem that can be solve using both methods, but to practice with the equation that will be
required for more complex calculations, solve the problem below using the equation method
(for review, see Lesson 17D).
Q. If ocean waves are traveling at 200. meters/minute and the crests pass a fixed point
at a rate of 15.0 waves per minute, what is the wavelength, in meters?
* * * * *
Write the one equation learned so far for waves.
Wave speed = λ υ
List those three terms in a data table. After each term, write the data in the problem that
corresponds to the term. Add a ? and the desired unit after the WANTED symbol.
* * * * *
Wave speed = 200 m/min. (speed units are distance over time)
λ = ? meters (the length of a wave is a distance)
―1 υ = 15.0 wave cycles/min. = 15.0 min. (frequency units are 1/time)
When solving frequency calculations using equations, “wave cycles” is usually omitted as
understood to be the object being measured.
Solve the equation in symbols for the WANTED symbol, then substitute the DATA.
Include the consistent units and check the unit cancellation.
* * * * *
Page 382
Module 19 — Light and Spectra

SOLVE: Since Wave speed = λ υ
λ in meters = speed = speed • 1 = 200. m • 1 = 13.3 meters
υ υ min. ―1 15.0 min.
Note in the unit cancellation in the denominator:
―1 1 ―1 0min.• min. = min. • min. = min. = 1 . Anything to the zero power equals one.

Practice A
1. Write the SI units for
a. Wavelength b. Frequency c. Energy d. Speed
2. Street lights containing sodium vapor lamps emit an intense yellow light at two close
14wavelengths. The more intense wave has a frequency of 5.09 x 10 Hz. If light travels
8 ―1at the speed of 3.00 x 10 m • s , what is the wavelength of this intense yellow wave
in meters? (Use the equation method to solve.)

Electromagnetic Waves
The movement of electric charge creates electromagnetic waves. The waves propagate:
they travel outward from the moved charge. The energy that was added to move the
charge is carried outward by the waves.
In a vacuum, all electromagnetic waves travel at the speed of light:
83.00 x 10 meters/second. The speed of light is the “speed limit of the universe:” the
fastest speed possible for energy or matter. In wave calculations, the speed of light is given
the symbol c.
Electromagnetic waves slow when they travel through a medium that is denser than a
vacuum, but when passing through air or other gases at normal atmospheric pressures, the
speed of light does not slow sufficiently to affect most calculations in chemistry.
For electromagnetic waves, this relationship will be true (and must be memorized):
8Speed of Light = c = λ υ = 3.00 x 10 m/s in vacuum or air
Since c is a constant, υ and λ are inversely proportional. As wavelength goes up, frequency
must go down. If υ goes up, λ must go down.
Further, as long as we work in consistent units and in air or vacuum, since c is constant, a
specific value for the frequency of an electromagnetic wave will always correlate to a
specific value for its wavelength.
The Regions of the Electromagnetic Spectrum
The electromagnetic spectrum goes from very high to very low wavelengths and
frequencies. Regions of the spectrum are assigned different names that help in predicting
the types of interactions that the energy will display. However, all of these forms of energy
Page 383
Module 19 — Light and Spectra

are electromagnetic waves. The difference among the divisions of the spectrum is the
length (or corresponding frequency) of the waves.
The following table (no need to memorize) summarizes some of the general divisions of the
electromagnetic spectrum.
―1Frequency (s ) Wavelength (m) Type of Electromagnetic Wave
24 ─16 10 3 x 10 Gamma Rays
21 ─13 10 3 x 10
18 ─10 10 3 x 10 X-rays
15 ─7 10 3 x 10 Ultraviolet, Visible, Infrared Light
12 ─4 10 3 x 10 Microwaves
9 ─1 10 3 x 10 UHF Television Waves
6 10 300 Radio Waves
Units For Frequency and Wavelength
Measurements of wavelengths and frequencies often Prefix Symbol Means
involve very large and very small numbers. Values
12tera T x 10 are often expressed using SI prefixes such as gigahertz
9(GHz) or nanometers (nm). Prefixes needed most giga- G x 10
often are those for powers of three.
6mega- M x 10
Engineering Notation
3kilo- k x 10
Scientific notation expresses a value as a significand
―3milli- m x 10 between 1 and 10 times a power of 10.
―6micro- μ x 10 Engineering notation expresses values as a
―9significand between 1 and 1,000 times a power of 10 nano- n x 10
that is divisible by 3. In wave calculations, answers
―12pico- p x 10 are often preferred in engi