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# 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

Exrait

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.
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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.
* * * * *
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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
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 engineering rather than
scientific notation to ease conversion to the metric
prefixes based on powers of three.
Examples: Converting scientific to engineering notation,
―4 ―6 5.35 x 10 m = 535 x 10 m in engineering notation ( = 535 micrometers = 535 μm)
10 99.23 x 10 Hz = 92.3 x 10 Hz in engineering notation ( = 92.3 GHz )
To convert any exponential notation to engineering notation, adjust the exponent and
decimal position until the exponent is divisible by 3 and the significand is between 1
and 1,000.
(To review moving the decimal, see Lesson 1A). Try this example.
―11 Q. Convert to engineering notation, then to metric-prefix notation: 5.27 x 10 m
* * * * *
Page 384
Module 19 — Light and Spectra

―11 ―12 A. 5.27 x 10 m = 52.7 x 10 m in engineering notation = 52.7 picometers or 52.7 pm
―12 52.7 x 10 is the only way to write the given quantity that results in both an
exponent divisible by 3 and a significand between 1 and 1,000.
Engineering notation, like scientific notation, results in one unique expression for
each numeric value, and this makes answers easy to compare and check.
During electromagnetic wave calculations, you should work in general exponential
notation, then, at the end, convert your answers to engineering notation and then prefix
nn if needed.

Practice B: Do every other question. Complete the rest during your next study session.
1. By inspection, convert these to units in engineering notation, without prefixes.
a. 5.4 GHz b. 720 nm c. 96.3 MHz
2. Convert these first to engineering notation, then to measurements with metric prefixes
in place of the exponential terms.
―7 4a. 47 x 10 m b. 347 x 10 Hz
―8 ―1c. 1.92 x 10 m d. 14,920 x 10 Hz
11 e. 0.25 x 10 Hz f. 7,320 m

Practice A
1. a. Wavelength is a distance, and the SI unit for distance is the meter (m).
b. Frequency is defined as 1/time, and the SI unit for time is the second, so the SI unit of υ is 1/s or
―1s , which is called a Hertz.
c. Energy The SI unit for energy is the joule (J)
―1d. Speed is defined as distance over time, so the SI units are meters/second (m • s )
2. Wave speed = λ υ
8 ―1Speed = 3.00 x 10 m • s
λ in meters = ?
14 ―1 ―1υ = 5.09 x 10 Hz s ( During calculations, write Hz as s )
8 ―1 ―6 ―7 ? = λ in meters = speed = 3.00 x 10 m • s = 0.589 x 10 m = 5.89 x 10 m
14 ―1 υ 5.09 x 10 s
Page 385
Module 19 — Light and Spectra

Practice B
9 ―1 ―9 6 ―11a. 5.4 GHz = 5.4 x 10 Hz or s 1b. 720 nm = 720 x 10 m 1c. 96.3 MHz = 96.3 x 10 s
―7 ―6 2a. 47 x 10 m = 4.7 x 10 m = 4.7 μm (if exponent is made larger, make significand smaller)
4 6 ―8 ―9 2b. 347 x 10 Hz = 3.47 x 10 Hz = 3.47 MHz 2c. 1.92 x 10 m = 19.2 x 10 m = 19.2 nm
―1 32d . 14,920 x 10 Hz = 1.492 x 10 Hz = 1.492 kHz (significand must be between 1 and 1,000)
11 9 3 2e. 0.25 x 10 Hz = 25 x 10 Hz = 25 GHz 2f. 7,320 m = 7.32 x 10 m = 7.32 km
* * * * *
Lesson 19B: Wave Calculations and Consistent Units
When using equations to solve science calculations, the units of measurements must be
consistent: for each quantity in the equation, one unit must be chosen. The rule is
To solve with equations, first convert to consistent units, solve in consistent units,
then convert the consistent WANTED unit to other units if needed.
 When using a specified value and units for the gas constant R, our rule was:
convert the DATA to the units used in the constant.
 When using specific heat capacities (c), our rule was: convert to the unit of the most
complex term in the DATA.
In wave calculations, we will observe both of these rules. Specific constants will be
required, but those constants will generally contain the most complex units in the problem.
To choose and convert to consistent units, our steps will be
1. Write the equation needed for the problem.
2. If the equation has constants, in the DATA table, first write each constant’s symbol,
value and units, then below write the symbols for the variables.
3. In the DATA table, after each variable symbol, write “ = ? “and then its chosen
consistent unit.
Example: DATA: λ = ? m = (a wavelength is a distance)
To choose the consistent units to write after each variable symbol, apply these steps.
a. If the equation has constants, after each variable write a unit that is appropriate for
the symbol and is consistent with the units in the constants.
Example: If c = λ υ is the equation that is needed, list
DATA:
8 ―1 c = 3.00 x 10 m ∙ s (list constants in the equation first)
λ = ? m = (wavelength is a distance; the distance unit in c is meters)
―1 υ = ? s = (υ = 1/time, the time unit used in the constant c is seconds)
Page 386
Module 19 — Light and Spectra

b. If there are no constants are in the equation, label each variable with an appropriate
unit matching the units used in the WANTED unit.
4. If the unit that is WANTED is not specified,
a. pick a WANTED unit to match the units in the constants of the equation.
b. If no constants are used, write after the WANTED unit the SI unit for the quantity.
5. In the DATA, write the data supplied for each symbol, then convert that DATA to the
consistent units if needed.
6. First solve for the WANTED symbol in the consistent unit, then convert to a different
WANTED unit if specified.
The problem below will help you to understand and remember the rules above.
Q. When neon gas at low pressure is subjected to high voltage electricity, it emits
waves of light. One of the more intense waves in the visible spectrum has a
wavelength of 640. nm, perceived by the eye as red light. What is the frequency of
this light in terahertz (THz)?
Solve using the steps above.
* * * * *
This problem involves a frequency (υ) and a wavelength (λ) for light. We know that
light travels at the speed of light (c), a constant. So far, we know only one equation that
relates those three symbols. So, to start, your paper should look like this:
c = λ υ
8 ―1 DATA: c = 3.00 x 10 m ∙ s
λ =
υ =
For the speed of light (c), in conversion calculations the unit m/s must be used as a ratio
and written in the top/bottom format, but in equations, it will simplify unit cancellation
8 ―1 if the units are written in the “on one line” format: 3.00 x 10 m ∙ s .
Now assign a consistent unit to each variable. Since this equation has a constant (c),
after each variable symbol write a unit that both measures the variable and matches one
of the units used in the constant.
Do that step, then check below.
* * * * *
Page 387
Module 19 — Light and Spectra

Your DATA should look like the Rule 2a Example above, minus the (comments).
After the = sign for each variable, write the data for that variable that is supplied in the
problem. Then, in the DATA table,
 convert the supplied units to the consistent units if needed.
 For the WANTED variable, after the assigned consistent unit, write the unit
WANTED in the problem if it is not the consistent unit.
* * * * *
Your paper should look like this.
c = λ υ
8 ―1 DATA: c = 3.00 x 10 m • s
―9 λ = ? m = 640. nm = 640. x 10 m
―1 υ = ? s but THz is WANTED
For λ , meters is the chosen consistent unit, so you must convert nm to m. The easy way
is to substitute what the prefix means.
To SOLVE,
 First solve the equation for the WANTED symbol in symbols.
 Substitute the DATA into the solved equation using the consistent units, and solve
including the units.
 If needed, convert to the unit WANTED in the problem.
Modify your work if needed and finish.
* * * * *
―1 8 ―1 14 ―1SOLVE: ? = υ in s then THz = c = 3.00 x 10 m • s = 4.69 x 10 s
―9 λ 640. x 10 m
That solves in the consistent unit. To finish, convert to the WANTED unit.
* * * * *
14 ―1 4.69 x 10 s Hz • 1 THz = 469 THz
12 10 Hz
Done!
This method of choosing to solve in the units of the constants, is arbitrary. You can use any
consistent units to solve. In physics, it is usually preferred (and simplifying) to solve all
problems in SI units. In many chemistry calculations, constants will be stated in SI units,
matching the physics practice.
However, “solving in the units of the constants” will save a few steps if data is provided in
mL, grams, kcals, electron volts, BTUs, or other non-SI units, as may be the case in some
science calculations.
Page 388

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