User:Doomed Rasher/sandbox

Liberapedia is an English-language wiki launched in 2007 in response to the creation of Conservapedia. The site's articles are intentionally written with a strong liberal bias, parodying Conservapedia's depiction of conservative points of view. Liberapedia's founder, WillH, noted that "most articles should take stereotypical liberal views and distort them to the extreme".

WillH was the only bureaucrat of the original site, albeit an infrequent contributor. Users of Liberapedia moved the site to central wikia in May 2008 and recasted themselves as bureaucrats when it was realized that WillH had become inactive. The old domain, liberapedia.com, was taken over by a cybersquatter after it had expired on January 1, 2009.

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Liberapedia contains satirical entries along with serious factual articles, although the former is present in large abundance. Liberapedia describes its mission as "...to create an wiki that is both relatively factual and funny at the same time. It is important to remember that Liberapedia is not just another joke wiki like Uncyclopedia". Discrepancies in article quality had been attributed to extensive vandalism in Spring 2008, which left a number of pages disfigured.

Chapter 6 Review

 * Wavelength: The distance between two adjacent peaks in a wave, expressed in metres.
 * Ångstrom: 10-10 of a meter.
 * Nanometer: 10-9 of a meter.
 * Micrometer: 10-6 of a meter.
 * Millimeter: 10-3 of a meter.
 * Meter: a meter. You know what this is :P.

_______________________

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 * Frequency: The number of cycles of the wave passed each second in a wave, expressed in Hertz or s-1 (1/s).

Wave equation for light:


 * $$v\lambda = c $$

where:


 * $$c\,\!$$ is the speed of light, or 299792458 metres per second (Use 3.00 x 108 for your calculations if it makes life easier).
 * $$\lambda\,\!$$ is the wavelength.
 * $$v\,\!$$ is the frequency.

Simply solve for the unknown value in the equation after substituting in the two other values. _______________________

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 * Visible light: 400 - 750 nm wavelength.
 * Quantum: Fixed amount; the smallest quantity of energy that can be emitted or absorbed as electromagnetic radiation.

The energy of a single quantum (or packet of energy) is evaluated in the following equation:


 * $$E = hv\,\!$$

where:


 * $$E\,\!$$ is the energy of the quantum in Joules.
 * $$h\,\!$$ is Planck's constant, equivalent to 6.62606896 x 10-34 Joule-seconds (J·s). (Just remember 6.626 x 10-34)
 * $$v\,\!$$ is the frequency of the radiation.

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 * Photoelectric effect: The process by which electrons are ejected from matter due to absorption of light.
 * Every metal has a critical frequency; before this frequency, electrons will not be emitted no matter how intense the light is. Only light after this frequency will cause the emission of electrons.
 * The energy required to liberate an electron is the work function, which is the energy of one quantum of light at the critical frequency.
 * Each electron has energy = E = hv (See above equation)

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 * Spectrum: The range of wavelengths in radiation from a polychromatic source.
 * Line spectrum: A spectrum containing only specific wavelengths, e.g. a wavelength with only yellow light of 582 nm, etc. (These can have more than one line)


 * Rydberg equation: The calculation of the wavelengths of all spectral lines of hydrogen.


 * $$\frac{1}{\lambda} = (R_{\mathrm{H}})\left(\frac{1}{n^2_1} - \frac{1}{n^2_2}\right)$$

where:


 * $$\lambda\,\!$$ is the wavelength.
 * $$R_{\mathrm{H}}$$ is the Rydberg constant for hydrogen, equivalent to 1.096776 x 107 m-1.
 * $$n_1$$ and $$n_2$$ are integers such that $$n_1 < n_2$$, representing the energy levels at the beginning and the end of the transition.

_______________________


 * Bohr model of atomic structure:
 * Only electron orbits of certain radii can exist in the hydrogen atom, with each corresponding to an energy level.
 * An electron in a permitted orbit has a specific energy level and is "stable".
 * Electrons emit energy when they decrease energy levels; e.g. going from 5 to 2, and absorb energy when they increase energy levels, e.g. from 3 to 4. Energy emitted/absorbed are in forms of photons (E = hv).

_______________________


 * Principal quantum number: The first value describing an atomic orbital, which represents the energy level of an electron.
 * It is represented by $$n\,\!$$, and can have integer values ranging from one to infinity, e.g. 1, 2, 3, 4...
 * The lowest energy state for hydrogen, where n=1, is the ground state of the atom.
 * Any state where n=2 or higher is an excited state.

_______________________


 * Azimuthal quantum number: The second value describing an atomic orbital, which represents the angular momentum of an electron orbital.
 * It is represented by $$l\,\!$$, and can have integer values ranging from zero to n-1, e.g. 0, 1, 2, 3...n-1
 * The number determines the shape of an atomic orbital.
 * l = 0: s orbital, 2 electrons, spherical (sharp)
 * l = 1: p orbital, 6 electrons, two lobes (principal''')
 * l = 2: d orbital, 10 electrons, four lobes (diffuse)
 * l = 3: f orbital, 14 electrons, eight lobes/complicated (fundamental)

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 * Magnetic quantum number: The third value describing an atomic orbital, which represents the quantum state of an electron orbital.
 * It is represented by $$m_l\,\!$$, and can have integer values ranging from negative l to l, e.g. -l...0...l
 * The number determines the energy levels in a subshell.

_______________________


 * Spin quantum number: The fourth value describing an electron in an atomic orbital, which represents the spin of an electron.
 * It is represented by $$m_s\,\!$$, and can have values of either -1/2 or 1/2.
 * electrons of +1/2 and -1/2 spins spin in opposite directions.

_______________________


 * De Broglie hypothesis: A hypothesis that all matter have a wave-like property, and that the characteristic wavelength of any item depends on its mass and velocity:


 * $$\lambda = \frac{h}{mv}$$

where
 * $$\lambda\,\!$$ is the wavelength.
 * $$h\,\!$$ is Planck's constant.
 * $$m\,\!$$ is the particle's rest mass.
 * $$v\,\!$$ is the particle's velocity.

The term $$mv\,\!$$ is the momentum of the object. The equation is applicable to all matter, although ordinary objects have wavelengths so tiny that they are out of range of any possible observation. _______________________
 * Electrons, with their small mass, exhibit visible wave-like behavior.


 * Heisenberg's uncertainty principle: The concept that we can never be completely certain of both the momentum and position of a particle; the product of the uncertainty of momentum and uncertainty of position can never be less than h/4π, where h is Planck's constant:
 * $$\Delta x \Delta p \ge \frac{h}{4\pi}$$

_______________________


 * The consequence of the uncertainty principle meant that description of the electron will not be based on position, but rather on probability.
 * Quantum mechanics deals with subatomic particles. Application requires advanced calculus (which will not be covered here).

_______________________

_______________________
 * ψ is the Schrödinger wave function. ψ2 is the electron density, or the probability that the electron will be found at a specific location.


 * Electron shells obey certain rules based on quantum numbers:
 * The shell with principal quantum number n has exactly n subshells.
 * Each subshell consists of a specific number of orbitals, corresponding to different ml.
 * The total number of orbitals in a shell is n2

_______________________


 * Pauli exclusion principle: No two electrons in an atom can have the same set of four quantum numbers. For a given orbital, the values of n, l, and ml are fixed.
 * As a consequence, an orbital can hold a maximum of two electrons and they must have opposite spins.

_______________________


 * Electron configuration: The electron configuration is the way that electrons are distributed among orbitals of an atom at ground state. A subshell is written in the form nxy, where n is the shell number, x is the subshell label and y is the number of electrons in the subshell. An atom's subshells are filled in order of increasing energy.
 * Hund's rule: Empty orbitals are filled first, before half-filled orbitals. Both electrons in an orbital must have opposite spins.

1. The orange-red associated with neon lights and the He-Ne laser is due in part to a line at 640.2 nm. Calculate the energy difference between the two states that are responsible for this line, in kilojoules per mole.


 * $$640.2 \mbox{ nm} \times 10^{-9} \mbox{ m}\cdot \mbox{nm}^{-1} = 6.402 \times 10^{-7} \mbox{ m}$$


 * $$f = \frac{299792458 \mbox{ m}\cdot\mbox{s}^{-1}}{6.402 \times 10^{-7} \mbox{ m}}$$


 * $$f = 4.682793783 \times 10^{14} \mbox{ s}^{-1}$$


 * $$E = (6.62606896 \times 10^{-37} \mbox{ kJ}\cdot\mbox{s}) \times (4.682793783 \times 10^{14} \mbox{ s}^{-1}) \times (6.02214179 \times 10^{23} \mbox{ mol}^{-1}) $$


 * $$E = 186.8581141 \mbox{ kJ}\cdot \mbox{mol}^{-1}$$

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2. In a sodium atom, there are two states that differ in energy by 203.1 kJ/mol. When an electron transition occurs from higher of these states to the lower, energy is given off as yellow light. For this light, calculate a. λ in nanometers b. Frequency v

b.
 * $$203.1 \mbox{ kJ}\cdot \mbox{mol}^{-1} = (6.62606896 \times 10^{-37} \mbox{ kJ}\cdot\mbox{s}) (v) (6.02214179 \times 10^{23} \mbox{ mol}^{-1})$$
 * $$5.087320627 \times 10^{14} \mbox{ s}^{-1}= v$$

a.
 * $$5.087320627 \times 10^{14} \mbox{ s}^{-1} = \frac{299792458 \mbox{ m}\cdot\mbox{s}^{-1}}{\lambda}$$
 * $$5.892934217 \times 10^{-7} \mbox{ m} = \lambda$$

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3. Find the wavelength, in nanometers of the line in the Balmer series that results from the transition from n=3 to n=2.

(Balmer's constant = 364.56 nm)


 * $$\lambda\ = 3.6456 \times 10^{-7} \mbox{ m}\left(\frac{3^2}{3^2 - 2^2}\right)$$


 * $$\lambda\ = 6.56208 \times 10^{-7} \mbox{ m} \times 10^{9} \mbox{ nm}\cdot \mbox{m}^{-1}$$


 * $$\lambda\ = 656.208 \times 10^{-7} \mbox{ nm}$$

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4. The frequency of strong yellow line in the spectrum of sodium is 5.09 x 10^14 s^-1. Calculate the wavelength in nanometers (c = 3 x 10^8 m/s)


 * $$5.09 \times 10^{14} \mbox{ s}^{-1} = \frac{3 \times 10^8 \mbox{ m}\cdot\mbox{s}^{-1}}{\lambda}$$


 * $$10^{9} \mbox{ nm}\cdot \mbox{m}^{-1} \times 5.89390963 \times 10^{-7} \mbox{ m} = \lambda$$


 * $$589.390963 \times 10^{-7} \mbox{ nm} = \lambda$$

$$ 1.\ \lim_{x \to 2} \frac {x-2}{x^2-x-2} = \frac{1}{3} $$

$$ 3.\ \lim_{x \to 3} \frac {\frac{1}{x+1} - \frac{1}{4}}{x-3} = -\frac{1}{16} $$

$$ 5.\ \lim_{x \to 0} \frac {\sin(x)}{x} = 1 $$

$$ 7.\ \lim_{x \to 0} \frac {e^x - 1}{x} = 1 $$

$$ 9.\ \lim_{x \to 0} \frac {\ln(x+1)}{x} = 1 $$

$$ 11.\ \lim_{x \to 3} (4-x) = 1 $$ $$ 12.\ \lim_{x \to 1} (x^2 + 2) = 3 $$ $$ 13.\ \lim_{x \to 3} \frac{|x-3|}{x-3} = \mbox{DNE: Left-right behavior different} $$ $$ 14.\ f(x) = \begin{cases} x^2 + 3, & x \neq 1 \\ 1, & x=1 \end{cases}$$

$$ {\color{white}14.\ }\lim_{x \to 1} f(x) = 4 $$ $$ 15.\ \lim_{x \to 1} \sqrt[3]{x} \ln|x-2| = 0 $$ $$ 16.\ \lim_{x \to 0} \frac{4}{2 + e^{\frac{1}{x}}} = \mbox{DNE: Left-right behavior different} $$ $$ 17.\ \lim_{x \to \frac{\pi}{2}} \tan(x) = \mbox{DNE: Left-right behavior different, approaches} \pm \infty $$ $$ 18.\ \lim_{x \to 0} 2\cos\left(\frac{1}{x}\right) = \mbox{DNE: infinite oscillating behavior} $$ $$ 19.\ \lim_{x \to 0} \sec(x) = 1 $$ $$ 20.\ \lim_{x \to 2} \frac{1}{x-2} = \mbox{DNE: Left-right behavior different, approaches} \pm \infty $$ $$ 21.\ $$ $$ 22.\ $$
 * $$f(1) = 2 $$
 * $$\lim_{x \to 1} f(x) = \mbox{DNE: Left-right behavior different} $$
 * $$f(4) = \mbox{undefined} $$
 * $$\lim_{x \to 4} f(x) = 2 $$
 * $$f(-2) = \mbox{undefined} $$
 * $$\lim_{x \to -2} f(x) = \mbox{DNE: Left-right behavior different, approaches} \pm \infty $$
 * $$f(0) = 4 $$
 * $$\lim_{x \to 0} f(x) = \mbox{DNE: Left-right behavior different} $$
 * $$f(2) = \mbox{undefined} $$
 * $$\lim_{x \to 2} f(x) = 0.5 $$
 * $$f(4) = 2 $$
 * $$\lim_{x \to 4} f(x) = \mbox{DNE: Approaches} +\infty $$

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