History

Sample method

2 MnSO₄(s) + O₂(aq) ¡æ 2 MnO(OH)₂(s)

4 Mn(OH)₂(s) + O₂(aq) + 2 H₂O ¡æ 4 Mn(OH)₃(s)

Mn(SO₄)₂ + 2 I^-(aq) ¡æ Mn²⁺(aq) + I₂(aq) + 2 SO₄^2-(aq)

2 S₂O₃^2-(aq) + I₂ ¡æ S₄O₆^2-(aq) + 2 I^-(aq)

Analysis

1 mole of O₂ ¡æ 2 moles of MnO(OH)₂ ¡æ 2 mole of I₂ ¡æ 4 mole of S₂O₃^2-

Limitations

BOD₅

See also

• Oxygen saturation

References

1. ^ Chiya Numako and Izumi Nakai (1995). "XAFS studies of some precipitation and coloration reaction used in analytical chemistry". Physica B: Condensed Matter 208-209: 387–388. doi:10.1016/0921-4526(94)00706-2. 

• Lajos Winkler (1888). "Die Bestimmung des in Wasser Gelösten Sauerstoffes". Berichte der Deutschen Chemischen Gesellschaft 21: 2843–2855. doi:10.1002/cber.188802102122. http://gallica.bnf.fr/ark:/12148/bpt6k90715s/f46.chemindefer. 

• Moran, Joseph M.; Morgan, Michael D., & Wiersma, James H. (1980). Introduction to Environmental Science (2nd ed.). W.H. Freeman and Company, New York, NY ISBN 0-7167-1020-X

• Standard Methods for the Examination of Water and Wastewater - 20th Edition ISBN 0-87553-235-7. This is also available on CD-ROM and online by subscription

• http://www.ecy.wa.gov/programs/wq/plants/management/joysmanual/4oxygen.html

• Y.C. Wong & C.T. Wong. New Way Chemistry for Hong Kong A-Level Volume 4, Page 248. ISBN 962-342-535-X

• http://ocw.mit.edu/NR/rdonlyres/Earth--Atmospheric--and-Planetary-Sciences/12-097January--IAP--2006/35FAFFC3-135B-4266-93B4-2144F7487004/0/dissolved_oxygen.pdf

• http://io.uwinnipeg.ca/~simmons/ysesp/ap12pg7.htm
  
• http://io.uwinnipeg.ca/~simmons/ysesp/ap12pg3.htm
  
• http://io.uwinnipeg.ca/~simmons/ysesp/ap12pg11.htm

The spherical harmonics are the angular portion of the solution to Laplace's equation in spherical coordinates where azimuthal symmetry is not present. Some care must be taken in identifying the notational convention being used. In this entry, is taken as the polar (colatitudinal) coordinate with , and as the azimuthal (longitudinal) coordinate with . This is the convention normally used in physics, as described by Arfken (1985) and Mathematica (in mathematical literature, usually denotes the longitudinal coordinate and the colatitudinal coordinate). Spherical harmonics are implemented in Mathematica as SphericalHarmonicY[l, m, theta, phi].

Spherical harmonics satisfy the spherical harmonic differential equation, which is given by the angular part of Laplace's equation in spherical coordinates. Writing in this equation gives

(1)

Multiplying by gives

(2)

Using separation of variables by equating the -dependent portion to a constant gives

(3)

which has solutions

(4)

Plugging in (3) into (2) gives the equation for the -dependent portion, whose solution is

(5)

where , , ..., 0, ..., , and is an associated Legendre polynomial. The spherical harmonics are then defined by combining and ,

(6)

where the normalization is chosen such that

(7)

(Arfken 1985, p. 681). Here, denotes the complex conjugate and is the Kronecker delta. Sometimes (e.g., Arfken 1985), the Condon-Shortley phase is prepended to the definition of the spherical harmonics.

The spherical harmonics are sometimes separated into their real and imaginary parts,

(8)

(9)

The spherical harmonics obey

			(10)
			(11)
			(12)

where is a Legendre polynomial.

Integrals of the spherical harmonics are given by

(13)

where is a Wigner 3j-symbol (which is related to the Clebsch-Gordan coefficients). Special cases include

			(14)
			(15)
			(16)
			(17)

(Arfken 1985, p. 700).

The above illustrations show (top), (bottom left), and (bottom right). The first few spherical harmonics are

			(18)
			(19)
			(20)
			(21)
			(22)
			(23)
			(24)
			(25)
			(26)
			(27)
			(28)
			(29)
			(30)
			(31)
			(32)
			(33)

Written in terms of Cartesian coordinates,

			(34)
			(35)
			(36)

so

			(37)
			(38)
			(39)
			(40)
			(41)
			(42)

The zonal harmonics are defined to be those of the form

(43)

The tesseral harmonics are those of the form

(44)

(45)

for . The sectorial harmonics are of the form

(46)

(47)

RELATED WOLFRAM SITES: http://functions.wolfram.com/Polynomials/SphericalHarmonicY/, http://functions.wolfram.com/HypergeometricFunctions/SphericalHarmonicYGeneral/

Abbott, P. "2. Schrodinger Equation." Lecture Notes for Computational Physics 2. http://physics.uwa.edu.au/pub/Computational/CP2/2.Schroedinger.nb.

Arfken, G. "Spherical Harmonics" and "Integrals of the Products of Three Spherical Harmonics." ¡×12.6 and 12.9 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 680-685 and 698-700, 1985.

Byerly, W. E. "Spherical Harmonics." Ch. 6 in An Elementary Treatise on Fourier's Series, and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics. New York: Dover, pp. 195-218, 1959.

Ferrers, N. M. An Elementary Treatise on Spherical Harmonics and Subjects Connected with Them. London: Macmillan, 1877.

Groemer, H. Geometric Applications of Fourier Series and Spherical Harmonics. New York: Cambridge University Press, 1996.

Hobson, E. W. The Theory of Spherical and Ellipsoidal Harmonics. New York: Chelsea, 1955.

Kalf, H. "On the Expansion of a Function in Terms of Spherical Harmonics in Arbitrary Dimensions." Bull. Belg. Math. Soc. Simon Stevin 2, 361-380, 1995.

MacRobert, T. M. and Sneddon, I. N. Spherical Harmonics: An Elementary Treatise on Harmonic Functions, with Applications, 3rd ed. rev. Oxford, England: Pergamon Press, 1967.

Normand, J. M. A Lie Group: Rotations in Quantum Mechanics. Amsterdam, Netherlands: North-Holland, 1980.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Spherical Harmonics." ¡×6.8 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 246-248, 1992.

Sansone, G. "Harmonic Polynomials and Spherical Harmonics," "Integral Properties of Spherical Harmonics and the Addition Theorem for Legendre Polynomials," and "Completeness of Spherical Harmonics with Respect to Square Integrable Functions." ¡×3.18-3.20 in Orthogonal Functions, rev. English ed. New York: Dover, pp. 253-272, 1991.

Sternberg, W. and Smith, T. L. The Theory of Potential and Spherical Harmonics, 2nd ed. Toronto: University of Toronto Press, 1946.

Wang, J.; Abbott, P.; and Williams, J. "Visualizing Atomic Orbitals." http://physics.uwa.edu.au/pub/Orbitals.

Weisstein, E. W. "Books about Spherical Harmonics." http://www.ericweisstein.com/encyclopedias/books/SphericalHarmonics.html.

Whittaker, E. T. and Watson, G. N. "Solution of Laplace's Equation Involving Legendre Functions" and "The Solution of Laplace's Equation which Satisfies Assigned Boundary Conditions at the Surface of a Sphere." ¡×18.31 and 18.4 in A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, pp. 391-395, 1990.

Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 129, 1997.

1. The electrons can only travel in certain orbits: at a certain discrete set of distances from the nucleus with specific energies.
2. The electrons of an atom revolve around the nucleus in orbits. These orbits are associated with definite energies and are also called energy shells or energy levels. Thus, the electrons do not continuously lose energy as they travel in a particular orbit. They can only gain and lose energy by jumping from one allowed orbit to another, absorbing or emitting electromagnetic radiation with a frequency ¥í determined by the energy difference of the levels according to the Planck relation:

   

   where h is Planck's constant.
3. The frequency of the radiation emitted at an orbit of period T is as it would be in classical mechanics; it is the reciprocal of the classical orbit period:

   

1. Like Einstein's theory of the Photoelectric effect, Bohr's formula assumes that during a quantum jump a discrete amount of energy is radiated. However, unlike Einstein, Bohr stuck to the classical Maxwell theory of the electromagnetic field. Quantization of the electromagnetic field was explained by the discreteness of the atomic energy levels; Bohr did not believe in the existence of photons.
2. According to the Maxwell theory the frequency ¥í of classical radiation is equal to the rotation frequency ¥í_rot of the electron in its orbit, with harmonics at integer multiples of this frequency. This result is obtained from the Bohr model for jumps between energy levels E_n and E_n−k when k is much smaller than n. These jumps reproduce the frequency of the k-th harmonic of orbit n. For sufficiently large values of n (so-called Rydberg states), the two orbits involved in the emission process have nearly the same rotation frequency, so that the classical orbital frequency is not ambiguous. But for small n (or large k), the radiation frequency has no unambiguous classical interpretation. This marks the birth of the correspondence principle, requiring quantum theory to agree with the classical theory only in the limit of large quantum numbers.
3. The Bohr-Kramers-Slater theory (BKS theory) is a failed attempt to extend the Bohr model which violates the conservation of energy and momentum in quantum jumps, with the conservation laws only holding on average.

Electron energy levels

• Classical mechanics

The electron is held in a circular orbit by electrostatic attraction. The centripetal force is equal to the Coulomb force.

where m_e is the mass, e is the charge of the electron and k_e is Coulomb's constant. This determines the speed at any radius:

It also determines the total energy at any radius:

The total energy is negative and inversely proportional to r. This means that it takes energy to pull the orbiting electron away from the proton. For infinite values of r, the energy is zero, corresponding to a motionless electron infinitely far from the proton. The total energy is half the potential energy, which is true for non circular orbits too by the virial theorem.

For positronium, m_e is replaced by its reduced mass (¥ì = m_e/2).

• Quantum rule

The angular momentum L = m_evr is an integer multiple of ©¤:

Substituting the expression for the velocity gives an equation for r in terms of n:

so that the allowed orbit radius at any n is:

The smallest possible value of r in the hydrogen atom is called the Bohr radius and is equal to:

The energy of the n-th level for any atom is determined by the radius and quantum number:

is the rest mass energy of the electron (511 keV)

is the fine structure constant

The Bohr formula properly uses the reduced mass of electron and proton in all situations, instead of the mass of the electron: . However, these numbers are very nearly the same, due to the much larger mass of the proton, about 1836.1 times the mass of the electron, so that the reduced mass in the system is the mass of the electron multiplied by the constant 1836.1/(1+1836.1) = 0.99946. This fact was historically important in convincing Rutherford of the importance of Bohr's model, for it explained the fact that the frequencies of lines in the spectra for singly ionized helium do not differ from those of hydrogen by a factor of exactly 4, but rather by 4 times the ratio of the reduced mass for the hydrogen vs. the helium systems, which was much closer to the experimental ratio than exactly 4.0.

(positronium)

Rydberg formula

Shell model of the atom

Moseley's law and calculation of K-alpha X-ray emission lines

Shortcomings

The Bohr model gives an incorrect value for the ground state orbital angular momentum. The angular momentum in the true ground state is known to be zero. Although mental pictures fail somewhat at these levels of scale, an electron in the lowest modern "orbital" with no orbital momentum, may be thought of as not to rotate "around" the nucleus at all, but merely to go tightly around it in an ellipse with zero area (this may be pictured as "back and forth", without striking or interacting with the nucleus). This is only reproduced in a more sophisticated semiclassical treatment like Sommerfeld's. Still, even the most sophisticated semiclassical model fails to explain the fact that the lowest energy state is spherically symmetric--- it doesn't point in any particular direction.

• Much of the spectra of larger atoms. At best, it can make predictions about the K-alpha and some L-alpha X-ray emission spectra for larger atoms, if two additional ad hoc assumptions are made (see Moseley's law above). Emission spectra for atoms with a single outer-shell electron (atoms in the lithium group) can also be approximately predicted. Also, if the empiric electron-nuclear screening factors for many atoms are known, many other spectral lines can be deduced from the information, in similar atoms of differing elements, via the Ritz-Rydberg combination principles (see Rydberg formula). All these techniques essentially make use of Bohr's Newtonian energy-potential picture of the atom.
• the relative intensities of spectral lines; although in some simple cases, Bohr's formula or modifications of it, was able to provide reasonable estimates (for example, calculations by Kramers for the Stark effect).
• The existence of fine structure and hyperfine structure in spectral lines, which are known to be due to a variety of relativistic and subtle effects, as well as complications from electron spin.
• The Zeeman effect - changes in spectral lines due to external magnetic fields; these are also due to more complicated quantum principles interacting with electron spin and orbital magnetic fields.
• The model also violates the uncertainty principle in that it considers electrons to have known orbits and definite radius, two things which can not be directly known at once.

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It is often used in titrations because of its clear colour change. Because it changes colour at the pH of a mid-strength acid, it is usually used in titrations for acids. Unlike a universal indicator, methyl orange does not have a full spectrum of colour change, but has a sharper end point.

[edit] Indicator colours

In a solution becoming less acidic, methyl orange moves from red to orange and finally to yellow with the reverse occurring for a solution increasing in acidity. The entire colour change occurs in acidic conditions.

In an acid it is reddish and in alkali it is yellow.

[edit] Other Indicators

• Modified (or screened) methyl orange, an indicator consisting of a solution of methyl orange and xylene cyanol, changes from grey to green as the solution becomes more basic.
• A number of other common indicators are tabulated in the article on pH indicators.

[edit] See also

• pH indicator
• Methyl red
• Litmus
• Phenolphthalein
• Bromothymol blue
• Universal indicator

[edit] External links

• Informative page on different titration indicators, including Methyl Orange

Bromothymol blue acts as a weak acid in solution. It can thus be in protonated or deprotonated form, appearing yellow and blue respectively. It is bluish green in neutral solution. It is typically sold in solid form as the sodium salt of the acid indicator. It also finds occasional use in the laboratory as a biological slide stain. At this point it is already blue, and a drop or two is used on a water slide. The cover slip is placed on top of the water droplet and the specimen in it, with the blue coloring mixed in. It is sometimes used to define cell walls or nuclei under the microscope.

Bromothymol blue is mostly used in measuring substances that would have relatively low acidic or basic levels (near a neutral pH). It is often used in managing the pH of pools and fish tanks, and for measuring the presence of carbonic acid in a liquid.

A common demonstration of BTB's pH indicator properties involves exhaling through a tube into a neutral solution of BTB. As carbon dioxide is absorbed from the breath into the solution, forming carbonic acid, the solution changes color from green to yellow. Thus, BTB is commonly used in middle school science classes to demonstrate that the more that muscles are used, the greater the CO₂ output.

Bromothymol is also used in obstetrics for detecting premature rupture of membranes. Amniotic fluid typically has a pH > 7.2, bromothymol will therefore turn blue when brought in contact with fluid leaking from the amnion. As vaginal pH normally is acidic, the blue color indicates the presence of amniotic fluid. The test may be false-positive in the presence of other alkaline substances such as blood, semen, or in the presence of bacterial vaginosis.

The pKa for bromothymol blue is 7.10.

Indicator colors

BTB indicator in acidic, neutral, and alkaline solutions (left to right).

See also

• pH indicator
• Litmus
• Phenolphthalein
• Methyl orange
• Universal indicator

External links

• J.T.Baker MSDS
• Bromothymol Blue (Sodium Salt) -Sigma Aldrich

Species	In+	H2In	In2−	In(OH)3−
Structure
Model
pH	<0	0−8.2	8.2−12.0	>12.0
Conditions	strongly acidic	acidic or near-neutral	basic	strongly basic
Color	orange	colorless	pink to fuchsia	colorless
Image

The rather slow fading reaction that produces the colorless InOH³⁻ ion is sometimes used in classes for the study of reaction kinetics.

Phenolphthalein is insoluble in water and is usually dissolved in alcohols for use in experiments. It is itself a weak acid, which can lose H⁺ ions in solution. The phenolphthalein molecule is colorless. However, the phenolphthalein ion is pink. When a base is added to the phenolphthalein, the molecule ⇌ ions equilibrium shifts to the right, leading to more ionization as H⁺ ions are removed. This is predicted by Le Chatelier's principle.

Synthesis

Phenolphthalein is synthesized by condensation of phthalic anhydride with two equivalents of phenol under acidic conditions (hence the name). It was discovered in 1871 by Adolf von Baeyer.

Uses

Phenolphthalein (pH indicator)
below pH 8.2		above pH 10.0
colorless	⇌	fuchsia

Phenolphthalein has been used for over a century as a laxative, but is now being removed from over-the-counter laxatives^[2] because of concerns over carcinogenicity.^[3][4]

Phenolphthalein is used to perform a presumptive blood test, and is commonly known as the Kastle-Meyer test. A dry sample is collected with a swab or filter paper. First a few drops of alcohol, then a few drops of phenolphthalein and finally a few drops of hydrogen peroxide are dripped onto the sample. If the sample turns pink then it is a positive test. This test is nondestructive to the sample; it can be kept and used in further tests at the lab. This test has the same reaction with blood from any animal, so further testing would be required to determine whether it originates from a human.

Phenolphthalein is used in toys, for example as a component of disappearing inks, or disappearing dye on the Hollywood Hair Barbie hair. In the ink it is mixed with sodium hydroxide, which reacts with carbon dioxide in the air. This reaction leads to the pH falling below the color change threshold as hydrogen ions are released via the reaction:

OH⁻ (aq) + CO₂ (g) ¡æ CO₃²⁻ (aq) + H⁺ (aq)

To develop the hair and "magic" graphical patterns, the ink is sprayed with a solution of hydroxide, which leads to the appearance of the hidden graphics by the same mechanism described above for color change in alkaline solution. The pattern will eventually disappear by the same reaction with carbon dioxide detailed above. Thymolphthalein is used for the same purpose and in the same way, when blue color is desired. ^[5]

Phenolphthalein in alkaline solution

Phenolphthalein is used as an acid or base indicator where in contact or presence of acid it will turn colorless and with base, it will turn into a pinkish violet color. It is also a component in universal indicator, a solution consisting of a mixture of pH indicators (usually phenolphthalein, methyl red, bromothymol blue, and thymol blue).^[6]

The acid-base indication abilities of phenolphthalein also make it useful for testing for signs of carbonation reactions in concrete. Concrete has naturally high pH due to the calcium hydroxide formed when Portland cement reacts with water. The pH of the ionic water solution present in the pores of fresh concrete may be over 14. Normal carbonation of concrete occurs as the cement hydration products in concrete react with carbon dioxide in the atmosphere, and can reduce the pH to 8¨ö to 9, although that reaction usually is restricted to a thin layer at the surface. When a 1% phenolphthalein solution is applied to normal concrete it will turn bright pink. If the concrete has undergone carbonation, no color change will be observed.

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