Here you can find formulas for a precalculus Course.
1.1 Trigonometric functions
For the trigonometric ratios for a point p on the unit circle holds:
cos(φ) = xp , sin(φ) = yp , tan(φ) = yp/xp
sin2 (x) + cos2 (x) = 1 and cos−2 (x) = 1 + tan2 (x).
cos(a ± b) = cos(a) cos(b) ∓ sin(a) sin(b) ,
sin(a ± b) = sin(a) cos(b) ± cos(a)sin(b)
tan(a ± b) =( tan(a) ± tan(b) )/( 1 ∓ tan(a) tan(b) )
The sum formulas are:
sin(p) + sin(q) = 2 sin( 1 / 2 (p + q)) cos( 1 / 2 (p − q))
sin(p) − sin(q) = 2 cos( 1 / 2 (p + q)) sin( 1 / 2 (p − q))
cos(p) + cos(q) = 2 cos( 1 / 2 (p + q)) cos( 1 / 2 (p − q))
cos(p) − cos(q) = −2 sin( 1 / 2 (p + q)) sin( 1 / 2 (p − q))
From these equations can be derived! that
2 cos2 (x) = 1 + cos(2x) , 2 sin2 (x) = 1 − cos(2x)
sin(Ï€ − x) = sin(x) , cos(Ï€ − x) = − cos(x)
sin( 1 / 2 Ï€ − x) = cos(x) , cos( 1 / 2 Ï€ − x) = sin(x)
Conclusions from equalities:
sin(x) = sin(a)
⇒ x = a ± 2kÏ€ or x = (Ï€ − a) ± 2kÏ€, k ∈ N
cos(x) = cos(a)
⇒ x = a ± 2kÏ€ or x = −a ± 2kÏ€
tan(x) = tan(a)
⇒ x = a ± kÏ€ and x =/= Ï€ / 2 ± kÏ€
The following relations exist between the inverse trigonometric functions:
arctan(x) = arcsin(x/ √x2 + 1)= arccos(1/√ x2 + 1)
sin(arccos(x)) = √ 1 − x2
precalculus
Tuesday, August 31, 2010
Thin layer chromatography - Wikipedia, the free encyclopedia
Thin layer chromatography - Wikipedia, the free encyclopedia
Tlc plate development sequence: a mixture of a red and blue compound is separated as the plate develops, i.e. as the light blue solvent moves up the plate.
polar solvents list
Thin layer chromatography
From Wikipedia, the free encyclopedia
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| Thin layer chromatography | |
 Separation of black ink on a TLC plate | |
| Acronym | TLC |
|---|---|
| Classification | Chromatography |
| Other Techniques | |
| Related | Agarose gel electrophoresis SDS-PAGE |
Thin layer chromatography (TLC) is a chromatography technique used to separate mixtures.[1] It involves a stationary phase consisting of a thin layer of adsorbent material, usually si! lica gel , aluminium oxide, or cellulose immobilized onto a flat, inert carrier sheet. A liquid phase consisting of the solution to be separated is then dissolved in an appropriate solvent and is drawn up the plate via capillary action, separating the experimental solution based on the polarity of the components of the compound in question.
Its wide range of uses include
- assaying radiochemical purity of radiopharmaceuticals
- determination of the pigments a plant contains
- detection of pesticides or insecticides in food
- analysing the dye composition of fibers in fore! nsics, or
- identifying compounds present in a given substance
It is a quick, generic method for organic reaction monitoring.
Contents[hide] |
[edit] Plate preparation
TLC plates are made by mixing the adsorbent, such as silica gel, with a small amount of inert binder like calcium sulfate (gypsum) and water. This mixture is spread as a thick slurry on an unreactive carrier sheet, usually glass, thic! k aluminum foil, or plastic, and the resultant plate is dried and activated by heating in an oven for thirty minutes at 110 °C. The thickness of the adsorbent layer is typically around 0.1–0.25 mm for analytical purposes and around 1–2 mm for preparative TLC. Every type of chromatography contains a mobile phase and a stationary phase.
[edit] Technique
The process is similar to paper chromatography with the advantage of faster runs, better separations, and the choice between different stationary phases. Because of its simplicity and speed TLC is often used for monitoring chemical reactions and for the qualitative a! nalysis of reaction products.
A small spot of solution containing the sample is applied to a plate, about one centimeter from the base. The plate is then dipped in to a suitable solvent, such as ethanol or water, and placed in a sealed container. The solvent moves up the plate by capillary action and meets the sample mixture, which is dissolved and is carried up the plate by the solvent. Different compounds in the sample mixture travel at diff! erent rates owing to differences in solubility in the solvent,! and owi ng to differences in their attraction to the stationary phase
Separation of compounds is based on the competition of the solute and the mobile phase for binding places on the stationary phase. For instance, if normal phase silica gel is used as the stationary phase it can be considered polar. Given two compounds which differ in polarity, the most polar compound has a stronger interaction with the silica and is therefore more capable to dispel the mobile phase from the binding places. Consequently, the less polar compound moves higher up the plate (resulting in a higher Rf value). If the mobile phase is changed to a more polar solvent or mixture of solvents, it is more capable of dispelling solutes from the silica binding places and all compounds on the TLC plate will move higher up the plate. Practically this means that if you use a mixture of ethyl acetate and heptane as the mobile phase, adding more ethyl acetate results in higher Rf values for all compounds on th! e TLC plate. Changing the polarity of the mobile phase will not result in reversed order of running of the compounds on the TLC plate. If a reversed order of running of the compounds is desired, an apolar stationary phase should be used, such as C18-functionalized silica.
The appropriate solvent in context of Thin layer chromatography will be one which differs from the stationary phase material in polarity. If polar solvent is used to dissolve the sample and spot is applied over polar stationary phase TLC, the sample spot will grow radially due to capillary action, which is not advisable as one spot may mix with the other. Hence, to restrict the radial growth of sample-spot, the solvent used for dissolving samples in order to apply them on plates should be as non-polar or semi-polar as possible when the stationary phase is polar, and vice-versa.
[edit] Analysis
As the chemicals being separated may be colorless, several methods exist to visualize the spots:
- Often a small amount of a fluorescent compound, usually manganese-activated zinc silicate, is added to the adsorbent that allows the visualization of spots under a blacklight (UV254). The adsorbent layer will thus fluoresce light green by itself, but spots of analyte quench this fluorescence.
- Iodine vapors are a general unspecific color
- Specific color reagents exist into which the TLC plate is dipped or which are sprayed onto the plate
- In the case of lipids, the chromatogram may be transferred to a PVDF membrane and then subjected to further analysis, for example mass spectrometry, a technique known as Far-Eastern blotting.
Once visible, the Rf value , or Retention factor, of each spot can be determined by dividing the distance traveled by the product by the total distance traveled by the solvent (the solvent front). These values depend on the solvent use! d, and the type of TLC plate, and are not physical constants.
[edit] Applications
In organic chemistry, reactions are qualitatively monitored with TLC. Spots sampled with a capillary tube are placed on the plate: a spot of starting material, a spot from the reaction mixture, and a "co-spot" with both. A small (3 by 7 cm) TLC plate takes a couple of minutes to run. The analysis is qualitative, and it will show if starting material has disappeared, product has appeared, and how many products are generated. Unfortunately, TLC! 's from low-temperature reactions may give misleading resu! lts, bec ause the sample is warmed to room temperature in the capillary. One such reaction is DIBALH reduction of ester to aldehyde.
As an example the chromatography of an extract of green leaves (for example spinach) in 7 stages of development. Carotene elutes quickly and is only visible until step 2. Chlorophyll A and B are halfway in the final step and lutein the first compound staining yellow.
 Step 1 |  Step 2 |  Step 3 |  Step 4 |
 Step 5 |  Step 6 |  Step 7 |
In one study TLC has been applied in the screening of organic reactions[2] for example in the fine-tuning of BINAP synthesis from 2-naphtol. In this method the alcohol and catalyst solution (for instance iron(III) chloride) are place separately on the base line, then reacted and then instantly analyzed.
[edit] References
- ^ Vogel's Textbook of Practical Organic Chemistry (5th Edition) (Hardcover) by A.I. Vogel (Author), A.R. Tatchell (Author), B.S. Furnis (Author), A.J. Hannaford (Author), P.W.G. Smith ISBN 0582462363
- ^ TLC plates as a convenient platform for solvent-free reactions Jonathan M. Stoddard, Lien Nguyen, Hector Mata-Chavez and Kelly Nguyen Chem. Commun., 2007, 1240 - 1241, doi:10.1039/b616311d
- Hand book of Thin Layer Chromatography,Sherma, J.and Fried, B. (authors) 3rd ed. Marcel Dekker, New York.
[edit] See also
polar solvents list
Online Math and Science Tutors from India
Using computer whiteboards and talking through the computer, many U.S. students are now finding that they can get knowledgeable math and science tutors from India.
Excerpt:"Both Growing Stars and Studyloft say that the majority of their tutors hold master's degrees or doctorates in their subjects; many also have degrees in education."
The idea is that tutors through the Internet relieve parents of commuting and scheduling, and cut the prices of established tutoring businesses like Sylvan Learning. If any of you have experiences with an online tutoring service(good or bad), please leave us a comment. Thanks in advance!
Online tutoring - The Boston Globe
Studyloft.com
Growingstars.com
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Factoring the time
I stumbled upon this comic that you might enjoy... factoring the time (from xkcd.com).
math help for free
Video Solution and Discussion of Twitter SAT Probability Question from 8-25-10
I decided to post a video solution of the Twitter problem I posted on 8-25-10:
4 red, 2 blue cards; 4 are chosen at random. What is the probability that 2 of the cards will be red?
Because of the 140 character restriction on Twitter, the questions are often highly abbreviated and I actually consider it a "fun" challenge to write the question both concisely and clearly. Of course, as we all know about human interpretation of word problems, "clear" is in the eye of the beholder!
There's no doubt that the question above needs some fleshing out and might appear on the SAT and other standardized tests something like this:
A set of six cards contains four red and two blue cards. If four cards are chosen at random, what is the probability that exactly two of these cards will be red?
I'm sure my astute readers can improve on this wording but we'll leave it at this.
A few questions naturally pop up:
(1) Could this really be an SAT/Standardized Test question? Well, as I state in the video below, a question quite similar to this appeared on the College Board website the other day as the Question of the Day.
(2) For whom is the video intended? Everyone who happens upon it! I certainly wrote it to be helpful to students who will be taking the PSAT/SAT in the near future. Rather than simply presenting a single quick efficient solution, I demo'd 2-3 methods and indicated some important strategies and reviewed key pieces of knowledge to be successful on these harder probability questions. By the way, someone who is comfortable with probability will surely not find this question so formidable, but we're talking here about high school students or even undergraduates who struggle mightily with these.
(3) I'm hoping that the video will also serve as a catalyst for dialog in your math department. From the inception of this blog, I've never even intimated that a suggested way of explaining a concept, skill or a problem solution is in any way prescriptive. I encourage you to continue using whatever instructional methods have worked for you and to share these with our readers! However, for novice teachers or those who wish to see other approaches, I hope it will have some benefit. Of course, the video is not in a classroom. There are no students asking or being asked questions. There are no interruptions and I have a captive audience (except for my dogs who bark incessantly!).
SOME KEY STRATEGIES/TIPS/FACTS FOR PROBABILITY QUESTIONS
(1) It is highly recommended that students begin by listing 2-3 possible outcomes and to include at least one that is NOT one of the desired outcomes! This will help you to decide on a plan: organized list vs more advanced counting/probability methods. Further, you can ask yourself the key question in all counting/probability problems: DOES ORDER COUNT!
(2) Although it appears difficult for most test-takers to be systematic when making a list under test-taking conditions, preparation is critical here. If one practices several of these in the weeks leading up to the test, the chances of success improve dramatically. Did I just suggest preparation and practice could make a difference!
Where do you find these problems? Any SAT/ACT review book or my Twitter Problems of the Day or my upcoming SAT Challenge Quiz book to name a few sources...
(3) The basic definition of probability should always be in the forefront of your mind:
P(an event) = TOTAL NUMBER OF WAYS FOR THAT EVENT TO OCCUR DIVIDED BY TOTAL NUMBER OF OUTCOMES.
As indicated in the video, one can and should think of this ratio as TWO SEPARATE COUNTING PROBLEMS! Do the denominator first, i.e., the TOTAL number of possible outcomes. In the Twitter problem it is 15 if order is disregarded. Whether you arrive at 15 by listing/counting or by combinations methods, the denominator is 15 and is a completely separate question from "How many ways are there to get 2 red and 2 blue cards?"
(4) Finally, there are other methods for solving this probability question using Laws of Probabilities and/or permutation methods. I was going to make a 2nd video but I'm not so sure about that now.
An important point about the video below: I used 4 Blue and 2 Red cards, the opposite of the original Twitter problem but that won't change the final result!
Look for my other videos on my YouTube channel MathNotationsVids. Look for all of my Twitter SAT Problems on twitter.com/dmarain.
As I develop my Facebook page further, I may start posting these questions there as well as my videos. Facebook allows up to 20 minutes videos, much less restrictive than YouTube's 10 minute limit.
"All Truth passes through Three Stages:
First, it is Ridiculed... Second, it is Violently Opposed... Third, it is Accepted as being Self-Evident." - Arthur Schopenhauer (1778-1860)
"You've got to be taught To hate and fear, You've got to be taught From year to year, It's got to be drummed In your dear little ear You've got to be carefully taught" --from South Pacific
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free practice worksheets on www.math-drills.com
Much of the material we will be covering this session has worksheets that are free to download on www.math-drills.com This is an illustration of what that website looks like and how to manuver your way through it while avoiding all the ads which wish to sell you their services. Actually, these ads probably pay for this site to remain free, so you might be nice to them and read one or two even if you do not purchase their product.
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