Rodrigues' rotation formula: Difference between revisions

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Derivation: Simplifying the derivation by removing the axes and similar cruft – see talk
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[[File:Interconversion between eclipsed and gauche conformations..png|thumb|300px|Rotation about single bond of butane to interconvert one conformer to another. Above: Newman projection; below: depiction of spatial orientation.]]
It is very common to have a dental emergency -- a fractured tooth, an abscess, or severe pain when chewing. Over-the-counter pain medication is just masking the problem. Seeing an emergency dentist is critical to getting the source of the problem diagnosed and corrected as soon as possible.<br><br><br><br>Here are some common dental emergencies:<br>Toothache: The most common dental emergency. This generally means a badly decayed tooth. As the pain affects the tooth's nerve, treatment involves gently removing any debris lodged in the cavity being careful not to poke deep as this will cause severe pain if the nerve is touched. Next rinse vigorously with warm water. Then soak a small piece of cotton in oil of cloves and insert it in the cavity. This will give temporary relief until a dentist can be reached.<br><br>At times the pain may have a more obscure location such as decay under an old filling. As this can be only corrected by a dentist there are two things you can do to help the pain. Administer a pain pill (aspirin or some other analgesic) internally or dissolve a tablet in a half glass (4 oz) of warm water holding it in the mouth for several minutes before spitting it out. DO NOT PLACE A WHOLE TABLET OR ANY PART OF IT IN THE TOOTH OR AGAINST THE SOFT GUM TISSUE AS IT WILL RESULT IN A NASTY BURN.<br><br>Swollen Jaw: This may be caused by several conditions the most probable being an abscessed tooth. In any case the treatment should be to reduce pain and swelling. An ice pack held on the outside of the jaw, (ten minutes on and ten minutes off) will take care of both. If this does not control the pain, an analgesic tablet can be given every four hours.<br><br>Other Oral Injuries: Broken teeth, cut lips, bitten tongue or lips if severe means a trip to a dentist as soon as possible. In the mean time rinse the mouth with warm water and place cold compression the face opposite the injury. If there is a lot of bleeding, apply direct pressure to the bleeding area. If bleeding does not stop get patient to the emergency room of a hospital as stitches may be necessary.<br><br>Prolonged Bleeding Following Extraction: Place a gauze pad or better still a moistened tea bag over the socket and have the patient bite down gently on it for 30 to 45 minutes. The tannic acid in the tea seeps into the tissues and often helps stop the bleeding. If bleeding continues after two hours, call the dentist or take patient to the emergency room of the nearest hospital.<br><br>Broken Jaw: If you suspect the patient's jaw is broken, bring the upper and lower teeth together. Put a necktie, handkerchief or towel under the chin, tying it over the head to immobilize the jaw until you can get the patient to a dentist or the emergency room of a hospital.<br><br>Painful Erupting Tooth: In young children teething pain can come from a loose baby tooth or from an erupting permanent tooth. Some relief can be given by crushing a little ice and wrapping it in gauze or a clean piece of cloth and putting it directly on the tooth or gum tissue where it hurts. The numbing effect of the cold, along with an appropriate dose of aspirin, usually provides temporary relief.<br><br>In young adults, an erupting 3rd molar (Wisdom tooth), especially if it is impacted, can cause the jaw to swell and be quite painful. Often the gum around the tooth will show signs of infection. Temporary relief can be had by giving aspirin or some other painkiller and by dissolving an aspirin in half a glass of warm water and holding this solution in the mouth over the sore gum. AGAIN DO NOT PLACE A TABLET DIRECTLY OVER THE GUM OR CHEEK OR USE THE ASPIRIN SOLUTION ANY STRONGER THAN RECOMMENDED TO PREVENT BURNING THE TISSUE. The swelling of the jaw can be reduced by using an ice pack on the outside of the face at intervals of ten minutes on and ten minutes off.<br><br>In case you have virtually any issues regarding where by and the best way to use [http://www.youtube.com/watch?v=90z1mmiwNS8 Washington DC Dentist], you are able to call us on our internet site.
In [[chemistry]], '''conformational isomerism''' is a form of [[stereoisomerism]] in which the [[isomer]]s can be interconverted exclusively by rotations about formally single bonds (refer to figure on single bond rotation).<ref>[http://goldbook.iupac.org/C01258.html IUPAC definition of a conformer].</ref> Such isomers are generally referred to as '''conformational isomers''' or '''conformers''' and, specifically, as '''rotamers'''.<ref>{{GoldBookRef |file=F02520 |title=Rotamer |year=1996}}</ref> Rotations about single bonds are restricted by a rotational energy barrier which must be overcome to interconvert one conformer to another. Conformational isomerism arises when the rotation about a single bond is relatively unhindered.  That is, the [[activation energy|energy barrier]] must be small enough for the interconversion to occur.
 
Conformational isomers are thus distinct from the other classes of [[Stereoisomerism|stereoisomers]] (i. e. [[Molecular configuration|configurational]] isomers) where interconversion necessarily involves breaking and reforming of chemical bonds.<ref>{{cite web|last=Hunt|first=Ian|title=Stereochemistry|url=http://www.chem.ucalgary.ca/courses/350/Carey5th/Ch07/ch7-1.html|work=University of Calgary|accessdate=28 October 2013}}</ref> For example, L- & D and R- & S- configurations of organic molecules have different handedness and optical activities, and can only be interconverted by breaking one or more bonds connected to the [[Chirality (chemistry)|chiral]] atom and reforming a similar bond in a different direction or spatial orientation.
 
The study of the energetics between different rotamers is referred to as '''conformational analysis'''.<ref name=dougherty>{{cite book|last=Anslyn|first=Eric|title=Modern Physical Organic Chemistry|year=2006|publisher=University Science|isbn=978-1891389313|page=95|coauthors=Dennis Dougherty}}</ref> It is useful for understanding the stability of different isomers, for example, by taking into account the spatial orientation and through-space interactions of substituents. In addition, conformational analysis can be used to predict and explain product(s) selectivity, mechanisms, and rates of reactions.<ref name="nobel lect">{{cite web|last=Barton|first=Derek|title=The Principles of Conformational Analysis.|url=http://www.nobelprize.org/nobel_prizes/chemistry/laureates/1969/barton-lecture.html|work=Nobel Media AB 2013|publisher=Elsevier Publishing Co.|accessdate=10 November 2013}}</ref>
 
== Types of conformational isomerism ==
[[File:Butane conformations.jpg|thumb|400px|Free energy diagram of butane as a function of dihedral angle.]]
The types of conformational isomers are related to the spatial orientations of the substituents between two vicinal atoms. These are eclipsed and staggered.  The staggered conformation includes the gauche (±60°) and anti (180°) conformations, depending on the spatial orientations of the two substituents.
 
For example, butane has three rotamers relating to its two methyl (CH<sub>3</sub>) groups: two gauche conformers, which have the methyls ±60° apart and are [[enantiomer]]ic, and an anti conformer, where the four carbon centres are coplanar and the substituents are 180° apart (refer to free energy diagram of butane).  The energy difference between gauche and anti is 0.9 kcal/mol associated with the [[Strain (chemistry)|strain]] energy of the gauche conformer.<ref name=dougherty /> The anti conformer is, therefore, the most stable (~0 kcal/mol). The three eclipsed conformations with dihedral angles of 0°,120° and 240° are not considered to be rotamers, but are instead transition states of higher energy.<ref name=dougherty /> Note that the two eclipsed conformations have different energies: at 0° the two methyl groups are eclipsed, resulting in higher energy (~5 kcal/mol) than at 120°, where the methyl groups are eclipsed with hydrogens (~3.5 kcal/mol).<ref>{{cite web|last=Bauld|first=Nathan|title=Butane Conformational Analysis|url=http://research.cm.utexas.edu/nbauld/teach/butane.html|work=University of Texas|accessdate=28 October 2013}}</ref>
 
While simple molecules can be described by these types of conformations, more complex molecules require the use of the [[Klyne-Prelog system]] to describe the different conformers.<ref name=dougherty />
 
More specific examples of conformational isomerism are detailed elsewhere:
# Ring conformation
#*[[Cyclohexane conformation]]s with chair and boat conformers.
#*[[Carbohydrate conformation]].
#[[Allylic strain]] - energetics related to rotation about the single bond between sp<sup>2</sup> and sp<sup>3</sup> carbons.
#[[Atropisomerism]]- due to restricted rotation about a bond, a molecule can become chiral.
#[[Folding (chemistry)|Folding]] of molecules, where some shapes are stable and functional, but others are not.
 
==Free energy and equilibria of conformational isomers==
 
===Equilibrium of conformers===
[[File:Equillibrium conformers.jpg|thumb|250px|Equilibrium distribution of two conformers at different temperatures given the free energy of their interconversion.]]
Conformational isomers exist in a [[dynamic equilibrium]], where the relative free energies of isomers determines the population of each isomer and the energy barrier of rotation determines the rate of interconversion between isomers:<ref name="eq conformer">{{cite web|last=Bruzik|first=Karol|title=Chapter 6: Conformation|url=http://tigger.uic.edu/~kbruzik/text/chapter6.htm|work=University of Illinois at Chicago|accessdate=10 November 2013}}</ref>
:<math> K = {e^{-\Delta G/RT}}</math>
where <math>K</math> is the equilibrium constant, <math>\Delta G</math> is the change in free energy for the interconversion of two conformers in kcal/mol, <math>R</math> is the universal [[gas constant]] (0.002 kcal/mol K), and <math>T</math> is the system's temperature in Kelvin (K).
 
Three isotherms are given in the diagram depicting the equilibrium distribution of two conformers at different temperatures.  Note that a 0 kcal/mol free energy change gives an equilibrium constant of 1, meaning that two conformers have equal stability and exist in a 1:1 ratio.  A negative change in free energy means that a conformer interconverts to a thermodynamically favored conformation, thus the equilibrium constant will always be greater than 1.  For example, the ΔG of butane from gauche to anti is -0.9 kcal/mol, therefore the equilibrium constant is 4.5, favoring the anti conformation. Also notice that at large positive ΔG (i.e. unlikely for interconversion to occur), the equilibrium constant between two conformers can be increased by increasing temperature.
 
===Population distribution of conformers===
[[Image:2ConfBoltzmannDist.png|right|thumb|350px|Boltzmann distribution % of lowest energy conformation in a two component equilibrating system at various temperatures (degrees Celsius, color) and energy difference in kcal/mol (x-axis)]]
The fractional population distribution of different conformers follows a [[Boltzmann distribution]]:<ref name="boltz dist">{{cite web|last=Rzepa|first=Henry|title=Conformational Analysis|url=http://www.ch.ic.ac.uk/local/organic/conf/c1_definitions.html|work=Imperial College London|accessdate=11 November 2013}}</ref>
:<math> \frac{N_i}{N_{total}}
=
\frac
  {e^{-E_{rel}/RT}}
  {\sum_{k=1}^{N_{total}} e^{-E_k/RT}  }
 
</math>
 
The left hand side is the equilibrium ratio of conformer ''i'' to the total.  <math>E_{rel}</math> is the relative energy of the ''i''-th conformer from the minimum energy conformer. <math>E_k</math> is the relative energy of the ''k''-th conformer from the minimum energy conformer. ''R'' is the molar ideal gas constant equal to 8.31&nbsp;J/(mol·K) and ''T'' is the temperature in [[kelvin]]s (K).  The denominator of the right side is the partition function.
 
===Factors contributing to the free energy of conformers===
The effects of [[electrostatics|electrostatic]] and [[steric effects|steric]] interactions of the substituents as well as orbital interactions such as [[hyperconjugation]] are responsible for the relative stability of conformers and their transition states. The contributions of these factors vary depending on the nature of the substituents and may either contribute positively or negatively to the energy barrier.  Computational studies of small molecules such as ethane suggest that electrostatic effects make the greatest contribution to the energy barrier; however, the barrier is traditionally attributed primarily to steric interactions.<ref>{{cite journal|last=Liu|first=Shubin|title=Origin and Nature of Bond Rotation Barriers: A Unified View|journal=The Journal of Physical Chemistry A|date=7 February 2013|volume=117|issue=5|pages=962–965|doi=10.1021/jp312521z}}</ref><ref>{{cite book|last=Carey|first=Francis A.|title=Organic chemistry|year=2011|publisher=McGraw-Hill|location=New York|isbn=0-07-340261-3|page=105|edition=8th ed.}}</ref>
[[File:Contributions to Rotational Energy Barrier.png|center|thumb|450px|Contributions to Rotational Energy Barrier]]
In the case of cyclic systems, the steric effect and contribution to the free energy can be approximated by [[A value]]s, which measure the energy difference when a substituent is in the axial or equatorial position.
 
==Isolation or observation of the conformational isomers==
 
The short timescale of interconversion precludes the separation of conformational isomers in many cases. [[Atropisomer]]s are conformational isomers which can be separated due to restricted rotation.<ref>{{cite book|last=McNaught|title=IUPAC Compendium of Chemical Terminology|year=1997|publisher=Blackwell Scientific Publications|location=Oxford|isbn=0967855098|url=http://goldbook.iupac.org/A00511.html}}</ref>
 
[[Protein folding]] also generates stable conformational isomers which can be observed. The [[Karplus equation]] relates the dihedral angle of [[Vicinal (chemistry)|vicinal]] protons to their [[J-coupling]] constants as measured by NMR.  The equation aids in the elucidation of protein folding as well as the conformations of other rigid [[Aliphatic compound|aliphatic]] molecules.<ref>{{cite web|last=Dalton|first=Louisa|title=Karplus Equation|url=http://pubs.acs.org/cen/science/8151/8151karplus.html|work=Chemical and Engineering News|publisher=American Chemical Society|accessdate=2013-10-27}}</ref>
 
The equilibrium between conformational isomers may also be observed using [[spectroscopy|spectroscopic techniques]]:
 
In [[Cyclohexane|cyclohexane derivatives]], the two chair conformers interconvert with rates on the order of 10<sup>5</sup>ring flips/sec, which precludes their separation.<ref name="eliel"/>  The conformer in which the substituent is equitorial crystallizes selectively, and when these crystals are dissolved at very low temperatures, one can directly monitor the approach to equilibrium by NMR spectroscopy.<ref name="Rotamers21stCentury">{{cite doi|10.1016/S0959-440X(02)00344-5}}</ref>
 
The ''dynamics'' of conformational (and other kinds of) isomerism can be monitored by [[NMR]] spectroscopy at varying temperatures.  The technique applies to barriers of 8–14 kcal/mol, and species exhibiting such dynamics are often called "[[Fluxional molecule|fluxional]]".
 
[[IR spectroscopy]] is ordinarily used to measure conformer ratios. For the axial and equatorial conformer of bromocyclohexane, ν<sub>CBr</sub> differs by almost 50&nbsp;cm<sup>−1</sup>.<ref name="eliel">Eliel, E. L.; Wilen, S. H.; Mander, L. N. "Stereochemistry Of Organic Compounds", J. Wiley and Sons, 1994. ISBN 0-471-01670-5.</ref>
 
==Conformation-dependent reactions==
Reaction rates are highly dependent on the conformation of the reactants. This theme is especially well elucidated in organic chemistry. One example is provided by the [[elimination reaction]]s, which involve the simultaneous removal of a proton and a [[leaving group]] from vicinal positions under the influence of a base.
[[Image:E2 Elimination Reaction.png|center|thumb|350px|Base-induced bimolecular dehydrohalogenation (an E2 type reaction mechanism). The optimum geometry for the transition state requires the breaking bonds to be antiperiplanar, as they are in the appropriate staggered conformation]]
 
The mechanism requires that the departing atoms or groups follow antiparallel trajectories. For open chain substrates this geometric prerequisite is met by at least one of the three staggered conformers.  For some cyclic substrates such as cyclohexane, however, an antiparallel arrangement may not be attainable depending on the substituents which might set a conformational lock.<ref>{{cite web|title=Cycloalkanes|url=http://www.ch.ic.ac.uk/local/organic/conf/c1_rings.html|publisher=Imperial College London|accessdate=28 October 2013}}</ref> Adjacent [[substituent]]s on a cyclohexane ring can achieve antiperiplanarity only when they occupy trans diaxial positions.
 
One consequence of this analysis is that ''trans''-4-tert-butylcyclohexyl chloride cannot easily eliminate but instead undergoes substitution (see diagram below) because the most stable conformation has the bulky ''t''Bu group in the equatorial position, therefore the chloride group is not antiperiplanar with any vicinal hydrogen.  The thermodynamically unfavored conformation has the ''t''Bu group in the axial position, which exhibits the high energetic 7-atoms interactions (see [[A value]]) of 4.7 - 4.9 kcal/mol.<ref name="dougherty a value">{{cite book|last=Dougherty|first=Eric V. Anslyn ; Dennis A.|title=Modern physical organic chemistry|year=2006|publisher=University Science Books|location=Sausalito, Calif.|isbn=978-1-891389-31-3|page=104|edition=Dodr.}}</ref> As a result, the ''t''Bu group "locks" the ring in the conformation where it is in the equatorial position and substitution reaction is observed. On the other hand, ''cis''-4-tert-butylcyclohexyl chloride undergoes elimination because antiperiplanarity of Cl and H can be achieved when the ''t''Bu group is in the favorable equatorial position.
 
{{double image|center|7atoms interaction trans-tBu-cyclohexyl Cl.png|350|E2 14tBuchexCl.png|350|Thermodynamically unfavored conformation of trans-4-tert-butylcyclohexyl chloride where the tBu group is in the axial position exerting 7-atoms interactions.|The trans isomer can attain antiperiplanarity only via the unfavored axial conformer; therefore, it does not eliminate.The cis isomer is already in the correct geometry in its most stable conformation; therefore,  it eliminates easily.}}
 
<!--the following is probably untrue:==Conditions==
Conformational isomerism only occurs around single bonds. Double  bonds have a [[pi bond]] that prevents rotation about the longitudinal axis. Forcing such a rotation to take place would entail breaking of the π bond, whereas conformational isomerism does not involve breaking/reforming bonds.<ref group="Note">C–C pi-bond energies typically are of the order of 200 kJ/mol. Activation energies of this magnitude are considered prohibitive within the usual temperature range of organic chemical reactions, so ''E''/''Z'' isomerization cannot be considered simply a conformational change.</ref>--> <!--not sure what this statement contributes, well intentioned the area is so great that there are many special cases:
Conformers sufficiently constrained to exhibit measurable isomerism are unique among the other kinds of stereoisomers in that inter-converting them does not involve either bond breaking or any significant change in the hybridization state of the atoms forming the rotating bond. Instead, the forces that constrain each particular conformer structure come mainly from through-space interactions of atoms or groups of atoms not directly bonded. Such interactions can include any of the usual steric and electronic interactions, such as Van der Waals forces, Coulombic forces, and dipolar interactions. For instance,the unfavourable conformer has the ''t''-butyl group axial on the ring, which is an unstable [[cyclohexane conformation]]-->
 
==See also==
* [[Isomer]]
* [[Steric effects]]
* [[Molecular configuration]]
* [[Macrocyclic stereocontrol]]
* [[Klyne-Prelog System]]
* [[Anomeric effect]]
 
== References ==
{{reflist}}
 
[[Category:Stereochemistry]]
[[Category:Physical organic chemistry]]
[[Category:Isomerism]]

Latest revision as of 23:08, 31 December 2014

It is very common to have a dental emergency -- a fractured tooth, an abscess, or severe pain when chewing. Over-the-counter pain medication is just masking the problem. Seeing an emergency dentist is critical to getting the source of the problem diagnosed and corrected as soon as possible.



Here are some common dental emergencies:
Toothache: The most common dental emergency. This generally means a badly decayed tooth. As the pain affects the tooth's nerve, treatment involves gently removing any debris lodged in the cavity being careful not to poke deep as this will cause severe pain if the nerve is touched. Next rinse vigorously with warm water. Then soak a small piece of cotton in oil of cloves and insert it in the cavity. This will give temporary relief until a dentist can be reached.

At times the pain may have a more obscure location such as decay under an old filling. As this can be only corrected by a dentist there are two things you can do to help the pain. Administer a pain pill (aspirin or some other analgesic) internally or dissolve a tablet in a half glass (4 oz) of warm water holding it in the mouth for several minutes before spitting it out. DO NOT PLACE A WHOLE TABLET OR ANY PART OF IT IN THE TOOTH OR AGAINST THE SOFT GUM TISSUE AS IT WILL RESULT IN A NASTY BURN.

Swollen Jaw: This may be caused by several conditions the most probable being an abscessed tooth. In any case the treatment should be to reduce pain and swelling. An ice pack held on the outside of the jaw, (ten minutes on and ten minutes off) will take care of both. If this does not control the pain, an analgesic tablet can be given every four hours.

Other Oral Injuries: Broken teeth, cut lips, bitten tongue or lips if severe means a trip to a dentist as soon as possible. In the mean time rinse the mouth with warm water and place cold compression the face opposite the injury. If there is a lot of bleeding, apply direct pressure to the bleeding area. If bleeding does not stop get patient to the emergency room of a hospital as stitches may be necessary.

Prolonged Bleeding Following Extraction: Place a gauze pad or better still a moistened tea bag over the socket and have the patient bite down gently on it for 30 to 45 minutes. The tannic acid in the tea seeps into the tissues and often helps stop the bleeding. If bleeding continues after two hours, call the dentist or take patient to the emergency room of the nearest hospital.

Broken Jaw: If you suspect the patient's jaw is broken, bring the upper and lower teeth together. Put a necktie, handkerchief or towel under the chin, tying it over the head to immobilize the jaw until you can get the patient to a dentist or the emergency room of a hospital.

Painful Erupting Tooth: In young children teething pain can come from a loose baby tooth or from an erupting permanent tooth. Some relief can be given by crushing a little ice and wrapping it in gauze or a clean piece of cloth and putting it directly on the tooth or gum tissue where it hurts. The numbing effect of the cold, along with an appropriate dose of aspirin, usually provides temporary relief.

In young adults, an erupting 3rd molar (Wisdom tooth), especially if it is impacted, can cause the jaw to swell and be quite painful. Often the gum around the tooth will show signs of infection. Temporary relief can be had by giving aspirin or some other painkiller and by dissolving an aspirin in half a glass of warm water and holding this solution in the mouth over the sore gum. AGAIN DO NOT PLACE A TABLET DIRECTLY OVER THE GUM OR CHEEK OR USE THE ASPIRIN SOLUTION ANY STRONGER THAN RECOMMENDED TO PREVENT BURNING THE TISSUE. The swelling of the jaw can be reduced by using an ice pack on the outside of the face at intervals of ten minutes on and ten minutes off.

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