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Lecture 02. Symmetry and Spectroscopy Pt 2.
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Title  Lecture 02. Symmetry and Spectroscopy Pt 2. 
Title of Series  Chem 131B: Molecular Structure & Statistical Mechanics 
Part Number  2 
Number of Parts  26 
Author 
Martin, Rachel

License 
CC Attribution  ShareAlike 3.0 Unported: You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal and noncommercial purpose as long as the work is attributed to the author in the manner specified by the author or licensor and the work or content is shared also in adapted form only under the conditions of this license. 
DOI  10.5446/18910 
Publisher  University of California Irvine (UCI) 
Release Date  2013 
Language  English 
Content Metadata
Subject Area  Chemistry 
Abstract  UCI Chem 131B Molecular Structure & Statistical Mechanics (Winter 2013) Lec 02. Molecular Structure & Statistical Mechanics  Symmetry and Spectroscopy  Part 2 Instructor: Rachel Martin, Ph.D. Description: Principles of quantum mechanics with application to the elements of atomic structure and energy levels, diatomic molecular spectroscopy and structure determination, and chemical bonding in simple molecules. Index of Topics: 0:03:12 Examples of Point Groups 0:13:55 Example: Low Symmetry 0:20:02 Matrix Representation 0:34:27 Matrix Multiplication 0:36:46 Transformation Matrices 0:41:42 Rotation Matrix 0:44:07 Matrix Representations of Operations 0:45:36 Inverse of a Matrix 
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Transcript
00:07
the morning everybody wants what's going on and it started it's about that time so but if you didn't already know that a character table Britain and this we have a bunch of friends like almost everybody did and in case of consumer Menard TVA's yet this is Jerry I think guys John Martin and who call you know from last quarter yes right so we have a lot to do today Parliament were going to do a few more examples of assigning molecules to .period groups so I hope everybody has urine flow charts and hopefully corrected there is a corrected 1 posted online a workable mistakes so sorry about that used to check out the corrected version ,comma also sees you have your flow chart and your character table there were handing out in order to be able to to go through the examples and so hopefully everybody tried the practice problems on signing things too .period groups and we're going to go through a few of those this morning and then we're going to continue to talking about matrix representation so summary also asking when the lecturers are going to be posted online and I actually don't know that the person who does know is astronomers who's doing something in the back to tolls moments in the just like and so on and so forth 1 of the best in the last half of the year and this year great thanks so in general on how long do you think it's going to take further steps after each lecture hall and great thank you very much so that's where the lead right of many more questions about logistics things like that 1 thing I should mention is that of my office my Wednesday office hours are part 3 before I'm going to have not a bunch people stop office hours yesterday we really good discussions so I encourage you to do that the Tuesday office hours I think the certainly moved to 11 to 12 from from now on to avoid conflicting with our discussions so is there some class that everyone has to take Tuesday from 11 to 12 at the moment about OK said that so that sounds like aware that case so let's talk about some pointed examples OK can can everybody see OK so unfortunately I don't have very much control over the lights are choices are like that and like that so raise your hand if you like it better the reason that if you like it better brighter you have a operators right so if we have a molecule that would be like this and we want to assign it to a point of order we wanna do we need to get our flow charts and look at Sun does this thing belong to any special groups so we have no symmetry high symmetry and dialing year so if we added 1 more substituent here then that would be you know something like sulfur hexafluoride a the Nymex fluoride then that would be enough to lead role molecule but we don't we have that's so funny thing to think that needed somebody to volunteer their assigned a single point group you just volunteer of the year OK so adjusted so everyone can can see what it looks like here so we've got this thing it has a square on the bottom and it's got 1 substituent sticking out and the thing that keeps it from being high symmetry that doesn't have another 1 on the bottom not too so it's not high symmetry of it so now what Knight now we need to find the principal axis would view of the world Federated said has a C for access OK so that's good so now the next question is do we have some see to Accies literary perpendicular to that C for access someone C 2 taxis How We Can we do some 180 degree workstations perpendicular to the C 4 parliament they know from the model of the Capitol be easier to see certain kinds traveling to cameras are remaining in the area but before the act so it was use of this war also that's your C for operations books so that they could look to the but perpendicular to that but indicated that it would be the so of the wrote on his father's family 180 agrees it's it's not the same year like that now so we think everybody agreed yeah so there's no knows there's
06:28
no C to Accies OK so now we've got we know that it's gotta be via a or as to any group so we think does it have a horizontal plane perpendicular to the axis India that Paul Mozer remember here's our 28 activists on the little 1 parallel gathers mothers in perilous to parallel but what you it's well it's vertical right because it contains the principal access is not to be died droll it would also have to bisect some see to act season and we don't have anything to axis of the political places so you're going to know that like this when urinary tract but the question of 4 following the flow chart is just doesn't have a horizontal plane we all know How can so the question is how do we know which cemetery planes there are vertical and which ones are horizontal and the definition of that in this context is that if you're a cemetery plane contains the principal axis its vertical we also have died legal claims those also contain the principal axis not every molecule has and they bisect the to access the perpendicular if there are in this case there are on a horizontal plane would be 1 that cuts perpendicular to the principal access so anything to this molecule have won now rank is if we thought it over and this substituent would be down instead of up and when we have something to say to you so it is you were very close it would be CQ means will be the right if it was easy to see if it has its principal axis was the to access but instead so great job next year it is gone and yes I'm dying he drove plane is also it's also it's like a vertical plane and that it contains the principal axes but it also bisects here see 2 planes that are perpendicular to its very we need to have 1 discussion going on and not many I'm really happy Guenter everybody's question but we need to do miniseries and not in imperil showcases question is what kind of player filets let's find a molecule that has not made a whole bunch of models which was really nice except that I can't find anything OK so so here's benzene so some roles for looking at symmetry in general were going to say that you know resonant structures for assigning things to appoint group going to assume the resident structures are fluctuating back and forth so quickly that done that we can't see the individual structures which of course is that's why we have risen structures anyway it's and if you have an average of these bond lengths so this molecule which you can go through that the .period group it belongs to the D 6 age group so belongs to 1 of Indian groups because it has a horizontal plane it has but its principle axis C 6 access so benzene is due 6 and so when we were talking about just now do we have to accede to axes perpendicular to the principal axis but in this case we do we can flip it over all kinds of different ways of no thank you so we can fulfill our looks like 3 different ways of perpendicular to that that actress and so then we also have some died legal planes with a rebel claims that bisect those C 2 axes and 1 thing that's really nice about the character tables is that some of the cemetery elements are really heard due to visualize the at least when you when you go through and try to count all the monetary didn't miss any and the good news is that you don't really have to do that because once you assign the molecule to appoint group if you open up here .period group table and look at the G 6 age group 1st thing you notice is that benzene has a lot of cemetery operations you like lists for you what they all are and so you don't necessarily have to go through and find although yourself once you get into my group then you can go back after the fact and and check out what all the cemetery elements are the character table gives you all lot of other information about the molecule going talk about a lot of that today but before we go on I do want to talk a little bit more about assigning things too .period groups this is a really important skill that if you have a hard time with it it's going to be challenging due to keep up later on so let's make sure that everyone gets it again if you don't even need more practice stay after class will be here answering questions from come to office hours at the in discussion you know it it is something that once you get it you do but if you can take the practice OK so what's look at this molecule can I have another victim Iormina volunteer so
12:50
that molecule has interesting
12:53
things going on the protest Florida so the 1st thing I know is it does it it's not linear we can tell that is at low symmetry a high cemetery OK this is a controversial molecule some people are saying it's high symmetry and other people are saying flow symmetry OK so I don't think it's high cemetery right because it's not it's Nico seeking droll and it's not Tetra role that would be like there's an octet role we saw a little while ago so well let's see if low symmetry so as to give examples of some of the low symmetry .period groups C 1 is the 1 that doesn't have any cemetery elements that's like that I do think it's like that is it have no cemetery elements I can see why and right it looks like it has an inversion center if I turned it inside out this company would go over here on the stage to withdraw over here in this muffled group would go over there so another assumption that we making I said that we assume the resonant structures are the average structure we also assume that there's free petition about single bonds was methyl groups at just spinning around if we're going to talk about it In terms were the word that's not gonna happen I'll tell you that the molecules really really cold so it's not moving otherwise we're going to assume that that's OK so it's not low symmetry police it's not see 1 so what about something like C were it just has the identity and mural plane where you know it's not as we said it has an inversion center 1 of the other low symmetry groups is CIA which means it only has an aversion center so anything thinks is that things have anything going on other than its inversions so if we can it like this we would have this Kerviel over here and that 1 over there and that wouldn't be the same rate Is there any way we can rotate at what he thinks yeah I think she's right it doesn't have anything else going on so that is the way that the CIA I think you OK so that some of those are some examples of unite you see what the Gullo Symmetry Groups look like and I think we're going to stop therefore examples although the I'm happy to do huge numbers more frequent office hours 1st day after classes and things like that and I guess another thing that I want to point out is that until you get good at doing a really fast in this looking at them it's best to go through the flow chart and assigned them so 1 thing that people get confused about is looking at the cemetery operations versus the names of the pointless so for instance I noticed that 1 1 mistake that people make sometimes looking at something and saying it has an improper rotation axis and so then they think it has to belong to 1 of the S 2 and groups and not necessarily lots of things can happen proper rotation access without wanting to those groups so it's good if you just if you just go to systematically and with the workers OK so I think that's so that's it for playing with the tanker Toys but so go on and do some other things from a switch back to power .period yes I'm sorry this speak up please sure so improper rotation again is when you read when you rotate by 360 degrees over and and then reflect about of it reflects for a plane that's perpendicular to the axis so that Warren S. threeaxis we were rotate by 120 degrees and reflect and I can show that shirt was so I still have that up there so if I were going to do an improper rotation I would rotate my 3rd of in and then reflect in your foot that so it's so it's so rudimentary return and then reflect through this plane to site so searing a cat I can't quite do it to the model but that's that's what it is OK so what some for more questions the that's a good question so that so the question is for it for ethane if you're looking for the .period group would be staggered a eclipsed so remotely said that by default if I don't tell you anything about it we're going to assume that there's free rotation about single dance seeking just assume that those methyl groups a rotating if I wanted you to do it for stagger eclipsed anything I would have to tell you that specifically otherwise you would know can he know helped of course that those configurations do existed at low temperatures it's just otherwise we're assuming things like never gratuitous pretending around all the time the In that case we adding that you just created as 1 big substituent it's just of the metal just freely the following you know that said we might see problems were at work we say that something is stagger eclipsed and you just have to pay attention to the description of the molecule OK so now let's talk about all the information that you get in the character table so so far we've done examples were we look at how to put things into a
19:17
particular point group and that leaves aside the question of whether we want to do this so the reason we want to do this is that once we do we get all kinds of information about the molecule for free somebody already collected it put it this character table we can use it the
19:38
case I would really like this to show my slides now it looking at the rate
20:03
of right so there's a flow chart
20:08
OK so now wants talk about the character Table Rock so everybody has this in front of them wants to look at the information many gives you OK so I have put some examples here on this she she'd just 2 it's just to show some fans according using this a lot so please bring it to class to follow along on that with the discussion and these are the same character tables that you'll be given on the exam so OK so we have to see to the character table so that's a familiar molecule belongs to that .period Europe's water just to visualize it and let's look at the information that we have here so so in the top left we have the name of the .period group and then going along the top and we have the names of the cemetery operations that belonged to that point so easily identity so that's do nothing c 2 is a 180 degree rotation and then we have these 2 planes Sigma the X z and Sigma the prime wisely so they're they're called Sigmund Sigma Prime just a distinguished that they're not equivalent to each other because if we have a water molecule misses a little bit harder see because it's small but you know everybody knows or water looks like that should be OK we have are 2 planes 1 is we can slice through the molecule like this so that 1 hydrogen ends up on either side and the other 1 is we can cut through the whole thing so that were slicing through the orchestra the hydrogen oxygen hydrogen bonds and those 2 planes are not equivalent to each other so that's why they except for entries in the table and then the next question is whether the XY Howard E. x y and z axis defined the principal axis is always busy access and you just use the the right role OK so that tells us the total number of operations and the number of operations that exists in the group is kind of a measure of the symmetry of the molecule and so Fauci to be there are only 4 of we have the identity we have the 180 degree rotation and then we've got these 2 reflection planes and so that's kind of older parade we're going to come back to where all the rest of stuff the looks look at C 3 The now so that molecules like ammonia question earlier in the year and we haven't gotten that yet were connector OK so right now we're just talking about the the cemetery operations in the group OK so if we think about ammonia that has the identity in this group would which everything does and then if we look at the next entry we have to see 3 and so what that means is that there are 2 C 3 operations that you can do so I have my ammonia molecule and I have 1 of the hydrogen sticking out toward you and the other 1 there are pointing off to the sides and what that 3 designation means is that I can rotate this once and that gives us an equivalent state as far as symmetry but it's not identical to work started out and then I can rotate it again and you know again it's it's symmetrically equivalent but if we could tag all these hydrants Simon imagine that we can isotopic we label them so that when the proton and 1 is tritium and the other one's deuterium so we can tell them apart we have to go around the 3rd time before we get back to the initial configuration and this is an important thing we have to be able to make a distinction between things that are valid cemetery operations which the says and being able to tell the difference between that and the original configuration so that's why we have to see 3 operations because we have 1 too before we get back to the original configuration that doesn't mean that it has to separate C 3 axes now don't get confused because in some other point groups it might mean something has multiple axes that are the same but the important lesson here is that when you have something listed as yeah that operation select Sigmund Sigma Prime that means they're not equivalent but if it's called to signal than those it's describing to operations of public said similarly we have 3 signal the survivor vertical plane contains the principal axis and we have 3 of us because we can cut through many of these bonds and that gives the symmetry operation in there all equivalent to each other so yet the question of of whether what the army the the molecule that had the the square on the bottom and something sticking up was and the sea for the yeah that's interesting questions so please do you have seen have succeeded 3 C 4 but only to see to rate because you can you can go on you know 1 way and then the other way OK so Tetra he drove where I can go through all that but notice it has a lot of symmetry operations and that should fit with your intuition that molecule like methane is more symmetric than than these other things another important characteristic is is the number that you get when you add up all of these
26:08
cemetery operations that's called H some .period group tables give it to you this 1 doesn't see it candidate yourself but but that's something you can do are a question that is To see the origins of the accident before you get back to the original molecule yeah this is Italy from you know by convention we usually do it counterclockwise but you know know if he did everything other as long are consistent you get the same answers but for for purposes of doing something class the convention is usually we do it counterclockwise yes it is and that was that we all tend to be out of the but also the broken about the principal axes you can do the C 4 3 times before you get back to the the 1st it the because in that case if he rotated it like this or like this you would get there there were 2 ways to do it so that made my point is just you know be careful because there there are these operations with coefficients in front of them indicating that you have multiple ways to do the same operation and sometimes it means just that you can do the operation a couple times before you get back to the original state and other times it means that you have different Accies on different planes that are equivalent Will will see more examples of this says it as it comes up I don't wanna spend the a whole bunch of time talking about every case because I think it's a little bit abstract what's what's waitandsee examples OK so now what is all the rest of the stuff on the character table that's a lot of what we're gonna spend time on today and Friday OK cities 80's and easing tease those are the irreducible representations or the cemetery species of the group and what those are all it's it's a complete description of objects that can behave in certain ways under these particular cemetery operations and we're going to talk about that With some concrete examples a little bit later on so some things to know about them the ones that are called a and B or simply generate the ones that are called are doubly degenerate and the ones that are called tear trickling and then once look at the other information that you get In this table which starts to give you some some hints about how you might be able to use this information and that is we have things like x y and z we have x y X the Wise these are linear and quadratic terms in terms of the Cartesian coordinates so 4 x y and z right now you can't think about that as either a little unit vector directed along the appropriate access or you can think about it as a P X P wire PC orbital in terms of how it transforms related dysentery those very intuitive concept for for chemists and chemical engineer so it helps if you visualize it as oral the X Y X zee etc export minus Weisberg you can think about those is the orbitals they're going to have other interpretations or we get into talking about infrared and Raman spectroscopy later in the course but for now you can think about the interest in terms of what OK so you can start to see what's useful about this table so once you sign something new .period group for 1 thing there's a limited number of objects that can behave a certain way under the cemetery operations we have a complete set of cemetery operations to work with and we can already see that we learned some information about how at least orbitals behave with respect to this the cemetery and this is already renowned for you in the table OK so having gone over that a little bit we are going to switch gears and talk about matrices and how to make matrix representations of operators and we're going to do a little review of how to deal with matrices hopefully this is review for everybody if not we're going to go over what he did know about it said don't worry if you need a little bit of extra practice a background please check out the Wikipedia page and or the wall from site on on matrices and matrix multiplication Caucasian operators things like that OK so if we have a matrix which we call a these entries are it's matrix elements and we can call those AID will
31:21
see that kind of terminology a lot so in this case a 1 1 is minus3 1 2 it's 6 etc. that's just how we label them and what is going to go through a quick review of thought I had a deal with with matrices so you can add value if they have the same number of rows and columns and if so if you can do it it's pretty easy you just add up the individual matrix elements and so here's what you get in this case we just add the individual matrix elements and get the souls so I know everybody's probably seen the stuff before but it doesn't hurt to have a little bit of a review especially since thought if he didn't really talk about nature's representations of operators last quarter it does it actually makes your life quite a bit easier I think I think it's so it's much easier to deal with the operators and that formalism OK so that's how we add them that actually doesn't come up terribly often in the kind of things that were going to do hear something that does if you want to find the trace of a matrix you just add the elements on the diagonal and ignore everything else that might be in the matrix of doesn't matter can add the elements that are along the diagonal the trees is also often called the character which gives you a hint as to what the character table was about and why we're talking about this right now so all those ones and minus ones and zeros and choose etc. on the character tables each 1 represents the character of the matrix that corresponds to that particular operation for particular cemetery species and we're gonna learn how to make a own if not by the end of the event if not today by Friday OK so the character is a lot of times given the symbol client In this case it some so that's a really important matrix operation fortunately it's easy we can also multiply them by Scala's in order to do that we just multiply each element in the matrix by the scalar and we can take the trace about 1 the the so again pretty straightforward stuff but it's good to go to go over adjusting case are a let's talk about matrix multiplication also in case you haven't seen in a while so when we go to multiply the matrices I'm going to operate this out once so we go through and multiply the role of the 1st row this 1 by the 1st column about 1 so we get 1 1 times 5 plus 2 times 8 as the 1st of matrix element in the new matrix and then we just go across so now we have onetime 6 plus 2 times 9 etc. and we build up her new matrix like that so pretty simple but you have to double check because it's easy to make a mistake how many people taken Camp 5 or otherwise no Mathematica it's a lot easier if you use mathematical so most of the examples that will do in class will be relatively simple and die you know you'll be able to do it in your head find enough but if you have to do this for matrices of any size is mathematical makes it makes a lot easier OK so here's what we get for this particular 1 and it's also worth pointing out that the matrix multiplications don't necessarily view so we multiply these 2 things together and then we do it any the other order you don't get the same answer N you know cost this relates to stuff that you learned last quarter in quantum mechanics a lot about a lot of operators don't necessarily commute and they can be represented as as matrices and will also see that in some point groups cemetery operations there new so
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other things to look at it from the product of 2 matrices equals 0 that doesn't necessarily imply that either of the matrices has all zeros in the debate ways to get that so that's ah are little review of matrix properties again if you need more review than that check out the wall from sight and on Wikipedia Wikipedia a really great resource on things like this letter of the noncontroversial of course things were theirs to where there are differences of opinion people can change it all the time and troll each other nobody really does that unsorted basic math and chemistry and physics topics of this so it's a good thing using the resources OK so now we talked about properties of matrices let's start looking at how to construct transformation matrices for actual operations that we might wanna do and we're going to do it in a twodimensional space to start with just make things easier OK so the way we're going to do this is we're going to think about it I want to accomplish some transformation and I'm going to apply it to a test vector vendors calling alphabeta and we need to think about what do we want alphabeta to transform into and then what matrix do we have to multiply by it to get that result so if we want a reflection about the Y axis from the lower in a twodimensional plane so we need to think about what to be multiplied by Alpha Beta In order to get it reflected about the Y axis so of course if we reflected about the Y axis leaders change and Alpha is going to change sign and so working backwards we have to think about what do we need to multiply by that vector in order to accomplish our transformation if year and as we're going to see the matrix you get depends on White what you're trying to do what object applying it to but wouldn't talk about the cases of just doing this in twodimensional and threedimensional space OK so what we wanted your production the xaxis so we only want component so what do we have to multiply by Alpha Beta To get just as the production on the xaxis the yet so I hear people following alongside so everyone gets at that school right for want to scale it by 3 so we just did something that has threes on the diagonal so this is why I like group theory in these kind of geometric transformations because it really gives intuition into how we can set up matrix representations of different operators the quantummechanical operators of course are all operators as you were in the last quarter of so they can be represented this way but doing this with these geometrical things helps give us an intuition for how to use it before we have to do to get into more complicated concepts of so in general if we have some vector and we wanna rotated so we had our 1st vector are 1 and now we move it into this position are too if we just set up by the how we wanted this rotation if we look at X 2 and N Y 2 we have are cosigned Alpha Plus theater and are a sign of a data and we can expand this out and that gives us the rotation matrix that we need to be able to performed this so this particular transformation rotation matrices are is something that we see a lot for him use them now we talk about group theory so hopefully it's clear how that's going to work and and how we can use that quite a bit were also going to use them we talk about a more spectroscopy and look at and how spins behave in a magnetic field and really they come up and all kinds of different areas of chemistry and physics useful thing Nunavut OK so having gotten this far you have enough information to definitely do practice problems which were posted online said operate them all down .period now I just want to point out that there so you go ahead and check these out online trading due them for for Friday having looked at that let's move on to 3 dimensions so we talked about our little twodimensional rotation matrix now let's look at this in 3 dimensions that basis there is little unit vectors pointing the x y and z dimensions and notice under trying to be really careful about telling you what basis I'm using and I don't you should ask me because it's a really important question that affects everything about the problem so right now it's just our our unit vectors OK so what if we wanted you AC to rotation so sold 180 degrees so if we have our x y and z unit vectors that's going to foot the signs of X and Y and wheezy alone and said this is tallest where matrixes yet you used last year and the it was used by 1 of the site well this is in 3 dimensions that we're doing it in 2 dimensions before that was the end of this year yet so that's so so that you raise a really important point which is why I said that I have to be very careful always tell you with the with the bases as they were using the changes everything about the problem so you know if we if we're starting with so before we were starting with 80 at that too we're starting with the 2 too because we had a 2 dimensional vector now we have a threedimensional back and so on think that you have this year because of the weather and things like before they have a lot of work to do and that I'm going to check it and right up the right up something about it sorry but that is just what I wanna get through a little bit more less before class and you know whatever it is confusing weakened we can go over later OK so let's talk about rotation matrix for c 4 so this one's all the more complicated because we flip the position of X and Y we made ex negative and again as he stays the same because we were rotating about the principal axis groups and so that's the admission matrix that we end up with 4 C 4 and so I want to point out is that here's what we get for the general rotation matrix about any angle we need to put in the signs in coastlines and so in Cartesian coordinates here are the general rotation matrices for some angle about the the x y and z axis and these are things that they're going to come up over and over again renews on so again you don't have to write down right now this is the it's available you can look it up but they're going to come up and it's important I also want to point out that the inverse of the matrix is the matrix that if you multiply matrix times its inverse you're going to get an identity matrix which has just once on the diagonal 0 is everywhere else sometimes it's called I for talking about the identity operation in in terms of the BIA character tables we call it the and give a represent some transformation than its inverse which is called a minus 1 and does it and returns it to its original state bank here that is written out so OK that's pretty good as far as where I wanted to get this time but next time retired altogether and see how to use this in terms of group theory yes
00:00
Aldehyde
Substituent
Schwefelhexafluorid
Walking
Tuberculosis
Tolllike receptor
Operon
Wursthülle
Fluoride
Molecule
Alcohol by volume
Taxis
Tiermodell
Functional group
Controller (control theory)
Lead
06:25
Substituent
Chemical element
Ageing
Bohrium
Resonance (chemistry)
Operon
Wursthülle
Man page
Molecule
VSEPR theory
Bond length
Computer animation
Tiermodell
Functional group
Benzene
Chemical structure
Periodate
Process (computing)
12:52
Chemical element
Potenz <Homöopathie>
Octane rating
Bohrium
Resonance (chemistry)
Operon
Chemical bond
Katalase
Wursthülle
Californium
Methylgruppe
Niacin
Reflexionsspektrum
Man page
Optische Aktivität
Molecule
Volumetric flow rate
Tiermodell
Functional group
Metallatom
Cryogenics
Chemical structure
Active site
Periodate
19:32
Lot <Werkstoff>
Leftwing politics
Bohrium
Operon
Wursthülle
Reflexionsspektrum
Man page
Optische Aktivität
Tritium
Molecule
Origin of replication
VSEPR theory
Methanisierung
Octane rating
Hydrierung
Hydrogen bond
Protonation
Ice front
Erdrutsch
Oxygen
Water
Alcohol by volume
Computer animation
Functional group
Rock (geology)
Deuterium
Stuffing
Stereoselectivity
Ammonia
26:07
Semiotics
Chemical element
Chemical engineering
Species
Expressionsvektor
Atomic orbital
Operon
Set (abstract data type)
Wursthülle
Molecule
Origin of replication
Lactitol
Spectroscopy
Active site
Periodate
MetallmatrixVerbundwerkstoff
Chemical element
Nitrogen fixation
Formaldehyde
Ice front
River source
Californium
Electronic cigarette
Trace element
Computer animation
Functional group
Optische Analyse
Azo coupling
Pharmacy
Stuffing
36:32
Weathering
Physical chemistry
Area
Expressionsvektor
Chemical property
Lipopolysaccharide
Topicity
Operon
Wursthülle
Reflexionsspektrum
Electronic cigarette
Chemistry
Alpha particle
Optische Aktivität
Origin of replication
Alcohol by volume
Sample (material)
Optische Aktivität
Computer animation
Functional group
Beta sheet
Active site
Base (chemistry)
MetallmatrixVerbundwerkstoff