 Hello everyone. Welcome to the module on intermolecular forces and potential energy surfaces. We are still discussing the interaction between molecules and what kind of form they are and what are the consequences of that. And in this particular lecture, we shall look at permanent dipole-permanent dipole interaction and hydrogen bonding and van der Waals interaction in little more detail. But before we jump into the permanent dipole-permanent dipole interaction, let us just quickly recap what we learnt in the previous class. In the previous lecture, we first looked at why do we need intermolecular interactions or why should one even study intermolecular interactions and what is the role of it in chemistry, biology and material science. Then we looked into the different classification of intermolecular interactions based on whether it is charged or whether it is polar and the different subclasses of that. And after that, we had looked at in a little more detail the ionic as well as the ion dipole interaction and we had looked at in specifically the one, the distance dependence or of the potential energy of interaction between the two species which are interacting. And we had said that the ionic interaction which goes as 1 by r is a long range interaction. So in this particular lecture, what we shall do is we shall go a step ahead that is going from an ion dipole to a two permanent dipoles and see how does the interaction potential of such a system look like. Let us start looking at a permanent dipole-permanent dipole interaction. So what we shall do is we shall have a couple of assumptions that is the both the interacting dipoles are point dipoles and they are interacting in plane and in the play and along a collinear line in the plane of the board. This would make us easier to derive the expression first and then we can add on certain conditions to look at a more realistic picture. So now let us say we have a one dipole moment which is I am showing it here, the two charges that is let us say Q1 and minus Q1 and they are separated by a distance L okay. And now I have another point dipole which is separated by again a similar distance L, I will call this Q2 and minus Q2 and this again separated by distance L and the both of these are separated from each other by a distance R and I am representing R from the midpoint of this interaction between the individual dipoles and this is R. Now what we will do is we shall apply our same equation of potential between interaction potential for two point charges that is V of R is equal to general formula for the Q1, Q2 divided by 4 pi epsilon not R. So this is the general equation right. Now what we will do is we will take Q1 and look at the interaction of Q1 with Q2 as well as minus Q2. Similarly minus Q1 interacting with Q2 and minus Q2. So now let us go ahead and look at how does that potential look like. So V of R is equal to Q1, Q2 divided by 4 pi epsilon not and if you now look at the distance between Q1 and Q2 it is actually R because you can displace the R from the center point to let us say the starting charge that is the Q2 and Q1 and you would see that that distance is equal to R. So this distance would be now R correct and now I am going to look at Q1 interacting with minus Q2. So the term would be minus Q1 Q2 divided by again 4 pi epsilon not. Now the distance would be slightly higher because I am looking from Q1 to the minus Q2 right and if you now look at the distance by geometric arguments you would see that this distance would be R plus L. So you can try and first look from Q1 to Q2 that distance is R and from Q2 to minus Q2 is L. So I am adding the distance L here. So then I will add a I will go and look at how does minus Q1 interact with Q2. So that would again be a minus term Q1 Q2 divided by 4 pi epsilon not and this time you would see that this distance would be R minus L because if I take a look at distance between say Q1 and minus Q1 and minus Q2 that would be R and to get distance between minus Q1 and plus Q2 I need to subtract the distance L which is the length of this dipole. So that is why I have a minus term here and finally I am looking at the interaction between minus Q1 and minus Q2 and that would be positive Q1 Q2 divided by 4 pi epsilon not and the distance would be R. I hope that is easy to understand alright. Once we have got this potential form then what we will do is we will try and simplify this a bit further and let us go ahead and see that. Now V of R or potential is equal to I can take out Q1 Q2 and 4 pi epsilon not out then I would be left with 1 by R minus 1 divided by R plus L minus 1 divided by R plus L minus 1 divided by R minus L plus 1 divided by R alright. I hope this is clear to everyone. So once we have done this now let us go ahead and look at a little more simplification of this. So I am going to write the same expression here V of R is equal to Q1 Q2 divided by 4 pi epsilon not and so now what I would do is I would take a minus out and I would also take out R from that previous expression then that would leave me with plus 1 divided by R plus L plus 1 divided by R minus L I think this will be minus 2 yes. So this would lead me to this particular term and let me just not take out the R yet so I would have 2 by R right. So then the next step what we will do is I have just taken out a minus sign so then I would get this particular expression and in next step I will take an R out from the denominator. So minus Q1 Q2 divided by 4 pi epsilon not I have taken the R out so this is minus 2 plus 1 divided by 1 plus L by R plus 1 divided by 1 minus L by R plus 1 divided by R okay. So we shall just now look at just these 2 terms and see if we can simplify them further. So I can write L by R is equal to X right so then this term would become I am just looking at these 2 terms okay I will come back and put it into this in a minute. So it would become 1 divided by 1 plus X and the second term would be 1 divided by 1 minus X and in order to understand or in order to expand this solve for this we shall make use of what are called as Taylor expansion and the Taylor expansion series for 1 divided by 1 plus X is equal to 1 minus X plus X square minus X cube and this is the second term and so on similarly for 1 divided by 1 minus X is equal to 1 plus X plus X square plus so on. If we apply these 2 to the terms here which I have written on top let us say these terms then what I would get is 1 divided by 1 plus X plus 1 divided by 1 minus X would be equal to 1 minus X plus X square minus X cube so on plus 1 plus X plus X square plus plus X cube plus so on. So I hope now you can see that I can these 2 terms actually gets cancelled the X terms and similarly the odd powers get cancelled and all you are left with is the even powers and we can neglect the higher terms because the L is much much smaller compared to R so this would actually reduce to 2 plus 2 X square 2 plus 2 X square right. So if I take this now and come back and you put it here then I would have the following minus Q 1 Q 2 divided by 4 pi epsilon not R minus 2 plus 2 plus 2 X square so these 2 term gets cancelled and here I will have left with 2 and this would be equal to minus Q 1 Q 2 divided by 2 pi epsilon not R times X square and I know that X is equal to this particular formula that is L divided by R so I am going to write as L square by R square right. And we can do a very similar thing as we did for the ion dipole that is if I have the length L multiplied by the charge Q 1 or multiplied by the charge Q 2 then I would end up in the corresponding dipole moment vector right that is mu 1 or mu 2. So if I take help of that equation then this would reduce to minus mu 1 mu 2 divided by 2 pi epsilon not R to the power 3. So this is I am just going to write it here as V of R and this is the key equation which ultimately governs the dipole-dipole interaction and the negative sign indicates that there is a attractive force between the 2 species and the most important striking thing to notice that it goes as 1 by R to the power 3 that is as the third power of the distance between the 2 dipoles or the center of the distance between the 2 interacting dipoles. So this is a very important thing to keep in mind and as I said previously this was all for a simple 2 dipoles actually interacting in a collinear manner on the plane of the paper but that is a bit of an idealized situation. So now let us see what happens if the 2 dipoles are at an angle. Now let us look at the angular dependence of it and in order to do that I will again go back and draw the dipole moment picture that is I have 1 dipole here Q 1 minus Q 1 with a distance L and the other dipole I am going to draw somewhere not collinear with it this is Q 2 and minus Q 2. So I hope you now see that there is actually the angular term or the angle between these 2 dipoles or the center of these 2 dipoles also becomes an important factor now not just the distance between them. So I shall try and draw the distance between these 2 that is so this is the R that is the distance between the 2 dipoles and now this theta or the angle which is the 2 dipoles are making with respect to one another is also an important factor or component to take when we want to look at the potential of interaction between these 2 systems. If we take that into account then the potential would look like the following that is V of R is equal to mu 1 mu 2 divided by 4 pi epsilon naught R cube the distance dependence does not change so the R cube remains however you will have a new term that is 1 minus cos square theta 1 minus 3 cos square theta this is an important part to take into account the angular dependence between the 2 dipole moments which are now actually within the same plane but at different angles acting at different angles with respect to each other. However you can also think that this is one possible condition or one possible scenario what could also happen is that I can have 2 dipole moments actually may not be in the same plane they could actually be in 2 different planes like this one is in this plane another is in this particular plane. So in this case you will have one more angular part which will actually come into picture and that would slightly make the equation for V of R a little more complex but what would remain same is the distance dependence which is this the 1 by R to the power 3 or V of R the proportionality of 1 to the power 1 by R to the power 3 would remain constant and that is something which is very significant and which we need to keep in mind okay. So having talked about dipole moment permanent dipole permanent dipole let us just give you an example of this to see what kind of systems one looks at say if I have a molecule of an acetone which is interacting so this is one particular example where I have 2 dipoles which are now interacting at a certain angle and this equation can be used to calculate the dipolar interaction between these 2 molecules which are actually non charged or uncharged but they are still polar or they have a permanent dipole moment alright. Now let us go ahead and try to look at hydrogen bonding as a special case of the dipole-dipole interaction or permanent dipole-permanent dipole interaction. So I am sure when I say hydrogen bonding you will start thinking about water or the system which we talked about earlier that I had shown you that we had oxygen which was bonded to 2 hydrogens and what I can also do is I can write another oxygen and I can write 2 more hydrogens here like this and you can draw this interaction as a hydrogen bond and so on and so forth you can keep on writing like this as a chain of hydrogen bonds right. This is what would keep the water molecules interacting with one another in the glass of water and there are different forms of hydrogen bonding as well that is you can also write for let us say for example if you take a benzoic acid okay. So here again what you see is that there is a interaction between the 2 systems that is this is the interaction which is the interaction between a hydrogen which is attached to the OH and a carbonyl oxygen similarly on top. So people typically observed that if a X H system which is interacting with a Y then and if given that X and Y are 2 electronegative atoms which are more electronegative than hydrogen then typically you would have what is called as a hydrogen bonding interaction and this the strength of the interaction would depend heavily on one the difference in electronegativity between X and Y and also the angle at which the X H and Y is coming in if they are coming in like at 180 degrees that is this angle then you would have the maximum interaction possible however if they are coming in at a slightly different angle then the interaction strength would actually decrease. So that this is a classic example of a hydrogen bond in a system such as water or a benzoic acid and people have actually argued and debated about the hydrogen bonding extensively in the literature and what has come out from this is the following definition that a system is said to as hydrogen bonding is said to be there in a system if there is an attractive interaction between a hydrogen atom from a molecule or a molecular fragment X H in which X is more electronegative than hydrogen and an atom or a group of atoms in a same or a different molecule and in which there is an evidence of bond formation. So this is a very sort of recent and accepted definition of hydrogen bond which is come about by a group of scientists from all over the world sitting together and debating and coming up with this definition. So all this tries to do is to capture many different flavors or many different forms of hydrogen bonding and the basic essence is that the hydrogen bond should be planked between two electronegative atoms that is X and Y and there should be an evidence of bonding. So at this point you might be thinking so how can I classify hydrogen bond should I call it as electrostatic, should I call it as a permanent dipole, permanent dipole or how do I classify it. So that is the reason why we have called it as a special case because in the literature in the scientific research some people attribute it as electrostatic in origin or it has primarily electrostatic origins and some people call it as a dipole-dipole origin. So that is actually not very clear to or easy to separate. Both factors actually do play a role that is electrostatic interaction as well as the dipole-dipole interaction do play a role in giving rise to hydrogen bond. So with this in mind we shall just call it as a special case and the strength of this interaction would depend heavily on what are the two different atoms X and Y and also what is the angle at which these two species are interacting with one another. So now we shall actually move on to the last class of interaction which are typically called as Van der Waals interaction or also called as London dispersion or induced dipole-induced dipole interaction. Now let us go ahead and try and see for which kind of systems this interaction plays a role. So previously we had seen that if we take a molecule like say CH4 non-polar so if you take a non-polar and also non-charge system like methane, ethane, argon or different kinds of inert systems so then there would not be any apparent interaction one could think of. However if you go back and look at your lectures and quantum mechanics what you would realize is that the electrons in a system in a molecule or in a particular bond is actually never stationary. It is always oscillating around the in a given space where the probability of finding the electron is maximum. So this movement of electron or the continuous let us say the oscillation of electron would give rise to what are called as induced dipoles or induced charges in a system. So if I take this particular system and this would probably give rise to something like say a delta plus and a delta minus in the same molecule itself right. And this is just instantaneous it is not something which is which you can find if you take a molecule of methane or in ethane it is just instantaneous which is formed on a time scale as the electrons redistribution takes place. So once this happens you will also have another neutral molecule which is actually floating around and if this comes in close contact with this what can happen is that this can start inducing the charge that this will give you a delta positive here and this part if it comes in close contact this would induce a delta negative and a delta positive here. So now what you have is the electron redistribution in a molecule is actually now giving rise to induced dipoles like what I have shown here by the partial charges and these induced dipoles actually interact with one another in a molecule and this is what one calls it as a van der Waals interaction or also called as a London dispersion forces or also called as induced dipole-induced dipole interaction. And the form of the potential is the following that is V of r is I will just write the distance dependence that is 1 minus 1 by r to the power 6 that is it goes as the sixth power of the distance between the two interacting molecules. For example if I take here this is the center and this is the center then this would be the r and the interaction potential would decrease very very rapidly. So I guess you can see that the van der Waals interactions are actually very close contact interactions or they would only come about or become significant when two molecules are nearby or close to each other and the moment they actually go a bit apart the interaction falls off very quickly. So this is what people would typically also call as London dispersion or induced dipole-induced dipole interaction okay. So so far we saw many different kinds of interactions ranging from interaction between two charge species to interaction between a charge species and a dipolar system to two dipolar system to two completely non-charged and non-polar system such as a methane or any such inert systems. So if I were to write a general formula to get the interaction potential as a function of r then that potential would be of the following form. So V of r would be 1 by r to the power n plus m minus 1. So here n and m are the the the the let us say the n and m correspond to the poles which are interacting that is if I have a dipole which is interacting that means there are two charges interacting. So n becomes 2 and then let us say I am having an interact dipole interacting with a charge species. So then the m is 1 so then 2 plus 1 that is 3 minus 1 then that would give me an r square which is exactly what we saw for an interaction between a point dipole and a dipole between a sorry between a point dipole and a charge species right. So n and m corresponds to the the dipole state of the two interacting species. So having looked at various systems now let us go ahead and try to summarize what we have learned in this lecture about intermolecular interactions. So I have shown you this particular diagram where there is a spectrum of interactions going from a completely charged species the interaction between two charged species to a completely to a interaction between two completely non-polar and uncharged species. So if we were to look at the interaction strength and the distance dependence of the potential then at the top we have the ionic interaction which we also called it as a long range interaction that is interaction between two ions and this is in the range of about 200 to 400 kilojoule per mole and it goes as 1 by r. And next what one can have is I can replace one of the ions by a dipole then I will have an ion dipole interaction and this would actually now have a significantly lower interaction strength because I no longer have a two charged species. I am looking at one charged and one uncharged species and this would be somewhere around 10 to about 80 kilojoule depending on the specific system we are interested in and that would give a 1 by r square dependence of the potential. And if you go it further and replace the first ion by another permanent dipole then I get a permanent dipole-permanent dipole interaction and this interaction strength would be somewhere between 5 to about 30 kilojoule per mole again depends on the particular system we are looking at and the interaction potential would go as 1 by r to the power of 3 which we looked at by looking at the derivation. And finally in the last slide we saw that if we now replace both the polar molecules by completely apolar molecules such as methane, ethane or argon then one would end up in what are called as London dispersion or van der Waals or induced dipole-induced dipole interactions and this interaction strength are very very small that is the order of 5 or 5 to 10 kilojoule per mole at the max and these are very short range interactions meaning they actually fall off as the sixth power of the distance between the two interacting systems. So what this tells us is that the induced dipole-induced dipole interactions or van der Waals interactions are very very short range very very short range interactions and the moment you actually go to a larger separation they actually fall off whereas the ionic interaction and ion dipole or to an extent dipole-dipole interactions are long range meaning they can interact with systems which are much further away from the parent system. So with this we shall stop our discussion on intermolecular interactions and in the next lecture we shall look at what are the consequences of these intermolecular interactions on the properties of the gases that is real gases and also critical phenomena. Thank you.