 Hello everyone, I once again welcome you all to MSB lecture series on interpretive spectroscopy. I hope you are all doing very well, if so let me continue from where I had stopped about discussing on infrared spectroscopy. In my previous lectures I was telling about how the stretching frequency of a diet molecule or of a bond can be related to its reduced mass and also the stretching force constant. The simplest equation one can think of is nu bar equals 130.3 into square root of F over mu and in some equations for stretching force constant K is also used as an abbreviation. It does not matter whether F or K you should remember this refers to stretching force constant and here F is stretching force constant and mu is the reduced mass and reduced mass can be easily calculated by taking ratio of M1 into M2 divided by M1 plus M2 and this equation is the simplest one and when you are using this equation what you should remember is a reduced mass is given in atomic mass unit and then force constant is given in newtons per meter and frequency in centimeter minus you should remember that one apart from this simple equation nu bar equals 130.3 into square root of F over mu. We have couple of more equations one is I have shown here of course this is from this one both the equations are derived here nu bar equals 1 over 2 pi c into square root of F by mu and if you simplify this one it leads to mu bar equals 4.12 into square root of F over mu and only thing is little bit variation will be there as far as the units of force constant are concerned. Let me tell you how this is derived and that is simplified into this equation here. The Hooke's law expression can be simply transferred into a very useful and simple or simplified equation in this manner. We know that nu bar equals 1 over 2 pi c into square root of k by mu this comes again by nu equals 1 by 2 pi into square root of so but nu equals nu bar by c. So therefore what we do is we are simply putting this value here and then this is what we get here. So then nu equals m 1 m 2 over m 1 plus m 2 is the reduced mass and nu bar is frequency in centimeter minus 1 and c velocity of light 3 into 10 raise to 10 centimeter per second and k is force constant in dines per centimeter. So that means in this one the force constant is given in dines per centimeter but in the simplified one where I showed it is 130.3 into square root of F over mu where we are considering force constant in terms of nu times per meter that conversion factor has to be added here if you want to use this equation and again mu equals m 1 m 2 and then we are also adding to denominator Avogadro number and then this will give masses of atoms in a mu and then if we simplify further by adding Avogadro number from the denominator of reduced mass expression and taking the square root we obtained this simplified equation here and then if we apply here add the constant such as pi 22 by 7 and c 3 into 10 raise to 10 it would simplify to this format here. So 4.12 into square root of k by u here as I mentioned force constant is represented by a dines per centimeter and then one dine is approximately 1.02 into 10 raise to minus 3 grams. So this equation may be used to calculate approximate position of a band in the infrared spectrum by assuming that k for single double and triple bond is 5 10 or 15 into 10 raise to 5 dines per centimeter. So that means for single bond you consider 5 into 10 raise to 5 and double bond consider 10 into 10 raise to 5 and then triple bond it is 15 into 10 raise to 5 or if it is 500 if it is 500 is given value it should be 500 into 10 raise to 3 it becomes. So we can simplify in this fashion. So if we know this conversion and if you know how to convert stretching force constant to dines per centimeter we can use it otherwise simply we can use the previous one what I showed is a much simplified equation. So let us try this method here for some simple molecules for example where we have C C double bond and C C double bond is there you can consider here 10 into 10 to 5 dines per centimeter or the value is also given here C double bond C 600 stretching force constant is 960 that means 960 into 10 raise to 5 you can take then if we apply here of course this can be calculated here we get 6 here C double bond C this is what I have shown here in that table here and we can verify all the data to see authenticity of all the three equations and now by calculating by putting mu equals 6 and 10 for stretching force constant value comes around 1682 centimeter minus 1 but if we use this one 9.60 it comes exactly 1648 very close to this experimental value and of course here as I mentioned when the values are given in newtons per meter so multiply this value into 10 raise to 3 and if it is 21 if you consider then it will be 21 into 10 raise to 5 and you can verify all these things since we know the equation now three equations are there you can just practice since all the data is there first you consider this as unknown entity there is no point in considering this as unknown entity because reducer mask can be easily calculated either you consider this as unknown entity or this one as unknown entity and you can verify from three equations to make yourself familiar in looking into the relationship between the stretching force constant and stretching frequency that data obtained from a spectroscopy so let me take now another example CH bond here and of course CH bond if you consider carbon atomic weight is 12 and hydrogen is 1 and if you put here what you get is 0.923 for CH and of course here it is the same value I have listed here and then if you apply simply this equation here what you get is calculated one will be 3032 centimeter minus one and but the experimental one is 3000 centimeter minus one so that means this is the one calculated from this equation how about considering CD instead of CH and then we have to take the exact value here and then this also 10 raise to 5 here single bond we are considering and then what we get here is 22 28 2228 centimeter minus one this is a calculated value we are getting and then here it is 22 0 6 experimental value of course if you take the exact value it is very easy to achieve the experimental value here exact D atomic weight if you consider carbon then the reducer mass would vary little bit so that we can get the experimental value here so now with this information let us try to solve some problems so here is a question calculate the force constant of the P double bond O group in POCl3 oxy phosphorous trichloride the strong infrared absorption band of the PO stretching vibration is observed at 1290 centimeter minus one so now let us look into the simplest one this one is 130.3 into square root of so here of course when we consider P atomic weight will be 31 and oxygen it is 16 so mu is P into O over P plus O atomic weights this is 31 into 16 over 31 plus 16 this comes around 10.55 we can calculate so this is the value we get and here mu is given here we have to calculate the stretching force constant so 1290 equals 130.3 k is the unknown entity and here 10.55 and then to obtain this one what we can do is we can square entire stuff so that we can remove the square root this one we can do by simply square we get this one by simplifying this we can get k rf we can 1.05 Newton per meter so this is how we can get the value here and of course one should also know one atomic mass unit equals 1.67377 into 10 raise to minus 27 kg so this also this one we should use if we want to use the next equation here we can do from this equation also for example we can take this equation here let us consider this equation now mu equals 1 by 2 pi c square root of k rf either of one we can consider and then mu equals we know now we have already calculated and now mu square if you do it this 1 over 2 pi c whole square into k by mu so k becomes now if we add all the value here in this one what we should do is this reduced mass they should be converted into grams so if you multiply this one by 1.67377 into 10 raise to minus 27 this we should do it and after simplifying here what we get is if you add all the values here one can get 1044.91 Newton's per meter so this is how you can get it of course here you can add all the everything is known except for stretching force constant you can calculate using this method so little difference comes that's okay that is all over and I would stay stick to this method this is rather easy I use both either f or k this is much simpler I have to remember and also to perform the calculation so now another example is there calculate the force constant of CO which shows a strong absorption at 2147 centimeter minus 1 so here one can also do the calculation by considering the previous one whatever I am considering here this equation to just to make you familiar with all the methods so CO mu equals m1 m2 over m1 plus m2 and whatever the value it comes that should be multiplied by this is 6.857 atomic mass unit it comes and this should be multiplied by 1.672377 into 10 raise 27 then it will be giving the value kg's so here force constant we have to calculate as we did earlier if you put all the value here so you can find out the stretching force constant easily this comes around 1707.1 Newton's per meter you can put here of course 22 by 7 3 10 raise to 10 to this value 6.857 10 raise to minus 27 so this will give you value shown here so this how you can use either this method or you can also go starting from a simple method such as 130.3 square root of as I said k or f both will tell you force constant you should not very much about this one as I said some books refers to k for force constant and some books refers here for force constant. Now let us look into one more example here if the force constant of HCL is 521 Newton's per meter calculates absorption frequency in centimeter minus 1 absorption frequency will be way number so on atomic mass unit I have given here 1.67377 10 raise to minus 27 kg so this is needed if you want to use this equation here you have to convert that one if you want to use this one if you want to use other equation there is no need you can apply directly the values in Newton's per meter. So now let us consider here 130.3 by 21 reduced mass is going to be 0.9801 if we take here HCL H if I consider 1.008 into this one is 35.453 and then 1.008 you can get directly here value or you can make it square so that we can remove the square root or if you want to use the other equation this equation what we should do is we should multiply this value obtained in atomic mass unit into 1.67377 10 raise to minus 27 and then we have to use this value in this one for mu here once when we do that one we can get the approximately 2990 centimeter minus 1 you can verify that one you can use both the methods and you should be able to get the correct value here. So now you can see in this table I have given as I mentioned three methods are given this is very convenient this is also convenient provided you convert reduced mass after calculating or multiply this by 1.67377 into 10 raise to minus 27 your job is done or you can use directly here only thing is in this one if you want to use mu bar equals 4.122 square root of f by mu what you should do is you should multiply force convert force constant from Newton's per meter to dine's per centimeter for that one you have to multiply by 10 raise to 3 if it is 2 digit number or if it is 4 digit number it should be multiplied by 10 raise to 3 if it is 2 digit number it should be multiplied by 10 raise to 5 for example you take here you can multiply this one by 10 raise to 3 and you can do it. We can do now for a simple one OH let us calculate mu 700 we have so from this one it is 130.3 square root of here f is given 700 by 0.94 so here if you multiply this one simply 27.288 it gives and then this will give you 3555.7 centimeter minus 1. So what we are getting is here 3600 what we are getting is 3555 centimeter minus 1 this looks fine and now let us look into other method if you use this method here we can use it what we should do is we can simplify this the value simplified this one is 3.5559 into 10 raise to 22 after considering square root and then square root of 700 over 0.94 into 1.67377 into 10 raise to 22 and then 10 raise to minus 27 looks like a little bit more complicated but nevertheless it is simple arithmetic we are performing here and this would give you 125.1194 into 10 raise to 5 and then this will be very close to 3537 centimeter minus 1. So this how we can do the calculation here let us look into C C bond here C C bond using this one and 30.3 square root of C C bond we are considering 425 by reduced mass is 6 so this will give you 130.3 into the product is 8.41625 and then multiplication will give you 1096.6 centimeter minus 1. So what we have is 1100 so we have experiment this is calculated and experimental is 1100 like this we can verify for all. So just make yourself familiar by considering either force constant or stretching frequency as unknown entities and use all the three equations to understand these things so that you will not do any mistakes while calculating what you should remember is in this equation nothing you have to do simply you have to consider force constant is Newton's per meter whereas here you have to consider again Newton's per meter no problem but reduced mass after getting it should be multiplied by 1.67377 into 10 raise to minus 27 to convert that into kg in the other equation where we are using 4.12 into square root of F by mu so F you multiply that one into 10 raise to 3 or 10 raise to 5 to make it dines per centimeter. So let me come up with more examples in my next lecture until then have a good time. Thank you.