1 00:00:15,070 --> 00:00:21,370 welcome back to the lectures on molecular spectroscopy and introductory chemistry in 2 00:00:21,370 --> 00:00:28,390 this short lecture let us look at the rotational and vibrational line intensities from an elementary 3 00:00:28,390 --> 00:00:35,580 point of view using the maxwell boltzmann statistics it was mentioned to you some lectures 4 00:00:35,580 --> 00:00:44,390 ago that there are three parameters which are of importance in any ah study using spectroscopy 5 00:00:44,390 --> 00:00:52,399 namely the line positions which are dealt with by quant mechanics the line intensities 6 00:00:52,399 --> 00:00:59,329 which require both quant mechanics and also statistical mechanics since we deal with populations 7 00:00:59,329 --> 00:01:06,490 of molecular system in different energy levels and the third was the line ah width we shall 8 00:01:06,490 --> 00:01:12,850 not consider the line width in this course but let us look at the line intensities here 9 00:01:12,850 --> 00:01:20,180 little more closely and particularly for the rotational microwave and infrared infrared 10 00:01:20,180 --> 00:01:28,780 ah frequency is an transitions ok now the ah basic ah formula or the basic prescription 11 00:01:28,780 --> 00:01:33,920 that you should remember is that molecules the number of molecules in any energy state 12 00:01:33,920 --> 00:01:42,740 means assume that we have n naught n one n two etcetera molecules in different energy 13 00:01:42,740 --> 00:01:56,320 levels e zero e one e two etcetera there is distribution in thermal equilibrium distribution 14 00:01:56,320 --> 00:02:19,260 of molecules in different energy levels due to thermal equilibrium 15 00:02:19,260 --> 00:02:23,520 what we need to know is the boltzmann formula which was also mentioned to you earlier in 16 00:02:23,520 --> 00:02:30,760 the course the boltzmanns man formula tells you that the number of molecules n at any 17 00:02:30,760 --> 00:02:42,730 given energy level e is approximately proportional to the degeneracy of that energy ah often 18 00:02:42,730 --> 00:02:55,630 it written using omega of e and to the boltzmann factor which tells you ah the waiting for 19 00:02:55,630 --> 00:03:08,299 that number for that given energy e by k b t in the case of rotational spectroscopy the 20 00:03:08,299 --> 00:03:17,260 molecular energy levels for the ah simple system of a diatomic they are proportional 21 00:03:17,260 --> 00:03:24,940 to ah the j in to they they are basically j into j plus one you recall the energy level 22 00:03:24,940 --> 00:03:36,660 formula for rotation for a given quantum number j it is b j in to j plus one times h c the 23 00:03:36,660 --> 00:03:47,389 quantum number j corresponds to two j plus one states which are given by different values 24 00:03:47,389 --> 00:03:55,180 of of the projection of j on the molecular axis and these are called the k quantum numbers 25 00:03:55,180 --> 00:04:06,060 and they go from minus j to plus j in integer steps 26 00:04:06,060 --> 00:04:12,120 in the case of diatomic molecule we have the symmetry axis which is the axis of the molecule 27 00:04:12,120 --> 00:04:17,070 about which there is no movement inertia therefore there is no rotational degree of freedom ah 28 00:04:17,070 --> 00:04:23,070 which would show up in that for that motion the other two axis which are perpendicular 29 00:04:23,070 --> 00:04:31,700 to the ah molecular axis the moment of inertia above both of those axis are identical therefore 30 00:04:31,700 --> 00:04:38,570 there is one moment of inertia and the moment of inertia is built in this constant b as 31 00:04:38,570 --> 00:04:46,700 h by eight pi square i c you have already solved such problems therefore what you have 32 00:04:46,700 --> 00:04:53,980 here is are there is only one energy level in the sense one energy value for a given 33 00:04:53,980 --> 00:05:13,320 j so if you write e j its b j in to j plus one h c but there are two j plus one states 34 00:05:13,320 --> 00:05:21,950 which are d generate because they all have the same value and these are indicated by 35 00:05:21,950 --> 00:05:27,540 the wave function the two quantum numbers j and k and the wave functions are usually 36 00:05:27,540 --> 00:05:41,500 written as using the dirac notation ket notation is called the bracket notation 37 00:05:41,500 --> 00:05:54,230 ah what do you see is that the states are represented as the states j k and for molecular 38 00:05:54,230 --> 00:06:00,710 systems where the rotational quantum numbers are integers these j k states can be identified 39 00:06:00,710 --> 00:06:09,920 with specific representation through spherical harmonics 40 00:06:09,920 --> 00:06:19,420 y j k of theta phi which you had come across in the case of hydrogen atom as contributing 41 00:06:19,420 --> 00:06:25,260 to the angular distribution of the wave function therefore the molecular wave functions for 42 00:06:25,260 --> 00:06:32,540 the rotational states or two j plus one whole the degenerate because all the sates starting 43 00:06:32,540 --> 00:06:44,450 from k is equal to j to j j minus one j j minus two and all the wave down to j minus 44 00:06:44,450 --> 00:06:54,730 j all have the same energy in the absence of any external field or in the absence any 45 00:06:54,730 --> 00:07:00,690 other consideration like movements of inertia being different and so on for a diatomic molecule 46 00:07:00,690 --> 00:07:05,530 the true movement of inertia are equal the third movement of inertia is zero therefore 47 00:07:05,530 --> 00:07:12,840 we have all the energy levels for a given j b ah degenerate therefore what does this 48 00:07:12,840 --> 00:07:19,150 tell you this tells you that if you are calculating the molecules the number of molecules in a 49 00:07:19,150 --> 00:07:24,730 given energy state corresponding to the quantum number j energy value corresponding to the 50 00:07:24,730 --> 00:07:32,990 quantum number j and the number of molecules in another energy corresponding to a different 51 00:07:32,990 --> 00:07:43,900 value j prime this tells you that is ratio is two j plus one by two j prime plus one 52 00:07:43,900 --> 00:07:50,460 and then you have e to the exponential to the minus the energy corresponding to the 53 00:07:50,460 --> 00:07:56,930 quantum number j and the energy corresponding to the quantum number j prime the difference 54 00:07:56,930 --> 00:08:03,490 between the two divided by the boltzmanns constant k b t so this is the formula this 55 00:08:03,490 --> 00:08:11,139 is what is called the fundamental formula for the calculation of elementary form of 56 00:08:11,139 --> 00:08:17,830 micro wave intensity without other consideration like ah ah reactions and ah i mean the interaction 57 00:08:17,830 --> 00:08:23,030 between rotation and vibrations and all those things not being considered pure micro wave 58 00:08:23,030 --> 00:08:28,060 transmission of rigid by atomic molecule that number of molecules in any given energy state 59 00:08:28,060 --> 00:08:36,430 j to the number of molecules in another energy state j prime is given by the ratio of the 60 00:08:36,430 --> 00:08:43,899 degeneracy's and the exponential or the boltzmanns man factor given by the energy difference 61 00:08:43,899 --> 00:08:53,570 so what is this if you write that you recall that e j is b j in to j plus one h c and e 62 00:08:53,570 --> 00:09:04,760 j prime s b j prime into j prime plus one h c and if you calculate for example n one 63 00:09:04,760 --> 00:09:13,300 by n zero n one meaning e one with j is equal to verses e with j is equal to zero e zero 64 00:09:13,300 --> 00:09:19,560 the grounds rotational state if you do that then this is three the two j plus one here 65 00:09:19,560 --> 00:09:25,750 is corresponding to the value j is equal to one and this is j is equal to zero therefore 66 00:09:25,750 --> 00:09:30,310 the degeneracy is here is one three by one and what do you have is the energy between 67 00:09:30,310 --> 00:09:42,000 e e one with the difference minus e one minus e not by k b t and that you know is two b 68 00:09:42,000 --> 00:09:53,610 h c therefore what you have is three exponential minus two b h c by k b t ok 69 00:09:53,610 --> 00:10:02,630 now this energy two b h c is very small compared to the factor k b t the thermal energy for 70 00:10:02,630 --> 00:10:11,920 any temperatures like greater than ah say ten kelvin and suddenly t three hundred kelvin 71 00:10:11,920 --> 00:10:23,240 this energy b h c is much smaller than k b t therefore you see that the number of molecules 72 00:10:23,240 --> 00:10:28,700 in the higher energy state to the number of molecules in the energy state in this particular 73 00:10:28,700 --> 00:10:36,260 case is actually that ratio is greater than one or n e one divided by n e not this factor 74 00:10:36,260 --> 00:10:42,390 is almost equal to one or very close to one therefore its is greater than one so what 75 00:10:42,390 --> 00:10:49,050 you see in the micro wave transfusion is since the degeneracy increases as j increases there 76 00:10:49,050 --> 00:10:56,580 is this accommodation of the higher energy state of the system in to many more levels 77 00:10:56,580 --> 00:11:04,870 all of which are degenerate verses the accommodation of the rotational states into that lower j 78 00:11:04,870 --> 00:11:11,490 where the degeneracy's are slightly lower therefore what you see is that the number 79 00:11:11,490 --> 00:11:18,839 of molecules in e one is usually greater than the number of molecules in the energy state 80 00:11:18,839 --> 00:11:24,800 e zero and the number of molecules in the energy state e two is again greater than the 81 00:11:24,800 --> 00:11:34,230 number of molecules in the energy state e one and so on so you can say n e j is greater 82 00:11:34,230 --> 00:11:44,410 than n e j minus one so on until greater than n e zero this is go on like thats forever 83 00:11:44,410 --> 00:11:50,240 no it doesnt because you see the rotational energy levels also increase very very fast 84 00:11:50,240 --> 00:11:57,410 because you see the lowest energy level is zero the next one is two b the next one is 85 00:11:57,410 --> 00:12:07,550 actually six b four b six b and the third is twelve b and this one is twenty b and so 86 00:12:07,550 --> 00:12:15,670 on therefore the energy gap between successive rotational quantum numbers the states of ah 87 00:12:15,670 --> 00:12:21,360 states with successive rotational quantum numbers the gap increases therefore at some 88 00:12:21,360 --> 00:12:30,390 point of time the numbers start decreasing therefore if you plot the micro wave rotational 89 00:12:30,390 --> 00:12:42,779 intensity as a function of j you start seeing for j is equal to say zero one two three etcetera 90 00:12:42,779 --> 00:12:52,490 you start seeing that well lets keep the zero out of picture start with one with respect 91 00:12:52,490 --> 00:12:59,110 to zero this is some number two increases three the j three increases and so on but 92 00:12:59,110 --> 00:13:07,350 after some j when it reaches some maximum it comes down to it starts becoming less and 93 00:13:07,350 --> 00:13:13,330 less and so on and what you see is that the micro wave intensities are like an envelope 94 00:13:13,330 --> 00:13:22,810 bit increasing and there is a maximum for a some value of j 95 00:13:22,810 --> 00:13:29,860 and that can be easily calculated because if you write the ah the number density or 96 00:13:29,860 --> 00:13:39,130 the number for that particular j b in proportional to two j plus one times exponential minus 97 00:13:39,130 --> 00:13:49,649 h c b j into j plus one by k b t because this is ah energy and the the correct unit is the 98 00:13:49,649 --> 00:13:52,750 h c times the b which is only a wave number unit 99 00:13:52,750 --> 00:13:57,940 so if you have this you want to find out what is the value of j for which the number is 100 00:13:57,940 --> 00:14:06,990 maximum you can do a simple calculus by taking the derivative of the number ah numbers with 101 00:14:06,990 --> 00:14:13,080 respect to j and then set that equal to zero you know its an proximate calculation j is 102 00:14:13,080 --> 00:14:18,180 not continuous therefore you know to take the derivative doesnt make much sense but 103 00:14:18,180 --> 00:14:22,920 to get a feel for this the energy levels ah gaps of skill very small compared to thermal 104 00:14:22,920 --> 00:14:27,790 energies therefore if you take a derivative like this you can see that immediately this 105 00:14:27,790 --> 00:14:33,721 gives you the following that the number density gives you the derivatives you take the derivatives 106 00:14:33,721 --> 00:14:43,350 of this it is two times e to the minus h c b j into j plus one by k b t thats the first 107 00:14:43,350 --> 00:14:49,230 term of the derivative the second is the derivative of this term which if you take it is minus 108 00:14:49,230 --> 00:15:00,040 two j plus one and this will give you the ah expression e to the minus h c b j in to 109 00:15:00,040 --> 00:15:09,850 j plus one by k b t times this derivative which is one by k b t and the minus sign is 110 00:15:09,850 --> 00:15:17,170 already here it will give you h c b in to two j plus one this is what you will get and 111 00:15:17,170 --> 00:15:28,820 if you set that equal to zero its easy to show that the j max is approximately square 112 00:15:28,820 --> 00:15:45,550 root of k b t by two h c b minus one half we call that this is a if you set this equal 113 00:15:45,550 --> 00:15:51,279 to zero the exponential factor goes away so its ah you can see immediately that it is 114 00:15:51,279 --> 00:16:03,339 two minus two j plus one whole square in to h c b by k b t thats equal to zero and that 115 00:16:03,339 --> 00:16:09,820 gives you the solution ok thats the j max the corresponding value of j is the j max 116 00:16:09,820 --> 00:16:14,350 and you can see that this is the ratio therefore microwave intensity is actually 117 00:16:14,350 --> 00:16:22,160 peak for some middle value of ray verses ah the starting from j is equal to zero to some 118 00:16:22,160 --> 00:16:29,750 other value j what about the infrared intensity for a diatomic molecule there is no problem 119 00:16:29,750 --> 00:16:37,640 because all vibrational states are non degenerate single vibrational states ah the degeneracy 120 00:16:37,640 --> 00:16:47,500 is one therefore in the case of vibrational motion if you write just n v ah for n v prime 121 00:16:47,500 --> 00:16:59,240 where v and v prime are the vibrational quantum numbers for two different states for states 122 00:16:59,240 --> 00:17:06,100 with ah two different states and then you can see that this is nothing other than simply 123 00:17:06,100 --> 00:17:15,819 exponential minus e v minus e v prime by k b t therefore there is no factor in front 124 00:17:15,819 --> 00:17:21,730 of the exponential which counters the decrease of the exponential you can see that vibrational 125 00:17:21,730 --> 00:17:28,919 intensities usually or ah maximum for the v is equal to zero to one transition or the 126 00:17:28,919 --> 00:17:36,870 v is equal to zero state itself and then its slightly lower and so on and lower and so 127 00:17:36,870 --> 00:17:42,600 on secondly with respect to microwaves and rotation spectroscopy the energy gap is very 128 00:17:42,600 --> 00:17:48,059 small compared to the thermal energy whereas in the case of vibrational states the energy 129 00:17:48,059 --> 00:17:54,460 gap is comparable or even more than thermal energy therefore the number of molecules even 130 00:17:54,460 --> 00:18:00,960 in the first excited vibrational state is growing to be a lots smaller fraction of the 131 00:18:00,960 --> 00:18:05,230 number of molecules in the ground state this is not so in the case of rotations in the 132 00:18:05,230 --> 00:18:10,559 case of rotation a large number of j values are populated fairly well whereas in the case 133 00:18:10,559 --> 00:18:15,730 of vibration unless the vibrational energies are very close to each other most molecules 134 00:18:15,730 --> 00:18:20,759 will be at any given temperature in the ground state and few were in the first exited state 135 00:18:20,759 --> 00:18:25,769 and even few were in the second exited state therefore you can see the drop in the intensity 136 00:18:25,769 --> 00:18:30,489 very very directly so these are things that you have to keep in mind in observing the 137 00:18:30,489 --> 00:18:35,710 intensity and also some of the spectra that i will show in the in one of these lectures 138 00:18:35,710 --> 00:18:40,820 or through the lecture notes we will continue this with the spectroscopy of the micro wave 139 00:18:40,820 --> 00:18:44,470 spectroscopy of poly atomic molecules in the next lecture until then 140 00:18:44,470 --> 00:18:44,879 thank you very much