1 00:00:00,110 --> 00:00:02,410 The following content is provided under a Creative 2 00:00:02,410 --> 00:00:03,820 Commons license. 3 00:00:03,820 --> 00:00:06,850 Your support will help MIT OpenCourseWare continue to 4 00:00:06,850 --> 00:00:10,520 offer high quality educational resources for free. 5 00:00:10,520 --> 00:00:13,390 To make a donation or view additional materials from 6 00:00:13,390 --> 00:00:17,590 hundreds of MIT courses, visit MIT OpenCourseWare at 7 00:00:17,590 --> 00:00:20,100 ocw.mit.edu. 8 00:00:20,100 --> 00:00:27,190 PROFESSOR: So, let me start here with the temperature 9 00:00:27,190 --> 00:00:37,400 dependence of k. 10 00:00:37,400 --> 00:00:39,800 And this turns out to be extremely important. 11 00:00:39,800 --> 00:00:52,560 And it's due to Arrhenius in 1889. 12 00:00:52,560 --> 00:00:56,980 And Mr. Arrhenius is famous for many reasons. 13 00:00:56,980 --> 00:00:59,130 Not just for his rate law. 14 00:00:59,130 --> 00:01:01,900 So I learned recently, turns out that he was also one of 15 00:01:01,900 --> 00:01:05,900 the first people to calculate the potential effect of carbon 16 00:01:05,900 --> 00:01:07,950 dioxide in the atmosphere. 17 00:01:07,950 --> 00:01:12,810 And and its potential effect on global warming. 18 00:01:12,810 --> 00:01:16,000 This was the Industrial Revolution and people were 19 00:01:16,000 --> 00:01:20,890 starting to use fossil fuels at increasing amounts. 20 00:01:20,890 --> 00:01:25,720 And back then, already, some people started to get worried. 21 00:01:25,720 --> 00:01:27,130 And so he did the calculation. 22 00:01:27,130 --> 00:01:29,890 It was a very crude calculation. 23 00:01:29,890 --> 00:01:34,590 And he extrapolated, assuming that the rate to fossil fuel 24 00:01:34,590 --> 00:01:37,670 use would keep going, he got his 25 00:01:37,670 --> 00:01:40,020 calculation pretty much right. 26 00:01:40,020 --> 00:01:42,610 What he didn't get right was the amount of fossil fuel that 27 00:01:42,610 --> 00:01:44,430 people would be using. 28 00:01:44,430 --> 00:01:47,230 And so when he did his calculation, which is again a 29 00:01:47,230 --> 00:01:50,560 very crude calculation, he got that we would get in trouble 30 00:01:50,560 --> 00:01:54,520 at about 2,000 years from the time of his calculation. 31 00:01:54,520 --> 00:01:56,830 So he said, no problem, 2,000 years, we've 32 00:01:56,830 --> 00:02:01,290 got a lot of time. 33 00:02:01,290 --> 00:02:04,130 And ever since then people have redone these calculations 34 00:02:04,130 --> 00:02:10,150 and more and more sophisticated. 35 00:02:10,150 --> 00:02:11,180 But his crude calculation was good enough. 36 00:02:11,180 --> 00:02:13,680 And as people redo the calculations, the time that 37 00:02:13,680 --> 00:02:16,050 they say we're in trouble to the time of the calculation 38 00:02:16,050 --> 00:02:17,190 gets closer and closer together. 39 00:02:17,190 --> 00:02:19,400 And the reason for that isn't that their calculations are 40 00:02:19,400 --> 00:02:21,810 wrong, it's that the rate at which we put out carbon 41 00:02:21,810 --> 00:02:26,520 dioxide just keeps getting faster and faster and faster. 42 00:02:26,520 --> 00:02:29,250 So it's interesting to go back to these calculations through 43 00:02:29,250 --> 00:02:37,760 the last century and a half and see that. 44 00:02:37,760 --> 00:02:42,360 Anyway, so Mr. Arrhenius predicted global warming. 45 00:02:42,360 --> 00:02:46,860 And he wrote his rate equation, k is equal to e to 46 00:02:46,860 --> 00:02:51,400 the minus Ea over RT. 47 00:02:51,400 --> 00:02:55,720 Famous Arrhenius rate equation. 48 00:02:55,720 --> 00:03:02,800 If you plot it as log of k is equal to log A minus Ea over 49 00:03:02,800 --> 00:03:06,920 RT, you find that it looks like a straight line. 50 00:03:06,920 --> 00:03:12,210 With an intercept at log A, where A is the pre-factor here 51 00:03:12,210 --> 00:03:16,100 and Ea is going to be the activation energy. 52 00:03:16,100 --> 00:03:21,530 And the slope is Ea, minus Ea over RT. 53 00:03:21,530 --> 00:03:23,460 R is the gas constant. 54 00:03:23,460 --> 00:03:26,490 T is the temperature. 55 00:03:26,490 --> 00:03:29,930 And if you put in some typical numbers, activation energies 56 00:03:29,930 --> 00:03:35,160 typically are on the order of a few tens to hundreds of 57 00:03:35,160 --> 00:03:36,820 kilojoules per mole. 58 00:03:36,820 --> 00:03:44,130 Let's say 50 to 300 kilojoules per mole. 59 00:03:44,130 --> 00:03:48,110 And typical activation factors, typical 60 00:03:48,110 --> 00:03:53,470 pre-exponential factors, are depending if it's a first 61 00:03:53,470 --> 00:03:54,220 order or second order. 62 00:03:54,220 --> 00:03:57,720 Let's say, first order 10 to the 12th, 10 to the 15th, per 63 00:03:57,720 --> 00:04:03,260 second, so the A carries the units for the rate here. 64 00:04:03,260 --> 00:04:08,210 So if you have A, so this is for first order, if you have a 65 00:04:08,210 --> 00:04:11,250 second order, then the units of A will be different. 66 00:04:11,250 --> 00:04:12,890 And typically they're going to be on the order of 10 to the 67 00:04:12,890 --> 00:04:16,890 11th or so, 10 to the 10th, 10 to the 12th, one over molar 68 00:04:16,890 --> 00:04:23,870 per second, for second order. 69 00:04:23,870 --> 00:04:30,080 And it's interesting to do sort of a back of the envelope 70 00:04:30,080 --> 00:04:42,420 example to figure out how important this rate is. 71 00:04:42,420 --> 00:04:48,980 And one of the interesting things to look at is, suppose 72 00:04:48,980 --> 00:04:52,430 that I'm at some temperature and I want to know how do I 73 00:04:52,430 --> 00:04:54,610 change the temperature. 74 00:04:54,610 --> 00:04:56,170 How much do I need to change the temperature 75 00:04:56,170 --> 00:04:58,460 to double the rate. 76 00:04:58,460 --> 00:05:01,860 And let's say my temperature's at, say, T1 is on the order of 77 00:05:01,860 --> 00:05:06,070 300 degrees Kelvin, room temperature. 78 00:05:06,070 --> 00:05:10,020 Room temperature, and let's take a typical Ea on the order 79 00:05:10,020 --> 00:05:15,800 of 100 kilojoules per mole. 80 00:05:15,800 --> 00:05:17,990 Room temperature, or body temperature are roughly the 81 00:05:17,990 --> 00:05:19,920 same, right? 82 00:05:19,920 --> 00:05:23,060 These things are. 83 00:05:23,060 --> 00:05:28,320 And so the question is, what does T2 need to be to double 84 00:05:28,320 --> 00:05:31,470 the rate connected with an activation energy which is 85 00:05:31,470 --> 00:05:33,790 fairly typical of 100 kilojoules per mole. 86 00:05:33,790 --> 00:05:37,370 So we want to know, what is, if I have k2 over k1 is equal 87 00:05:37,370 --> 00:05:40,970 to two, what's the new T2. 88 00:05:40,970 --> 00:05:46,780 Alright so A e to the minus Ea over R T2 divided by A e to 89 00:05:46,780 --> 00:05:51,250 the minus Ea over R T1, we want that equal to two, the 90 00:05:51,250 --> 00:05:55,660 A's cancel out. 91 00:05:55,660 --> 00:06:02,590 So log k2 over k1, which is equal to log two, is equal to 92 00:06:02,590 --> 00:06:07,880 Ea over R, one over T1 minus one over T2. 93 00:06:07,880 --> 00:06:12,520 We know what this is, we know what this is, we solve for T2 94 00:06:12,520 --> 00:06:19,380 and we find that T2 is 305 degrees kelvin. 95 00:06:19,380 --> 00:06:25,560 It's a pretty small change to double the rate. 96 00:06:25,560 --> 00:06:27,670 So it's pretty important that your body temperature doesn't 97 00:06:27,670 --> 00:06:30,760 change very much. 98 00:06:30,760 --> 00:06:36,110 If you have a fever and your temperature goes up, rates 99 00:06:36,110 --> 00:06:36,700 start going up. 100 00:06:36,700 --> 00:06:42,710 In your body, you could cause a lot of problems if the rates 101 00:06:42,710 --> 00:06:50,180 of some reactions go up by a factor of two. 102 00:06:50,180 --> 00:06:52,640 So, very sensitive to temperature. 103 00:06:52,640 --> 00:06:57,890 Rates are very sensitive to temperature. 104 00:06:57,890 --> 00:07:00,790 What are these things physically? 105 00:07:00,790 --> 00:07:03,840 Ea and and the pre-exponent factor. 106 00:07:03,840 --> 00:07:04,970 Let's take a look at that. 107 00:07:04,970 --> 00:07:09,380 So physically what's going on is, you have your two, let's 108 00:07:09,380 --> 00:07:17,890 say it's a bimolecular reaction, A plus B goes to C. 109 00:07:17,890 --> 00:07:20,190 So mechanistically what we think is happening is that the 110 00:07:20,190 --> 00:07:22,080 molecules come together and collide, right? 111 00:07:22,080 --> 00:07:29,880 You have A and B getting together and colliding. 112 00:07:29,880 --> 00:07:32,750 And the hypothesis is that when they 113 00:07:32,750 --> 00:07:34,680 collide they form a complex. 114 00:07:34,680 --> 00:07:37,600 They sort of bind together momentarily. 115 00:07:37,600 --> 00:07:41,120 And form this complex that has all the kinetic energy, or 116 00:07:41,120 --> 00:07:44,940 part of the kinetic energy of this collision as part of it. 117 00:07:44,940 --> 00:07:47,740 So A and B are bound together in some 118 00:07:47,740 --> 00:07:50,410 highly excited complex. 119 00:07:50,410 --> 00:07:55,380 Which then falls apart to give the product by rearranging its 120 00:07:55,380 --> 00:08:00,670 atoms around. 121 00:08:00,670 --> 00:08:13,060 And if you plot the energy of this process as a function of 122 00:08:13,060 --> 00:08:28,690 the reaction, we call this the reaction coordinate, then 123 00:08:28,690 --> 00:08:34,040 we're going to start at some delta G, or some energy. 124 00:08:34,040 --> 00:08:40,440 For the reactants. 125 00:08:40,440 --> 00:08:42,920 We're going to end up with some energy for the products. 126 00:08:42,920 --> 00:08:44,240 And it could be higher or lower. 127 00:08:44,240 --> 00:08:46,260 Let me make them higher here, just because I don't have 128 00:08:46,260 --> 00:08:47,870 enough room on the board. 129 00:08:47,870 --> 00:08:51,620 Products. 130 00:08:51,620 --> 00:08:56,340 So if this is an endothermic reaction, that's the energy 131 00:08:56,340 --> 00:08:59,660 for C, this is the energy for A plus B, and then along the 132 00:08:59,660 --> 00:09:05,470 way we have this complex that captures some of the energy of 133 00:09:05,470 --> 00:09:06,970 the collision. 134 00:09:06,970 --> 00:09:10,650 Up here somewhere. 135 00:09:10,650 --> 00:09:18,320 And this is the energy, then, at A B star, which we call the 136 00:09:18,320 --> 00:09:25,310 activated complex. 137 00:09:25,310 --> 00:09:28,180 So for the reaction to happen, then, we have to go over the 138 00:09:28,180 --> 00:09:31,920 barrier and then back to the product. 139 00:09:31,920 --> 00:09:38,400 And this distance from the energy of the reactants to the 140 00:09:38,400 --> 00:09:41,500 top of the barrier, that's Ea. 141 00:09:41,500 --> 00:09:42,970 That's the energy of activation. 142 00:09:42,970 --> 00:09:45,410 How much extra energy you have to put in there 143 00:09:45,410 --> 00:09:47,310 to go over the barrier. 144 00:09:47,310 --> 00:09:49,790 So it's very clear that as you increase the temperature and 145 00:09:49,790 --> 00:09:53,530 you increase the amount of energy that the reactants 146 00:09:53,530 --> 00:09:58,010 thermally have, or in terms of their kinetic energy, as you 147 00:09:58,010 --> 00:10:00,040 raise the temperature you get more and more kinetic energy, 148 00:10:00,040 --> 00:10:02,510 you're going up higher and higher in the Boltzmann 149 00:10:02,510 --> 00:10:03,820 distribution. 150 00:10:03,820 --> 00:10:07,420 And the number of reactants that can make it over the 151 00:10:07,420 --> 00:10:10,330 barrier clearly goes up exponentially. 152 00:10:10,330 --> 00:10:14,700 As the temperature goes up. 153 00:10:14,700 --> 00:10:17,870 So this is Ea for the forward reaction. 154 00:10:17,870 --> 00:10:22,520 Clearly there's going to be an equivalent activation energy 155 00:10:22,520 --> 00:10:23,890 for the backward reaction. 156 00:10:23,890 --> 00:10:26,390 The two are related. 157 00:10:26,390 --> 00:10:29,640 The difference between the forward and backward 158 00:10:29,640 --> 00:10:32,970 reactions, so E backwards minus Ea forward gives you the 159 00:10:32,970 --> 00:10:37,330 delta G for the reaction itself. 160 00:10:37,330 --> 00:10:46,040 So, Ea backwards minus Ea forwards gives you delta E for 161 00:10:46,040 --> 00:10:49,130 the reaction. 162 00:10:49,130 --> 00:10:55,220 Where E could be enthalpy or it could be free energy. 163 00:10:55,220 --> 00:10:58,260 So that's the physical origin of this Ea, 164 00:10:58,260 --> 00:11:00,980 this activation energy. 165 00:11:00,980 --> 00:11:05,740 Now, A, the pre-exponential factor, that's the rate of 166 00:11:05,740 --> 00:11:11,770 attempt for the molecules to try to go over the barrier. 167 00:11:11,770 --> 00:11:27,170 So it has units that make sense for that. 168 00:11:27,170 --> 00:11:29,070 And this A is different than this molecule here. 169 00:11:29,070 --> 00:11:33,430 Rate of attempt. 170 00:11:33,430 --> 00:11:37,110 How many times per unit time, or how many times per second, 171 00:11:37,110 --> 00:11:40,080 do these two molecules try to collide? 172 00:11:40,080 --> 00:11:43,050 They never collide, they never try, they'll 173 00:11:43,050 --> 00:11:47,720 never make it over. 174 00:11:47,720 --> 00:11:58,260 So it's one over second, or one over mole second. 175 00:11:58,260 --> 00:12:01,790 So this is the rate of attempt. 176 00:12:01,790 --> 00:12:14,580 Then e to the minus Ea over RT is the probability of success, 177 00:12:14,580 --> 00:12:18,470 of making it over the barrier. 178 00:12:18,470 --> 00:12:21,480 So the rate of attempt times the probability of success 179 00:12:21,480 --> 00:12:24,430 gives you the rate per unit time of 180 00:12:24,430 --> 00:12:31,920 making it over the barrier. 181 00:12:31,920 --> 00:12:35,420 Any questions on the dissection of this 182 00:12:35,420 --> 00:12:40,090 Arrhenius rate law? 183 00:12:40,090 --> 00:12:45,380 Alright, so here's some examples, a few examples that 184 00:12:45,380 --> 00:12:46,480 you know of. 185 00:12:46,480 --> 00:12:57,310 Let's look at OH minus plus methyl bromide. 186 00:12:57,310 --> 00:13:01,390 Displacement reaction going to this OH minus attacks the 187 00:13:01,390 --> 00:13:03,490 carbon right here. 188 00:13:03,490 --> 00:13:11,350 To form an activated complex H, H, H, with the OH 189 00:13:11,350 --> 00:13:12,760 coming in like this. 190 00:13:12,760 --> 00:13:16,840 Let me go like this. 191 00:13:16,840 --> 00:13:19,030 OK, there's the intermediate right here. 192 00:13:19,030 --> 00:13:23,950 The activated complex in your reaction. 193 00:13:23,950 --> 00:13:25,100 Which then falls apart. 194 00:13:25,100 --> 00:13:26,390 It can fall apart two ways. 195 00:13:26,390 --> 00:13:30,560 The OH can be spit out again, forming back to the reactants, 196 00:13:30,560 --> 00:13:34,310 or the bromine can be spit out, forming the product. 197 00:13:34,310 --> 00:13:41,220 Which in this case here is methanol. 198 00:13:41,220 --> 00:13:48,030 H, H, H, OH, plus Br minus. 199 00:13:48,030 --> 00:13:51,150 Typical example. 200 00:13:51,150 --> 00:13:56,440 So, given this picture here, which you're probably already 201 00:13:56,440 --> 00:14:03,350 somewhat familiar with, we can then move on to talk about how 202 00:14:03,350 --> 00:14:04,620 to affect this barrier. 203 00:14:04,620 --> 00:14:07,320 How to change the rate. 204 00:14:07,320 --> 00:14:08,910 Oh, I should say a couple more things. 205 00:14:08,910 --> 00:14:10,390 One more thing, too. 206 00:14:10,390 --> 00:14:15,180 Now, the rate of, so clearly the Ea's are related. 207 00:14:15,180 --> 00:14:19,010 Because the difference of the two Ea's is the delta E for 208 00:14:19,010 --> 00:14:21,030 the reaction. 209 00:14:21,030 --> 00:14:25,210 But the pre-exponential factors, these A's, for the 210 00:14:25,210 --> 00:14:27,530 forward reaction and the backward reactions aren't 211 00:14:27,530 --> 00:14:29,490 necessarily related. 212 00:14:29,490 --> 00:14:31,720 You can imagine that in one case you have, first of all 213 00:14:31,720 --> 00:14:34,830 they don't even have to have the same units. 214 00:14:34,830 --> 00:14:37,610 You could have a bimolecular reaction one way. 215 00:14:37,610 --> 00:14:42,430 And a unimolecular reaction the other way. 216 00:14:42,430 --> 00:14:45,790 So you can't really say anything about the forward 217 00:14:45,790 --> 00:14:49,880 versus the backward rate this way. 218 00:14:49,880 --> 00:14:54,430 All you know is about the energies. 219 00:14:54,430 --> 00:15:01,950 OK, catalysis. 220 00:15:01,950 --> 00:15:05,560 So now we have this barrier we have to overcome. 221 00:15:05,560 --> 00:15:14,550 And, so suppose that you have two motivators. 222 00:15:14,550 --> 00:15:15,680 Suppose you have an equilibrium 223 00:15:15,680 --> 00:15:17,460 case which is slow. 224 00:15:17,460 --> 00:15:19,210 And a reaction which is slow in both 225 00:15:19,210 --> 00:15:23,580 directions, k1, k minus one. 226 00:15:23,580 --> 00:15:25,710 And you don't want to wait for this to happen. 227 00:15:25,710 --> 00:15:28,720 And this could be in a biological environment. 228 00:15:28,720 --> 00:15:30,100 Or it could be an industrial process, 229 00:15:30,100 --> 00:15:33,370 like the Haber process. 230 00:15:33,370 --> 00:15:34,620 You don't want to wait. 231 00:15:34,620 --> 00:15:36,590 And so one of the ways that you can speed it up, you know 232 00:15:36,590 --> 00:15:38,960 is by changing the temperature. 233 00:15:38,960 --> 00:15:41,190 You change the temperature, the rate goes up. 234 00:15:41,190 --> 00:15:43,590 You change it by five degrees, the rate goes up 235 00:15:43,590 --> 00:15:44,690 by a factor of two. 236 00:15:44,690 --> 00:15:47,790 You can make a huge change by changing the temperature. 237 00:15:47,790 --> 00:15:56,220 But, so if you raise T, speeds up. 238 00:15:56,220 --> 00:16:02,750 But, K equilibrium also changes. 239 00:16:02,750 --> 00:16:04,950 And we saw that when we did the Haber process. 240 00:16:04,950 --> 00:16:08,020 We raised the temperature. 241 00:16:08,020 --> 00:16:11,110 The rate speeds up, but the equilibrium 242 00:16:11,110 --> 00:16:12,610 switches to the reactants. 243 00:16:12,610 --> 00:16:15,970 That's no good. 244 00:16:15,970 --> 00:16:18,090 So changing the temperature is not always the best thing to 245 00:16:18,090 --> 00:16:21,900 do if you want to change the rate. 246 00:16:21,900 --> 00:16:26,990 Instead, what you can do is use a catalyst. 247 00:16:26,990 --> 00:16:30,330 if you find one. 248 00:16:30,330 --> 00:16:38,240 A molecule that reacts with your reactant to form a 249 00:16:38,240 --> 00:16:42,900 product, that's B, spitting back that molecule C again 250 00:16:42,900 --> 00:16:45,410 without using it up. 251 00:16:45,410 --> 00:16:49,450 And now with the activation energy, we can understand what 252 00:16:49,450 --> 00:16:50,960 the catalyst does. 253 00:16:50,960 --> 00:16:55,820 What the catalyst does is speeds up those rates, k2 and 254 00:16:55,820 --> 00:17:02,110 k minus one, by changing the activation energy. 255 00:17:02,110 --> 00:17:04,590 By making the E smaller. 256 00:17:04,590 --> 00:17:08,630 So now I have a catalyst, and I can 257 00:17:08,630 --> 00:17:10,690 make this energy smaller. 258 00:17:10,690 --> 00:17:26,270 So this would be A plus A C activated, getting ready to 259 00:17:26,270 --> 00:17:27,630 spit out B. 260 00:17:27,630 --> 00:17:31,480 So in my example here where I have A plus B goes to 261 00:17:31,480 --> 00:17:37,000 products, so I would have A, some combination of A, B, and 262 00:17:37,000 --> 00:17:42,150 C together, to give out the products. 263 00:17:42,150 --> 00:17:45,280 In this case, the difference in energies doesn't change. 264 00:17:45,280 --> 00:17:51,850 All that you're changing is that the hump, in both ways. 265 00:17:51,850 --> 00:17:53,230 Equilibrium constant doesn't change. 266 00:17:53,230 --> 00:17:58,190 Just the rate changes through the Arrhenius rate law. 267 00:17:58,190 --> 00:18:00,470 And this is extremely powerful. 268 00:18:00,470 --> 00:18:04,330 Especially in biology. 269 00:18:04,330 --> 00:18:10,300 So let me give you some examples here. 270 00:18:10,300 --> 00:18:16,580 This is sort of a typical example of increasing rates. 271 00:18:16,580 --> 00:18:18,910 Let's say you have the reaction, hydrogen peroxide, 272 00:18:18,910 --> 00:18:28,270 goes to water plus oxygen. 273 00:18:28,270 --> 00:18:31,530 If you take a little bit of hydrogen peroxide and you put 274 00:18:31,530 --> 00:18:33,060 it on your skin, it starts to bubble. 275 00:18:33,060 --> 00:18:37,660 But if you let it sit on the bottle nothing happens to it. 276 00:18:37,660 --> 00:18:40,890 Put it on your hair, your hair turns white. 277 00:18:40,890 --> 00:18:43,500 Blond. 278 00:18:43,500 --> 00:18:45,670 But again, if you just let it sit in the 279 00:18:45,670 --> 00:18:47,400 bottle very little happens. 280 00:18:47,400 --> 00:18:51,570 Well, it happens but very, very slowly over time. 281 00:18:51,570 --> 00:18:56,690 So if you look at the rate of this reaction here, if the 282 00:18:56,690 --> 00:19:05,700 rate, moles per second, with no catalyst at all, the rate 283 00:19:05,700 --> 00:19:10,580 is 10 to the minus 8 molar per second. 284 00:19:10,580 --> 00:19:12,570 Which is very slow. 285 00:19:12,570 --> 00:19:21,740 And the activation energy in kilojoules per mole, in this 286 00:19:21,740 --> 00:19:25,720 case here is 71 kilojoules per mole. 287 00:19:25,720 --> 00:19:27,110 Now we can start adding catalysts. 288 00:19:27,110 --> 00:19:29,730 We can start adding inorganic catalysts 289 00:19:29,730 --> 00:19:31,850 like hydrogen bromide. 290 00:19:31,850 --> 00:19:35,710 Increase the rate at 10 to the minus 4. 291 00:19:35,710 --> 00:19:40,320 So this creates a complex with the hydrogen peroxide, which 292 00:19:40,320 --> 00:19:45,910 lowers the barrier to 50 kilojoules per mole, a small 293 00:19:45,910 --> 00:19:46,720 amount of lowering. 294 00:19:46,720 --> 00:19:51,740 But because the energy is in the exponent there, it makes a 295 00:19:51,740 --> 00:19:52,860 big change in the rate. 296 00:19:52,860 --> 00:19:53,180 Yes. 297 00:19:53,180 --> 00:20:01,270 STUDENT: [INAUDIBLE] 298 00:20:01,270 --> 00:20:03,420 PROFESSOR: The rate is independent of the catalyst 299 00:20:03,420 --> 00:20:04,700 concentration. 300 00:20:04,700 --> 00:20:05,300 STUDENT: [INAUDIBLE] 301 00:20:05,300 --> 00:20:09,980 PROFESSOR: The rate would have to change. 302 00:20:09,980 --> 00:20:19,070 You're right. 303 00:20:19,070 --> 00:20:24,530 That's a good question and I'm not prepared to answer it. 304 00:20:24,530 --> 00:20:27,890 I'm going to have to think about this. 305 00:20:27,890 --> 00:20:31,840 Let's pretend now that we are at per unit concentration of 306 00:20:31,840 --> 00:20:33,550 the catalyst. 307 00:20:33,550 --> 00:20:37,410 And then, and I'll look into it. 308 00:20:37,410 --> 00:20:40,700 OK, so this is what happens with an inorganic catalyst. 309 00:20:40,700 --> 00:20:46,890 And instead if you use a generic biological catalyst, 310 00:20:46,890 --> 00:20:50,690 an enzyme catalase, it's a sort ubiquitous enzyme which 311 00:20:50,690 --> 00:20:53,510 is why you have it on your skin, et cetera, then this 312 00:20:53,510 --> 00:20:57,680 rate become 10 to the 7th. 313 00:20:57,680 --> 00:21:05,170 And the activation energy drops to eight 314 00:21:05,170 --> 00:21:08,420 kilojoules per mole. 315 00:21:08,420 --> 00:21:12,170 So your question really has to do with the units of A. Of the 316 00:21:12,170 --> 00:21:17,030 pre-factor right there. 317 00:21:17,030 --> 00:21:21,940 OK, and so there are all these examples of reactions. 318 00:21:21,940 --> 00:21:24,430 That are very important biologically. 319 00:21:24,430 --> 00:21:28,010 Where with an inorganic catalyst, an organic catalyst, 320 00:21:28,010 --> 00:21:30,780 you can change the rate by maybe a 321 00:21:30,780 --> 00:21:32,340 few orders of magnitude. 322 00:21:32,340 --> 00:21:35,610 But as soon as you put in an enzyme, the rate changes by 323 00:21:35,610 --> 00:21:37,060 ten orders of magnitude. 324 00:21:37,060 --> 00:21:40,050 Or in this case eight plus seven, 325 00:21:40,050 --> 00:21:42,100 fifteen orders of magnitude. 326 00:21:42,100 --> 00:21:46,500 Humongous change in the rate. 327 00:21:46,500 --> 00:21:50,080 So you've probably done some enzyme catalysis before. 328 00:21:50,080 --> 00:22:04,740 But it's probably a good idea to quickly do it again. 329 00:22:04,740 --> 00:22:07,010 Because it's just so important. 330 00:22:07,010 --> 00:22:11,430 And it ties together our approximations that we've 331 00:22:11,430 --> 00:22:15,510 learned about. 332 00:22:15,510 --> 00:22:19,990 So enzyme catalysis, so enzymes can be either, could 333 00:22:19,990 --> 00:22:22,700 be heterogeneous catalysis, can be homogeneous catalysis. 334 00:22:22,700 --> 00:22:25,960 Enzymes are ubiquitous in the biological environment. 335 00:22:25,960 --> 00:22:31,450 They serve to regulate the cellular activity in very 336 00:22:31,450 --> 00:22:35,550 complicated ways. 337 00:22:35,550 --> 00:22:40,050 The cell will up-regulate or down-regulate the 338 00:22:40,050 --> 00:22:43,780 concentration of enzymes as it needs to make 339 00:22:43,780 --> 00:22:45,440 more or less products. 340 00:22:45,440 --> 00:22:47,470 And there's whole cascades of events that 341 00:22:47,470 --> 00:22:52,040 happen in this way. 342 00:22:52,040 --> 00:22:57,500 And they're in very small concentrations. 343 00:22:57,500 --> 00:22:59,560 But they play an extremely important role. 344 00:22:59,560 --> 00:23:01,190 Because they're also extremely specific. 345 00:23:01,190 --> 00:23:06,740 So you can have an enzyme that will only act on one part of 346 00:23:06,740 --> 00:23:09,070 the biological cycle. 347 00:23:09,070 --> 00:23:11,590 And affect it by 10 orders of magnitude or 348 00:23:11,590 --> 00:23:12,590 15 orders of magnitude. 349 00:23:12,590 --> 00:23:16,960 But not affect any other protein that's around. 350 00:23:16,960 --> 00:23:20,540 And that's amazing. 351 00:23:20,540 --> 00:23:29,790 And it turns out that a lot of the diseases of old age like, 352 00:23:29,790 --> 00:23:34,920 what I'm about to face, or am facing already, you, not yet, 353 00:23:34,920 --> 00:23:44,360 but, are the result of some of these enzyme up-regulation, 354 00:23:44,360 --> 00:23:46,250 down-regulation, beginning to break down. 355 00:23:46,250 --> 00:23:52,560 So we have all these reactions going on that need to be 356 00:23:52,560 --> 00:23:53,840 essentially perfect. 357 00:23:53,840 --> 00:23:58,330 When you have DNA replication or protein folding, 358 00:23:58,330 --> 00:24:01,240 or things like this. 359 00:24:01,240 --> 00:24:02,500 And errors are made. 360 00:24:02,500 --> 00:24:06,280 Errors are made all the time in these processes. 361 00:24:06,280 --> 00:24:09,770 And errors cause diseases. 362 00:24:09,770 --> 00:24:15,160 And so, even as a baby, your biological 363 00:24:15,160 --> 00:24:17,120 processes make errors. 364 00:24:17,120 --> 00:24:20,260 But you have these processes, these enzymes especially based 365 00:24:20,260 --> 00:24:25,060 on enzymes, that can go in there and sort of fix things. 366 00:24:25,060 --> 00:24:29,350 Fix things and make sure that you don't end up getting 367 00:24:29,350 --> 00:24:33,260 Alzheimer's at age six months. 368 00:24:33,260 --> 00:24:40,360 But as we get older, for some reason, these repair processes 369 00:24:40,360 --> 00:24:41,740 lose their bearings. 370 00:24:41,740 --> 00:24:43,850 Just like we lose our bearings. 371 00:24:43,850 --> 00:24:48,090 And and they can't repair things any more. 372 00:24:48,090 --> 00:24:50,460 They don't do it very well. 373 00:24:50,460 --> 00:24:54,320 And that causes all sorts of diseases. 374 00:24:54,320 --> 00:24:57,380 Cancer is probably one of the diseases, due to the lack of 375 00:24:57,380 --> 00:24:59,270 being able to repair problems. 376 00:24:59,270 --> 00:25:00,000 Alzheimer's. 377 00:25:00,000 --> 00:25:02,990 All sorts of dementias. 378 00:25:02,990 --> 00:25:06,620 MS. Just name a chronic disease and it's probably due 379 00:25:06,620 --> 00:25:09,170 to a problem with up-regulation or 380 00:25:09,170 --> 00:25:12,315 down-regulation of some proteins, some enzymes, that 381 00:25:12,315 --> 00:25:17,550 are due to a repair process. 382 00:25:17,550 --> 00:25:25,360 So anyway, these enzymes are big proteins. 383 00:25:25,360 --> 00:25:29,090 10 to the 4th, 10 to the 6th molecular weight proteins. 384 00:25:29,090 --> 00:25:30,810 On that order or so. 385 00:25:30,810 --> 00:25:35,310 They tend to be fairly large in size. 386 00:25:35,310 --> 00:25:36,060 On the order of nanometers. 387 00:25:36,060 --> 00:25:39,570 Let's say ten nanometers to 100 nanometers. 388 00:25:39,570 --> 00:25:41,620 That could be, that's a little bit on the big side. 389 00:25:41,620 --> 00:25:44,590 Probably closer to ten nanometers. 390 00:25:44,590 --> 00:25:46,180 Ten nanometers is kind of big, for any sort 391 00:25:46,180 --> 00:25:51,860 of biological molecule. 392 00:25:51,860 --> 00:25:54,750 And they always end with their name ase. 393 00:25:54,750 --> 00:26:00,810 Like catalase, uriase, rnase. 394 00:26:00,810 --> 00:26:04,210 Esterase, clips ester bonds. 395 00:26:04,210 --> 00:26:06,040 Your liver is full of esterases. 396 00:26:06,040 --> 00:26:09,450 Because it likes to break things down into smaller and 397 00:26:09,450 --> 00:26:12,190 smaller pieces and lots of ester bonds and things that 398 00:26:12,190 --> 00:26:14,380 are not very biologically interesting. 399 00:26:14,380 --> 00:26:22,100 And that's one way of the liver breaking things down. 400 00:26:22,100 --> 00:26:29,450 So the way it works is that you have your reactants, which 401 00:26:29,450 --> 00:26:39,970 in the biological language are called your substrate, come 402 00:26:39,970 --> 00:26:44,230 in, into an enzyme. 403 00:26:44,230 --> 00:26:50,700 Gets bound up in a pocket of, there's 404 00:26:50,700 --> 00:26:52,430 the substrate, reactant. 405 00:26:52,430 --> 00:26:55,520 There's the enzyme here. 406 00:26:55,520 --> 00:26:58,740 Forms a complex. 407 00:26:58,740 --> 00:27:06,470 And then product gets spit out. 408 00:27:06,470 --> 00:27:08,400 And the product floats off. 409 00:27:08,400 --> 00:27:09,230 And does its thing. 410 00:27:09,230 --> 00:27:12,430 It probably gets bound to another enzyme, which makes 411 00:27:12,430 --> 00:27:13,110 another product. 412 00:27:13,110 --> 00:27:15,030 Et cetera, and the cascade goes on. 413 00:27:15,030 --> 00:27:18,400 The signaling cascade goes on. 414 00:27:18,400 --> 00:27:22,220 And this enzyme is in very small concentration. 415 00:27:22,220 --> 00:27:25,260 The product goes away, so that's going to stay as a very 416 00:27:25,260 --> 00:27:32,350 small concentration as well. 417 00:27:32,350 --> 00:27:44,220 So let's observe experimentally, then. 418 00:27:44,220 --> 00:27:50,620 Experimentally, what's seen is that the substrate makes 419 00:27:50,620 --> 00:27:55,130 products in the presence of the enzyme. 420 00:27:55,130 --> 00:28:03,030 With a rate dP/dt, that's, at t equals zero this is the 421 00:28:03,030 --> 00:28:05,200 initial rate is proportional to the 422 00:28:05,200 --> 00:28:06,310 concentration of the enzyme. 423 00:28:06,310 --> 00:28:10,890 This is initial rate. 424 00:28:10,890 --> 00:28:14,250 And in the language of enzymatic kinetics, this is 425 00:28:14,250 --> 00:28:20,130 called the velocity. dP/dt is also called the velocity. 426 00:28:20,130 --> 00:28:23,830 Velocity, moles per unit time. 427 00:28:23,830 --> 00:28:26,100 And this would be called, then, the initial velocity of 428 00:28:26,100 --> 00:28:30,070 v initial. v initial's proportional to the enzyme 429 00:28:30,070 --> 00:28:33,040 concentration. 430 00:28:33,040 --> 00:28:37,090 And what else is seen? 431 00:28:37,090 --> 00:28:41,800 For fixed, if I fix my concentration of enzyme, I 432 00:28:41,800 --> 00:28:46,390 look at the velocity over time, let's say minus dS/dt, 433 00:28:46,390 --> 00:28:58,250 which is dP/dt, I find that that's proportional to S. For 434 00:28:58,250 --> 00:29:03,940 small S. Small concentration of substrate. 435 00:29:03,940 --> 00:29:04,710 And then at large 436 00:29:04,710 --> 00:29:13,750 concentration, it's a constant. 437 00:29:13,750 --> 00:29:22,690 So it's not a straight line. 438 00:29:22,690 --> 00:29:28,870 In fact, I don't want to do it on this board here. 439 00:29:28,870 --> 00:29:41,870 I'll do it on this board here. 440 00:29:41,870 --> 00:29:44,070 Plot the concentration of substrate on this axis, and I 441 00:29:44,070 --> 00:29:48,120 plot the velocity, or the rate, of the reaction 442 00:29:48,120 --> 00:29:49,650 on that axis here. 443 00:29:49,650 --> 00:29:53,210 What I find is that it's a constant. 444 00:29:53,210 --> 00:29:55,690 So it's going to saturate, the rate is going to saturate to 445 00:29:55,690 --> 00:29:59,580 some number. 446 00:29:59,580 --> 00:30:03,420 And it's going to start linear with 447 00:30:03,420 --> 00:30:04,570 substrate at the beginning. 448 00:30:04,570 --> 00:30:07,160 So it's going to be a straight line at the beginning. 449 00:30:07,160 --> 00:30:09,210 And eventually it will saturate. 450 00:30:09,210 --> 00:30:09,670 That sort of slope. 451 00:30:09,670 --> 00:30:14,050 And the saturation point, that's the maximum velocity, 452 00:30:14,050 --> 00:30:16,050 or maximum rate that it can have. 453 00:30:16,050 --> 00:30:20,240 So we call this v max. 454 00:30:20,240 --> 00:30:34,010 And this part here, the velocity is proportional to S. 455 00:30:34,010 --> 00:30:39,230 And somehow we have to find, explain this. 456 00:30:39,230 --> 00:30:43,140 Using what we know from kinetics. 457 00:30:43,140 --> 00:30:45,430 So we have to come up with a mechanism, and 458 00:30:45,430 --> 00:30:48,150 then solve the mechanism. 459 00:30:48,150 --> 00:30:53,230 And make sure that it reproduces the data. 460 00:30:53,230 --> 00:30:54,800 So we're not the first ones to do this, obviously. 461 00:30:54,800 --> 00:31:02,830 Michaelis and Menton did this many years ago. 462 00:31:02,830 --> 00:31:04,420 For this mechanism. 463 00:31:04,420 --> 00:31:10,200 And the idea is, you have the enzyme plus a substrate react, 464 00:31:10,200 --> 00:31:15,010 k1, k minus one, to form a complex. 465 00:31:15,010 --> 00:31:16,510 Enzyme substrate complex. 466 00:31:16,510 --> 00:31:20,500 But unlike what we've drawn before in terms of this hump, 467 00:31:20,500 --> 00:31:23,140 where there's an activated complex which is not stable, 468 00:31:23,140 --> 00:31:28,520 in this case here this enzyme substrate combination lasts a 469 00:31:28,520 --> 00:31:36,700 long enough time that it's basically a stable complex. 470 00:31:36,700 --> 00:31:41,290 So we're going to write this as a real intermediate that 471 00:31:41,290 --> 00:31:44,150 lasts long enough for you to be able to fish it out. 472 00:31:44,150 --> 00:31:47,300 And characterize it. 473 00:31:47,300 --> 00:31:52,900 And then eventually, that, k2, k minus two, goes to product 474 00:31:52,900 --> 00:31:54,130 plus enzyme. 475 00:31:54,130 --> 00:31:58,500 So the enzyme is a catalyst that forms 476 00:31:58,500 --> 00:31:59,940 a long-lived complex. 477 00:31:59,940 --> 00:32:04,650 And if you were to draw this, then, in our energy diagram, 478 00:32:04,650 --> 00:32:10,500 where we have the reaction coordinate here, you start out 479 00:32:10,500 --> 00:32:16,590 with your enzyme plus substrate. 480 00:32:16,590 --> 00:32:23,000 Go up, and you form your intermediate up here. 481 00:32:23,000 --> 00:32:23,700 ES. 482 00:32:23,700 --> 00:32:26,860 And I put a little dimple in there, because it's stable 483 00:32:26,860 --> 00:32:31,040 enough that it's not on the top of the hump, but it lives 484 00:32:31,040 --> 00:32:32,880 long enough. 485 00:32:32,880 --> 00:32:38,080 And it comes back down to form the product. 486 00:32:38,080 --> 00:32:39,710 Plus the enzyme. 487 00:32:39,710 --> 00:32:42,160 Without the enzyme in there, this hump would 488 00:32:42,160 --> 00:32:44,660 be way, way up there. 489 00:32:44,660 --> 00:32:47,550 Would be maybe a factor of ten higher. 490 00:32:47,550 --> 00:32:51,760 So this really lowers it a lot. 491 00:32:51,760 --> 00:32:54,510 Now we can start to solve this mechanism. 492 00:32:54,510 --> 00:32:56,650 And I forgot one arrow. 493 00:32:56,650 --> 00:33:00,070 There which is the arrow going back. 494 00:33:00,070 --> 00:33:01,680 Which you usually don't see, but we might as 495 00:33:01,680 --> 00:33:03,180 well keep it there. 496 00:33:03,180 --> 00:33:15,490 For the sake of completeness. 497 00:33:15,490 --> 00:33:16,190 So what do we know? 498 00:33:16,190 --> 00:33:22,210 We know that this intermediate concentration is very small. 499 00:33:22,210 --> 00:33:24,340 The enzyme concentration itself is very small. 500 00:33:24,340 --> 00:33:26,420 Intermediate is very small. 501 00:33:26,420 --> 00:33:29,420 And it doesn't change very much. 502 00:33:29,420 --> 00:33:36,540 Not changing much. 503 00:33:36,540 --> 00:33:38,860 So that means that we need to use a steady state 504 00:33:38,860 --> 00:33:39,500 approximation. 505 00:33:39,500 --> 00:33:46,210 So let's write down the rate for this 506 00:33:46,210 --> 00:33:49,200 intermediate, d[ES]/dt. 507 00:33:49,200 --> 00:33:53,760 It gets formed through the forward process. 508 00:33:53,760 --> 00:33:57,160 I'm going to put my brackets back in because E and ES would 509 00:33:57,160 --> 00:33:59,280 look the same otherwise. 510 00:33:59,280 --> 00:34:02,990 Gets destroyed. 511 00:34:02,990 --> 00:34:03,910 Through the backward ways. 512 00:34:03,910 --> 00:34:05,570 And I'm going to use a steady state approximation. 513 00:34:05,570 --> 00:34:07,950 So I'm already going to start adding steady state here every 514 00:34:07,950 --> 00:34:09,460 time I see an intermediate. 515 00:34:09,460 --> 00:34:15,490 Get it destroyed to make products. 516 00:34:15,490 --> 00:34:17,030 It gets recreated through the backwards 517 00:34:17,030 --> 00:34:24,320 reaction from the products. 518 00:34:24,320 --> 00:34:26,030 And I'm going to set that equal to zero for the steady 519 00:34:26,030 --> 00:34:28,710 state approximation. 520 00:34:28,710 --> 00:34:33,240 Now, we don't really want to have this E 521 00:34:33,240 --> 00:34:34,050 floating around here. 522 00:34:34,050 --> 00:34:37,840 Because this is something that is very hard to measure. 523 00:34:37,840 --> 00:34:40,700 It's much easier to measure the initial concentration of 524 00:34:40,700 --> 00:34:41,900 the enzyme. 525 00:34:41,900 --> 00:34:44,240 What you put in there, instead of the amount that's 526 00:34:44,240 --> 00:34:46,330 not being bound up. 527 00:34:46,330 --> 00:34:51,980 So we're going to solve, instead, in terms of [E] is 528 00:34:51,980 --> 00:34:55,550 equal to [E]0 minus [ES], where this is the initial 529 00:34:55,550 --> 00:34:57,040 concentration and this is the amount 530 00:34:57,040 --> 00:34:59,620 that's binding substrate. 531 00:34:59,620 --> 00:35:03,710 And this is the amount of free enzyme here then. 532 00:35:03,710 --> 00:35:08,500 And when you do that, and you plug in here, and you plug in 533 00:35:08,500 --> 00:35:14,300 here, you get your result. 534 00:35:14,300 --> 00:35:33,950 Which is that [E] steady state, [ES] steady state, is 535 00:35:33,950 --> 00:35:42,250 this ratio. k1 times [S] plus k minus two times the product 536 00:35:42,250 --> 00:35:52,520 divided by k minus one plus k2 plus k1 times [S] plus k minus 537 00:35:52,520 --> 00:35:58,680 two times the product, times proportional to the initial 538 00:35:58,680 --> 00:36:05,330 concentration of enzyme. 539 00:36:05,330 --> 00:36:10,430 And then you can take your steady state approximation for 540 00:36:10,430 --> 00:36:30,220 the intermediate and plug it back into your rate equation. 541 00:36:30,220 --> 00:36:35,950 So the velocity, we defined as the rate which is minus 542 00:36:35,950 --> 00:36:46,200 d[S]/dt, which is k1 times [E] times [S], this is the 543 00:36:46,200 --> 00:36:52,400 destruction of the substrate minus k minus one, times [ES] 544 00:36:52,400 --> 00:36:57,760 steady state, the backwards process. 545 00:36:57,760 --> 00:37:02,160 So we put in, instead of this [E] we put in [E] is equal to 546 00:37:02,160 --> 00:37:06,010 [E]0 minus [ES] 547 00:37:06,010 --> 00:37:10,910 steady state. 548 00:37:10,910 --> 00:37:12,340 And then we put in for [ES] 549 00:37:12,340 --> 00:37:15,680 steady state, we put in what we found here. 550 00:37:15,680 --> 00:37:19,990 We turn the crank on the algebra. 551 00:37:19,990 --> 00:37:27,590 And we find that the velocity, then, is k1 k2 times the 552 00:37:27,590 --> 00:37:29,800 substrate concentration. 553 00:37:29,800 --> 00:37:34,730 Minus k minus one times k minus two times the product 554 00:37:34,730 --> 00:37:36,780 concentration. 555 00:37:36,780 --> 00:37:44,760 The whole thing times the initial enzyme concentration. 556 00:37:44,760 --> 00:37:48,320 And then k minus one plus k2 on the bottom. 557 00:37:48,320 --> 00:37:57,870 Plus k1 [S] plus k minus two times the product. 558 00:37:57,870 --> 00:38:07,380 So, as I mentioned, this product here usually just 559 00:38:07,380 --> 00:38:08,070 floats away. 560 00:38:08,070 --> 00:38:13,640 And so locally, where you're doing the, and then it gets 561 00:38:13,640 --> 00:38:17,070 used by the next step in the cycle. 562 00:38:17,070 --> 00:38:20,430 So this is pretty much zero. 563 00:38:20,430 --> 00:38:22,740 We can pretty much ignore this. 564 00:38:22,740 --> 00:38:25,680 We can ignore this guy here too. 565 00:38:25,680 --> 00:38:27,920 Because it's not, you don't have any steady state or an 566 00:38:27,920 --> 00:38:28,300 equilibrium. 567 00:38:28,300 --> 00:38:31,480 You have a steady state but not an equilibrium situation. 568 00:38:31,480 --> 00:38:40,810 And so in that case here, you can rewrite this then as k1 k2 569 00:38:40,810 --> 00:38:49,860 times the substrate times [E]0 divided by k minus one plus k2 570 00:38:49,860 --> 00:38:53,710 plus k1 times the substrate. 571 00:38:53,710 --> 00:39:00,820 And now we can look at these experimental observations and 572 00:39:00,820 --> 00:39:05,070 see whether they match our mechanism. 573 00:39:05,070 --> 00:39:12,880 If we can understand something. 574 00:39:12,880 --> 00:39:15,220 So let's look at the initial rate. 575 00:39:15,220 --> 00:39:19,760 Initial rate is supposed to be proportional to the substrate. 576 00:39:19,760 --> 00:39:22,640 Initial rate is supposed to be proportional to the substrate. 577 00:39:22,640 --> 00:39:34,110 So the initial rate, k1 k2 plus k1, if at 578 00:39:34,110 --> 00:39:43,770 early times, plus one. 579 00:39:43,770 --> 00:39:56,370 So the initial rate is going to be, somehow I've got the 580 00:39:56,370 --> 00:40:08,540 product missing here. 581 00:40:08,540 --> 00:40:09,670 No, I got it backwards here. 582 00:40:09,670 --> 00:40:11,400 This is not the initial rate. 583 00:40:11,400 --> 00:40:19,460 This is the rate where [S] is small. 584 00:40:19,460 --> 00:40:23,400 The initial rate is when you don't have any products made. 585 00:40:23,400 --> 00:40:25,830 Or when the product concentration is very low. 586 00:40:25,830 --> 00:40:28,960 And because we're making the assumption that the product 587 00:40:28,960 --> 00:40:31,660 concentration is basically equal to zero here, this is 588 00:40:31,660 --> 00:40:33,490 basically the initial rate here. 589 00:40:33,490 --> 00:40:36,740 So by taking the product equal to zero here, we're also 590 00:40:36,740 --> 00:40:41,720 saying that this is the same thing as the initial rate. 591 00:40:41,720 --> 00:40:44,610 So this is what it looks like here.. 592 00:40:44,610 --> 00:40:45,970 The initial rate here. 593 00:40:45,970 --> 00:40:56,050 And this is, this, you rewrite as v initial is equal to k2 594 00:40:56,050 --> 00:40:58,700 times [S] times [E]0. 595 00:40:58,700 --> 00:41:02,040 You divide by k1 up and down. 596 00:41:02,040 --> 00:41:09,750 And you have this k minus one plus k2 over k1 and then plus 597 00:41:09,750 --> 00:41:11,500 [S] sitting down there. 598 00:41:11,500 --> 00:41:17,930 And you define this ratio of rates, k minus one plus k2 599 00:41:17,930 --> 00:41:23,130 over k1 as the KM, the Michaelis constant, by 600 00:41:23,130 --> 00:41:25,180 definition. 601 00:41:25,180 --> 00:41:26,460 And this is an interesting ratio. 602 00:41:26,460 --> 00:41:30,260 This is the rate, k minus one is the rate of destroying the 603 00:41:30,260 --> 00:41:34,870 enzyme substrate complex by going back to the reactants. 604 00:41:34,870 --> 00:41:36,990 k2 is the destruction of the complex by 605 00:41:36,990 --> 00:41:38,470 going to the product. 606 00:41:38,470 --> 00:41:40,980 And k1's the creation of the complex. 607 00:41:40,980 --> 00:41:43,500 So this is the rate of destruction of the complex 608 00:41:43,500 --> 00:41:47,420 divided by the rate of creation of the complex. 609 00:41:47,420 --> 00:41:50,130 So if the rate of destruction of the complex is much faster 610 00:41:50,130 --> 00:41:56,250 than the rate of creation, meaning that this is a large 611 00:41:56,250 --> 00:41:59,900 number, then you're not going to pile up any complex. 612 00:41:59,900 --> 00:42:02,930 It's going to be destroyed as soon as you create it. 613 00:42:02,930 --> 00:42:11,280 So if KM is large, then the concentration, [ES], is going 614 00:42:11,280 --> 00:42:13,050 to be very small. 615 00:42:13,050 --> 00:42:15,600 Compared to [E]0. 616 00:42:15,600 --> 00:42:19,830 But if the rate of destruction of the complex is small 617 00:42:19,830 --> 00:42:21,970 compared to the rate of creation, you create 618 00:42:21,970 --> 00:42:24,450 complexes, you create complexes but you don't keep 619 00:42:24,450 --> 00:42:26,540 up in terms of destroying them, in terms of making 620 00:42:26,540 --> 00:42:29,060 products, or going back to reactants. 621 00:42:29,060 --> 00:42:33,430 And so you end up saturating your enzyme. 622 00:42:33,430 --> 00:42:38,280 Every enzyme ends up having substrate bound to it. 623 00:42:38,280 --> 00:42:51,780 So when KM is very small, then [ES] goes to saturation. 624 00:42:51,780 --> 00:42:56,740 Basically, when KM is very small, then you're limited by 625 00:42:56,740 --> 00:43:01,040 the rate, the second rate in the process, of the enzyme 626 00:43:01,040 --> 00:43:01,800 falling apart. 627 00:43:01,800 --> 00:43:03,130 To form the product. 628 00:43:03,130 --> 00:43:06,010 You have to wait until that happens. 629 00:43:06,010 --> 00:43:09,710 Because then the product just floats away. 630 00:43:09,710 --> 00:43:14,650 And that becomes your rate limiting step. 631 00:43:14,650 --> 00:43:20,070 Another way that you also will see this written is as this, 632 00:43:20,070 --> 00:43:27,860 then, is equal to, we'll define k cat is equal to k2. k 633 00:43:27,860 --> 00:43:33,220 cat times the enzyme concentration times the 634 00:43:33,220 --> 00:43:39,520 substrate, [E]0, divided by KM plus but the substrate 635 00:43:39,520 --> 00:43:46,620 concentration. 636 00:43:46,620 --> 00:43:48,720 So let's look at a few limiting cases then. 637 00:43:48,720 --> 00:43:59,070 First limiting case is, suppose, that [S] is large. 638 00:43:59,070 --> 00:44:03,600 Let's take [S] to be much larger than KM. 639 00:44:03,600 --> 00:44:07,590 Because you look at the denominator, and you see it's 640 00:44:07,590 --> 00:44:10,610 this ratio of rates but there's this concentration 641 00:44:10,610 --> 00:44:12,410 that's important here. 642 00:44:12,410 --> 00:44:14,730 So in one case KM is going to dominate, and in the other 643 00:44:14,730 --> 00:44:16,570 limiting case the substrate 644 00:44:16,570 --> 00:44:17,920 concentration is going to dominate. 645 00:44:17,920 --> 00:44:19,480 So let's say that the substrate 646 00:44:19,480 --> 00:44:20,600 concentration dominates. 647 00:44:20,600 --> 00:44:22,690 Meaning K sub M is small. 648 00:44:22,690 --> 00:44:23,630 In which case we already saw. 649 00:44:23,630 --> 00:44:27,620 If K sub M is small then you reach saturation. 650 00:44:27,620 --> 00:44:33,730 And in the equation, then, the velocity, KM is small, [S] is 651 00:44:33,730 --> 00:44:38,860 large, the [S]'s cancel out and the velocity is equal to k 652 00:44:38,860 --> 00:44:43,180 cat times the initial substrate concentration. 653 00:44:43,180 --> 00:44:44,740 Both of these are constants. 654 00:44:44,740 --> 00:44:46,290 The velocity is constant. 655 00:44:46,290 --> 00:44:50,850 And the rate is constant, it's that limit up here. 656 00:44:50,850 --> 00:44:55,880 That's experimentally seen. 657 00:44:55,880 --> 00:44:58,670 And the rate is depending on the initial substrate 658 00:44:58,670 --> 00:45:01,430 concentration. 659 00:45:01,430 --> 00:45:02,600 Initial enzyme concentration. 660 00:45:02,600 --> 00:45:04,550 All the enzymes have a substrate in there. 661 00:45:04,550 --> 00:45:07,160 The more enzymes you have to begin with, the more 662 00:45:07,160 --> 00:45:08,860 intermediates you're going to have, the faster you going to 663 00:45:08,860 --> 00:45:11,000 make products. 664 00:45:11,000 --> 00:45:13,860 And then it's going to depend on this on the rate, k cat, 665 00:45:13,860 --> 00:45:17,450 which is just k2, which is the rate of formation of products. 666 00:45:17,450 --> 00:45:24,530 That becomes the rate limiting step here. 667 00:45:24,530 --> 00:45:29,710 The other special case is if substrate concentration is 668 00:45:29,710 --> 00:45:33,400 very small compared to KM. 669 00:45:33,400 --> 00:45:38,725 And in that case here, and the other thing that we're going 670 00:45:38,725 --> 00:45:43,830 to do is, we're going to, because we now understand this 671 00:45:43,830 --> 00:45:46,920 as this maximum rate up there, we're going to call this v 672 00:45:46,920 --> 00:45:52,480 max. k cat times [E]0, we're going to call it v max. 673 00:45:52,480 --> 00:46:01,180 And so when [S] is very small, v, then, is equal to v max, 674 00:46:01,180 --> 00:46:03,040 which is k cat times [E]0. 675 00:46:03,040 --> 00:46:04,380 This is v max here now. 676 00:46:04,380 --> 00:46:12,130 So we can rewrite this as v max times substrate divided by 677 00:46:12,130 --> 00:46:14,130 KM plus the substrate concentration. 678 00:46:14,130 --> 00:46:15,880 Another way of writing it. 679 00:46:15,880 --> 00:46:19,690 Capital K. 680 00:46:19,690 --> 00:46:23,550 v max times the substrate concentration. 681 00:46:23,550 --> 00:46:25,510 And we have KM plus [S]. 682 00:46:25,510 --> 00:46:27,560 But [S] is very small. 683 00:46:27,560 --> 00:46:28,810 So we drop it. 684 00:46:28,810 --> 00:46:32,520 And this is then proportional to the substrate 685 00:46:32,520 --> 00:46:33,280 concentration. 686 00:46:33,280 --> 00:46:35,560 And then that's small substrate concentration 687 00:46:35,560 --> 00:46:37,460 compared to KM, we're sitting right here. 688 00:46:37,460 --> 00:46:40,800 Where it's linear, where the rate is linear. 689 00:46:40,800 --> 00:46:44,690 And there's a third place on the graph which is 690 00:46:44,690 --> 00:46:46,350 interesting. 691 00:46:46,350 --> 00:46:52,000 Which is when the substrate concentration is equal to KM. 692 00:46:52,000 --> 00:46:57,410 In that case there, you plug [S] equal to KM, and you end 693 00:46:57,410 --> 00:47:04,810 up with the velocity then is equal to v max divided by two. 694 00:47:04,810 --> 00:47:10,860 So when [S] is equal to KM, you are halfway up. 695 00:47:10,860 --> 00:47:18,160 There's v max over two and there's KM sitting here. 696 00:47:18,160 --> 00:47:24,690 When [S] is equal to KM, you're at v max over two. 697 00:47:24,690 --> 00:47:27,930 And so enzymes, then, are labeled by their KM's. 698 00:47:27,930 --> 00:47:31,280 Because then it becomes very important to know how strongly 699 00:47:31,280 --> 00:47:34,900 they bind the substrate. 700 00:47:34,900 --> 00:47:37,440 Sometimes you want the enzyme to bind it very strongly. 701 00:47:37,440 --> 00:47:40,170 Sometimes you don't, you want it to be fleeting. 702 00:47:40,170 --> 00:47:45,820 Depends on the role that the enzyme plays. 703 00:47:45,820 --> 00:47:52,480 Now there's a way to plot this that extracts out these 704 00:47:52,480 --> 00:47:54,540 important numbers. k cat and KM. 705 00:47:57,280 --> 00:48:00,770 And that's the Lineweaver-Burk plot.. 706 00:48:00,770 --> 00:48:06,560 Lineweaver-Burk plot.. 707 00:48:06,560 --> 00:48:08,630 And I just looked up this morning to see if Mr. 708 00:48:08,630 --> 00:48:09,600 Lineweaver was still alive. 709 00:48:09,600 --> 00:48:12,730 And as far as I can tell he's still alive. 710 00:48:12,730 --> 00:48:17,900 He was 97, in 2003. 711 00:48:17,900 --> 00:48:22,220 So as of 2007 he was still alive. 712 00:48:22,220 --> 00:48:25,140 He's getting up there. 713 00:48:25,140 --> 00:48:27,970 One of the most cited papers that you have in your notes is 714 00:48:27,970 --> 00:48:31,580 the in Jack's, was the paper that showed how to go from 715 00:48:31,580 --> 00:48:38,690 this curved line to a straight line by plotting one over v 716 00:48:38,690 --> 00:48:42,660 versus [S], instead of v versus [S]. 717 00:48:42,660 --> 00:48:51,660 So if you take your equation and massage it, one over v KM 718 00:48:51,660 --> 00:48:57,740 over v max times [S], plus one over v max, we haven't done 719 00:48:57,740 --> 00:49:00,720 anything except rewrite the equation in terms of one of v 720 00:49:00,720 --> 00:49:05,540 versus one over [S]. 721 00:49:05,540 --> 00:49:08,400 So it becomes linear. 722 00:49:08,400 --> 00:49:14,260 In one over [S], and there's one over v sitting here. 723 00:49:14,260 --> 00:49:18,890 And you get a straight line. 724 00:49:18,890 --> 00:49:23,080 With an intercept here that's one over v max. 725 00:49:23,080 --> 00:49:29,610 And if you keep going, extrapolate out, you get this 726 00:49:29,610 --> 00:49:40,010 point here to be minus 1 over KM and the slope 727 00:49:40,010 --> 00:49:45,500 is KM over v max. 728 00:49:45,500 --> 00:49:48,950 And v max was equal to k cat times [E]0, so you get k cat 729 00:49:48,950 --> 00:49:53,100 out of this. 730 00:49:53,100 --> 00:49:57,850 So it turned out to be a very useful plot. 731 00:49:57,850 --> 00:49:59,460 It's very easy to plot a straight line, especially 732 00:49:59,460 --> 00:50:00,500 before computers. 733 00:50:00,500 --> 00:50:02,120 In the age of computers. 734 00:50:02,120 --> 00:50:06,950 And the referees, there were six referees that got this 735 00:50:06,950 --> 00:50:09,880 paper and pretty much turned it down because they didn't 736 00:50:09,880 --> 00:50:11,640 think there was any new chemistry in it. 737 00:50:11,640 --> 00:50:12,250 Which is true, there's no new chemistry. 738 00:50:12,250 --> 00:50:15,040 It's just a way of rewriting the plot. 739 00:50:15,040 --> 00:50:18,980 But it was very important nevertheless. 740 00:50:18,980 --> 00:50:24,500 OK, any questions on catalysis? 741 00:50:24,500 --> 00:50:27,670 Enzymes? 742 00:50:27,670 --> 00:50:30,710 Arrhenius? 743 00:50:30,710 --> 00:50:35,250 Alright, next time we'll oscillating reactions and 744 00:50:35,250 --> 00:50:36,990 recap the course.