1 00:00:00,000 --> 00:00:01,250 2 00:00:01,250 --> 00:00:04,169 I've already made one video on it, and at least it attempts 3 00:00:04,169 --> 00:00:07,000 to give you the intuition behind how the equilibrium 4 00:00:07,000 --> 00:00:09,960 constant formula is derived or where it comes from. 5 00:00:09,960 --> 00:00:12,610 From maybe the probabilities of different molecules 6 00:00:12,609 --> 00:00:15,219 interacting, if they're in some small volume. 7 00:00:15,220 --> 00:00:17,699 But I think I was a little hand wavy with it, and it 8 00:00:17,699 --> 00:00:20,410 might not have been clear how probabilities and 9 00:00:20,410 --> 00:00:21,510 concentrations relate. 10 00:00:21,510 --> 00:00:24,070 So what I thought I would do in this video is kind of do 11 00:00:24,070 --> 00:00:27,789 the same exercise, but do it with real numbers and a real 12 00:00:27,789 --> 00:00:29,759 reaction, so just a's, b's and c's. 13 00:00:29,760 --> 00:00:34,000 So what I wrote here is the Haber process. 14 00:00:34,000 --> 00:00:39,070 This is how we get ammonia in the world and feed everyone. 15 00:00:39,070 --> 00:00:40,829 Ammonia is a very important fertilizer, but 16 00:00:40,829 --> 00:00:42,670 that's beside the point. 17 00:00:42,670 --> 00:00:44,480 The Haber process, which you see right here, is in 18 00:00:44,479 --> 00:00:47,739 equilibrium, which doesn't mean that the concentrations 19 00:00:47,740 --> 00:00:48,100 are the same. 20 00:00:48,100 --> 00:00:50,780 In fact, this is an equilibrium concentration that 21 00:00:50,780 --> 00:00:52,380 I worked out before starting this video. 22 00:00:52,380 --> 00:00:56,540 Notice, the concentrations of nitrogen and hydrogen are very 23 00:00:56,539 --> 00:00:58,079 different than the concentration of ammonia, 24 00:00:58,079 --> 00:00:59,079 which is much less. 25 00:00:59,079 --> 00:01:03,189 What equilibrium tells us is that once we get to this 26 00:01:03,189 --> 00:01:07,950 concentration of nitrogen and hydrogen, the rate of reaction 27 00:01:07,950 --> 00:01:10,719 of going in the rightward direction is the same as the 28 00:01:10,719 --> 00:01:13,179 rate of reaction of going in the leftward direction, when 29 00:01:13,180 --> 00:01:14,920 we have this much ammonia. 30 00:01:14,920 --> 00:01:17,510 So let's just think about what that rate of reaction means. 31 00:01:17,510 --> 00:01:20,359 And then I'll tell you how I think about it. 32 00:01:20,359 --> 00:01:23,769 At least how I think about it is that if you have some small 33 00:01:23,769 --> 00:01:29,560 volume-- Let's call that dV. 34 00:01:29,560 --> 00:01:32,299 You could kind of arbitrarily pick how small. 35 00:01:32,299 --> 00:01:33,569 So dV. 36 00:01:33,569 --> 00:01:36,719 And the way it works in my head is that if you pick some 37 00:01:36,719 --> 00:01:39,890 small volume in this solution-- we don't know how 38 00:01:39,890 --> 00:01:42,049 large of a solution we actually have. We just have 39 00:01:42,049 --> 00:01:45,230 the concentrations-- that in this volume, you're just as 40 00:01:45,230 --> 00:01:47,719 likely to have a reaction going into that direction as 41 00:01:47,719 --> 00:01:49,090 you are to have a reaction going in 42 00:01:49,090 --> 00:01:50,064 the backwards direction. 43 00:01:50,064 --> 00:01:53,569 So let's think about what the probability is of having a 44 00:01:53,569 --> 00:01:55,659 reaction in this volume. 45 00:01:55,659 --> 00:01:59,179 So the probability, let's say of the forward reaction. 46 00:01:59,180 --> 00:02:07,370 Probability of N2 plus 3H2, going in 47 00:02:07,370 --> 00:02:08,819 this forward direction. 48 00:02:08,819 --> 00:02:10,758 And whatever I do for this direction, then you just have 49 00:02:10,758 --> 00:02:12,589 to use the same logic for the backward direction. 50 00:02:12,590 --> 00:02:14,979 I just want to give you the intuition that it's equal to 51 00:02:14,979 --> 00:02:18,189 some constant related to their concentration. 52 00:02:18,189 --> 00:02:27,189 So the probability of going in that direction to 2NH3 in our 53 00:02:27,189 --> 00:02:34,300 little box, I claim-- and I think this will hopefully make 54 00:02:34,300 --> 00:02:40,320 some sense-- it's equal to first, the probability that 55 00:02:40,319 --> 00:02:46,699 they react given in the box. 56 00:02:46,699 --> 00:02:48,649 So if you know that if you have the constituent 57 00:02:48,650 --> 00:02:53,000 particles, you have one nitrogen molecule-- which has 58 00:02:53,000 --> 00:02:57,000 two nitrogen atoms in it-- and three hydrogen molecules. 59 00:02:57,000 --> 00:02:59,530 If you know you have those, there's some probability that 60 00:02:59,530 --> 00:03:02,009 they're going to react based on their configurations and 61 00:03:02,009 --> 00:03:03,939 their kinetic energy and how they're approaching each other 62 00:03:03,939 --> 00:03:05,740 and all of these different types of things. 63 00:03:05,740 --> 00:03:07,900 So this is the probability they react given that they're 64 00:03:07,900 --> 00:03:10,039 in this little box of dV. 65 00:03:10,039 --> 00:03:11,900 And then, of course, you're going to have to multiply that 66 00:03:11,900 --> 00:03:18,000 times the probability in the box, that you have the 67 00:03:18,000 --> 00:03:19,979 constituent particles in the box. 68 00:03:19,979 --> 00:03:22,519 Now, my claim is that this piece right 69 00:03:22,520 --> 00:03:24,770 here, this is a constant. 70 00:03:24,770 --> 00:03:27,180 If you know at a certain temperature, the Haber 71 00:03:27,180 --> 00:03:30,599 process, these concentrations, this would happen at 300 72 00:03:30,599 --> 00:03:33,229 degrees Celsius-- I just looked that up. 73 00:03:33,229 --> 00:03:34,949 No need to memorize something like this. 74 00:03:34,949 --> 00:03:37,509 But equilibrium constants hold at a certain temperature. 75 00:03:37,509 --> 00:03:41,159 So I'm claiming that if I give you a temperature-- say, 300 76 00:03:41,159 --> 00:03:46,389 degrees Celsius-- and if I tell you that I have one 77 00:03:46,389 --> 00:03:51,039 nitrogen and three molecules of hydrogen in your box, that 78 00:03:51,039 --> 00:03:54,500 there's some constant probability that they react. 79 00:03:54,500 --> 00:03:56,770 I mean, it depends on their configuration and all of that. 80 00:03:56,770 --> 00:04:01,870 So I'll just call this the constant-- I'll just make up a 81 00:04:01,870 --> 00:04:05,200 constant probability, whatever it is, or in the box. 82 00:04:05,199 --> 00:04:06,519 I could write anything there. 83 00:04:06,520 --> 00:04:08,520 So what we should be concerned with is what is the 84 00:04:08,520 --> 00:04:13,360 probability that we have those four molecules-- three 85 00:04:13,360 --> 00:04:15,190 molecules of hydrogen and two molecules of 86 00:04:15,189 --> 00:04:17,088 nitrogen-- in the box. 87 00:04:17,088 --> 00:04:21,829 So this is equal to some constant. 88 00:04:21,829 --> 00:04:25,139 I'll call it the constant of probability of react-- or let 89 00:04:25,139 --> 00:04:26,419 me say react. 90 00:04:26,420 --> 00:04:29,150 That's a good one: react. 91 00:04:29,149 --> 00:04:32,709 The constant of reaction-- if you have it in the box-- times 92 00:04:32,709 --> 00:04:35,764 the probability that they're in the box. 93 00:04:35,764 --> 00:04:37,319 Let me draw the box. 94 00:04:37,319 --> 00:04:41,389 So we want to know the probability, where this box is 95 00:04:41,389 --> 00:04:45,904 just some volume, that I have three hydrogen molecules. 96 00:04:45,904 --> 00:04:50,449 So one, two, three. 97 00:04:50,449 --> 00:04:53,029 And one nitrogen molecule. 98 00:04:53,029 --> 00:04:56,989 And we should pick a box that's small enough so that 99 00:04:56,990 --> 00:04:59,590 that would be indicative of how close the molecules need 100 00:04:59,589 --> 00:05:01,189 to get to actually react. 101 00:05:01,189 --> 00:05:05,719 So I'm just going to pick my dV to be-- I don't know. 102 00:05:05,720 --> 00:05:12,680 Let's pick my dV to be-- I looked up the diameter of an 103 00:05:12,680 --> 00:05:13,240 ammonia molecule. 104 00:05:13,240 --> 00:05:16,600 It was about 1/10 of a nanometer. 105 00:05:16,600 --> 00:05:19,189 106 00:05:19,189 --> 00:05:21,949 If this was a nanometer box, you could put 10 in each 107 00:05:21,949 --> 00:05:24,500 direction, so you can almost fit 1,000 if you packed them 108 00:05:24,500 --> 00:05:25,569 really tightly. 109 00:05:25,569 --> 00:05:30,279 So let's make this half a nanometer in each direction. 110 00:05:30,279 --> 00:05:33,609 So if I pick my dV-- and remember, I don't know if this 111 00:05:33,610 --> 00:05:35,180 is the right distance. 112 00:05:35,180 --> 00:05:36,629 I'm just trying to give you the intuition behind the 113 00:05:36,629 --> 00:05:38,180 equilibrium formula. 114 00:05:38,180 --> 00:05:44,699 But if I pick this as being 0.5 nanometers by 0.5 115 00:05:44,699 --> 00:05:51,769 nanometers by 0.5 nanometers, what is my volume? 116 00:05:51,769 --> 00:05:58,979 So my little volume is going to be 0.5 times 10 to the 117 00:05:58,980 --> 00:06:04,030 minus 1/9 meters-- that's a nanometer-- to the third 118 00:06:04,029 --> 00:06:06,549 power, because we're dealing with cubic meters. 119 00:06:06,550 --> 00:06:09,319 So this is equal to 0.5 to the 1/3 power. 120 00:06:09,319 --> 00:06:09,750 That's what? 121 00:06:09,750 --> 00:06:17,500 0.5 times 0.5 is 0.25 times 0.5 is 0.125. 122 00:06:17,500 --> 00:06:19,490 I want to do the math right, so let me just make sure I got 123 00:06:19,490 --> 00:06:20,810 that right. 124 00:06:20,810 --> 00:06:22,889 0.5 to the 1/3 power. 125 00:06:22,889 --> 00:06:29,629 Right, 0.125 times-- negative 9 to the 1/3 power is minus 126 00:06:29,629 --> 00:06:34,649 27-- 10 to the minus 27 meters cubed. 127 00:06:34,649 --> 00:06:36,449 So that's my volume. 128 00:06:36,449 --> 00:06:37,870 Now, we know the concentration. 129 00:06:37,870 --> 00:06:39,920 Let's figure out what's the probability. 130 00:06:39,920 --> 00:06:42,350 So this is the probability in the box, right? 131 00:06:42,350 --> 00:06:43,712 That's what we're concerned with, the 132 00:06:43,711 --> 00:06:45,189 probability in the box. 133 00:06:45,189 --> 00:06:48,199 Well, the probability in the box, that's the probability 134 00:06:48,199 --> 00:06:52,979 that I have one hydrogen in the box, times the probability 135 00:06:52,980 --> 00:06:55,939 that I have another hydrogen in the box, times the 136 00:06:55,939 --> 00:06:58,490 probability that I have another hydrogen in the box-- 137 00:06:58,490 --> 00:07:01,050 these are all in-the-box probabilities-- times the 138 00:07:01,050 --> 00:07:05,710 probability that I have a nitrogen in the box. 139 00:07:05,709 --> 00:07:08,430 I'll do the nitrogen in a different color just to ease-- 140 00:07:08,430 --> 00:07:11,060 oh, I should've done these in the orange because those are 141 00:07:11,060 --> 00:07:12,750 the color of the molecules up there. 142 00:07:12,750 --> 00:07:14,639 And I'll do this one in purple. 143 00:07:14,639 --> 00:07:17,899 What's the probability of having hydrogen in the box? 144 00:07:17,899 --> 00:07:22,099 Well, we know hydrogen's concentration at equilibrium 145 00:07:22,100 --> 00:07:24,810 is 2 Molar. 146 00:07:24,810 --> 00:07:29,740 So concentration of hydrogen, we know hydrogen's 147 00:07:29,740 --> 00:07:37,030 concentration is equal to 2 Molar, which is 2 Moles per 148 00:07:37,029 --> 00:07:42,539 liter, which is equal to-- 2 Moles is just 2 times 6 times 149 00:07:42,540 --> 00:07:46,819 10 to the twenty-third power-- Moles is just a number-- 150 00:07:46,819 --> 00:07:48,529 divided by liters. 151 00:07:48,529 --> 00:07:56,759 So 1 liter is-- we could write it in meters cubed, or we 152 00:07:56,759 --> 00:07:59,560 could just make the conversion. 153 00:07:59,560 --> 00:08:00,730 Actually, let me just do this for you. 154 00:08:00,730 --> 00:08:06,200 1 liter is equal to 1 times 10 to the minus 3 meters cubed. 155 00:08:06,199 --> 00:08:07,889 If you actually take a meter cubed, you can actually put 156 00:08:07,889 --> 00:08:12,479 1,000 liters in there. 157 00:08:12,480 --> 00:08:20,720 So the other way you could say this is 1 times 10 to the 158 00:08:20,720 --> 00:08:26,560 minus 3 meters cubed, and then if we want to figure out our 159 00:08:26,560 --> 00:08:33,440 dV times-- how many dV's do we have per meter cubed? 160 00:08:33,440 --> 00:08:36,190 Or how many meter cubes are their per dV? 161 00:08:36,190 --> 00:08:41,760 So we know that already, so it's 0.125 times 10 to the 162 00:08:41,759 --> 00:08:49,620 minus 27 meters cubed per our volume, right? 163 00:08:49,620 --> 00:08:53,220 I just got that from up here, that I have a small fraction 164 00:08:53,220 --> 00:08:55,710 of a meter cubed per my volume. 165 00:08:55,710 --> 00:08:58,879 And now, I just have to do some math. 166 00:08:58,879 --> 00:09:02,939 So let's see, I can cancel out some things first, because 167 00:09:02,940 --> 00:09:05,116 there's a lot of exponents here. 168 00:09:05,115 --> 00:09:09,879 So let's see, if I take the twenty-third-- so let me write 169 00:09:09,879 --> 00:09:11,899 it out here. 170 00:09:11,899 --> 00:09:18,539 So my hydrogen per box-- So my concentration of hydrogen per 171 00:09:18,539 --> 00:09:26,099 dV, is equal to 12 times 10-- whoops! 172 00:09:26,100 --> 00:09:28,942 That's not helping when my pen malfunctions. 173 00:09:28,942 --> 00:09:31,259 Let me get that right. 174 00:09:31,259 --> 00:09:39,220 12 times 10 to the twenty-third power times 0.125 175 00:09:39,220 --> 00:09:42,110 times 10 to the minus twenty-seventh power. 176 00:09:42,110 --> 00:09:47,060 All of that divided by 10 to the minus 3, right? 177 00:09:47,059 --> 00:09:49,329 That's 1 times 10 to the minus 3. 178 00:09:49,330 --> 00:09:50,690 So let's cancel out some exponents. 179 00:09:50,690 --> 00:09:55,840 If we get rid of the minus 3 here, you divide by minus 3, 180 00:09:55,840 --> 00:10:00,180 then this becomes minus 24. 181 00:10:00,179 --> 00:10:03,609 And then the minus 24 and the minus-- so this is equal to-- 182 00:10:03,610 --> 00:10:05,720 what's 12 times 1.25? 183 00:10:05,720 --> 00:10:12,149 So times 12 is equal to 1.5. 184 00:10:12,149 --> 00:10:17,169 So the 12 times the 1.25 is equal to 1.5 times-- and then 185 00:10:17,169 --> 00:10:21,019 10 to the twenty-third times 10 to the minus twenty-fourth 186 00:10:21,019 --> 00:10:25,789 is equal to 10 to the minus 1, right? 187 00:10:25,789 --> 00:10:28,740 So it's just divided by 10. 188 00:10:28,740 --> 00:10:35,310 So on average, your concentration of hydrogen in a 189 00:10:35,309 --> 00:10:39,509 little cube that's half a nanometer in each direction is 190 00:10:39,509 --> 00:10:50,649 equal to 0.15 molecules-- not Moles anymore-- of hydrogen 191 00:10:50,649 --> 00:10:55,860 molecule per my little dV, my little box. 192 00:10:55,860 --> 00:10:57,590 And so this is a probability, right? 193 00:10:57,590 --> 00:11:01,160 This is a probability, because obviously I can't have 0.15 194 00:11:01,159 --> 00:11:02,480 molecules in every box. 195 00:11:02,480 --> 00:11:06,269 This is just saying, on average, there's a 0.15 chance 196 00:11:06,269 --> 00:11:09,429 that I have a hydrogen molecule in my box. 197 00:11:09,429 --> 00:11:15,009 So if I want to go back here to this, this is 0.15, this is 198 00:11:15,009 --> 00:11:18,360 0.15, this is 0.15. 199 00:11:18,360 --> 00:11:21,060 But how did we get this 0.15? 200 00:11:21,059 --> 00:11:28,349 We multiplied the concentration of hydrogen, 201 00:11:28,350 --> 00:11:31,290 which was this right here. 202 00:11:31,289 --> 00:11:37,909 That's the concentration of the hydrogen-- I should've 203 00:11:37,909 --> 00:11:40,139 written it in a more vibrant color-- times just a bunch of 204 00:11:40,139 --> 00:11:41,840 scaling factors, right? 205 00:11:41,840 --> 00:11:44,170 We could just say that, well, this was just equal to the 206 00:11:44,169 --> 00:11:47,120 concentration of hydrogen times, based on how I picked 207 00:11:47,120 --> 00:11:49,519 my dV, I had to do all of this scaling. 208 00:11:49,519 --> 00:11:54,699 But it was times some constant of scaling, scaling to my 209 00:11:54,700 --> 00:11:56,250 appropriate factor. 210 00:11:56,250 --> 00:12:00,000 So if we want to figure out each of these, this is just 211 00:12:00,000 --> 00:12:05,909 the concentration of hydrogen times some scaling factor. 212 00:12:05,909 --> 00:12:07,679 And this is going to be the same thing. 213 00:12:07,679 --> 00:12:09,299 We could do the same exercise right here. 214 00:12:09,299 --> 00:12:12,000 We figured out the exact value with the hydrogen, but you 215 00:12:12,000 --> 00:12:14,429 could do the same thing with the nitrogen. 216 00:12:14,429 --> 00:12:19,309 In fact, nitrogen's concentration is just half of 217 00:12:19,309 --> 00:12:20,629 the hydrogen, so we know it. 218 00:12:20,629 --> 00:12:23,759 It's going to be half of that 0.15, so it's going to be 219 00:12:23,759 --> 00:12:28,600 0.075, which is just equal to the concentration of nitrogen 220 00:12:28,600 --> 00:12:30,779 times some scaling factor. 221 00:12:30,779 --> 00:12:33,600 It's actually going to be the same scaling factor. 222 00:12:33,600 --> 00:12:36,810 So let's go back to our original problem. 223 00:12:36,809 --> 00:12:40,589 So our probability that the forward reaction is going to 224 00:12:40,590 --> 00:12:43,519 occur in the box is going to be equal to some probability 225 00:12:43,519 --> 00:12:46,139 that is going to react-- given that you're on the box, that's 226 00:12:46,139 --> 00:12:48,870 some constant value-- times the probability that they're 227 00:12:48,870 --> 00:12:49,919 in the box. 228 00:12:49,919 --> 00:12:54,029 And I'm making the claim that that's equal to all of these 229 00:12:54,029 --> 00:12:55,399 things multiplied by each other. 230 00:12:55,399 --> 00:12:59,009 So that's the concentration of hydrogen times some scaling 231 00:12:59,009 --> 00:13:01,509 factor, some other scaling factor-- I'll call it K sub 232 00:13:01,509 --> 00:13:04,990 s-- times the concentration of hydrogen times some scaling 233 00:13:04,990 --> 00:13:07,440 factor, times the concentration of hydrogen 234 00:13:07,440 --> 00:13:12,230 times some scaling factor, times the concentration of 235 00:13:12,230 --> 00:13:15,389 nitrogen times some scaling factor. 236 00:13:15,389 --> 00:13:17,189 And what is that equal to? 237 00:13:17,190 --> 00:13:20,890 Well, if you combine all the constants, a bunch of scaling 238 00:13:20,889 --> 00:13:22,590 constants times the constant out here, that all just 239 00:13:22,590 --> 00:13:24,310 becomes a constant. 240 00:13:24,309 --> 00:13:31,639 So you get the probability of the forward reaction in the 241 00:13:31,639 --> 00:13:37,449 box is going to be equal to just some constant-- let's 242 00:13:37,450 --> 00:13:40,500 just call it constant forward-- times the 243 00:13:40,500 --> 00:13:44,779 concentration of the hydrogen to the third power-- I 244 00:13:44,779 --> 00:13:47,129 multiplied it three times-- times the 245 00:13:47,129 --> 00:13:49,820 concentration of nitrogen. 246 00:13:49,820 --> 00:13:54,160 Now, if you wanted to go in the reverse direction, 247 00:13:54,159 --> 00:13:59,059 probability of reverse, you could use the exact same 248 00:13:59,059 --> 00:14:01,679 argument that I just used, and I'm not going to do it just 249 00:14:01,679 --> 00:14:04,759 for the sake of time, but it'll be some constant. 250 00:14:04,759 --> 00:14:07,789 This is the constant that the ammonia will react in the 251 00:14:07,789 --> 00:14:12,349 reverse direction on its own, times the scaling factor, and 252 00:14:12,350 --> 00:14:12,800 all of that. 253 00:14:12,799 --> 00:14:14,689 But it's the same exact idea. 254 00:14:14,690 --> 00:14:18,000 So, times the reverse, which is just going to be-- How many 255 00:14:18,000 --> 00:14:19,799 Moles of ammonia do we have? 256 00:14:19,799 --> 00:14:21,000 Or how many molecules? 257 00:14:21,000 --> 00:14:23,090 What's its stoichiometric coefficient? 258 00:14:23,090 --> 00:14:25,100 It's 2. 259 00:14:25,100 --> 00:14:28,860 So the reverse direction is going to be concentration of 260 00:14:28,860 --> 00:14:31,580 ammonia to the second power. 261 00:14:31,580 --> 00:14:33,860 And when we're in equilibrium, these two things, the 262 00:14:33,860 --> 00:14:36,050 probability of having a forward reaction in the box, 263 00:14:36,049 --> 00:14:38,029 is going to be equal to the probability of a reverse 264 00:14:38,029 --> 00:14:39,490 reaction the box. 265 00:14:39,490 --> 00:14:42,220 So these two things are going to equal each other. 266 00:14:42,220 --> 00:14:45,320 So this is going to equal-- if I could just copy and paste 267 00:14:45,320 --> 00:14:47,640 it-- that up there. 268 00:14:47,639 --> 00:14:50,689 269 00:14:50,690 --> 00:14:51,990 There you go. 270 00:14:51,990 --> 00:14:56,740 Then if you set the constants equal to each other, and then 271 00:14:56,740 --> 00:14:59,700 you could pick what the-- you normally put the products on 272 00:14:59,700 --> 00:15:01,129 the right-hand side of the equation. 273 00:15:01,129 --> 00:15:05,620 So I'll take these and divide them into this, and I'll 274 00:15:05,620 --> 00:15:14,970 divide that into that, and you're left with KF/KR is 275 00:15:14,970 --> 00:15:23,540 equal to the concentration of ammonia to the second power, 276 00:15:23,539 --> 00:15:27,079 divided by the concentration of hydrogen to the third 277 00:15:27,080 --> 00:15:31,730 power, times the concentration of nitrogen. 278 00:15:31,730 --> 00:15:35,310 And you could call that the equilibrium constant. 279 00:15:35,309 --> 00:15:38,099 And there you have it. 280 00:15:38,100 --> 00:15:41,129 A pseudo-derived formula for the equilibrium constant. 281 00:15:41,129 --> 00:15:43,789 It's all, at least in my mind, coming from common sense, from 282 00:15:43,789 --> 00:15:46,490 the probability that if you have a small volume, things 283 00:15:46,490 --> 00:15:48,690 are actually going to react. 284 00:15:48,690 --> 00:15:49,267