1 00:00:01,716 --> 00:00:03,685 - Here we have a graph of exponential decay. 2 00:00:03,685 --> 00:00:08,158 Where N refers to the number of radioactive nuclei, 3 00:00:08,622 --> 00:00:10,633 alright as a function of time. 4 00:00:10,760 --> 00:00:12,459 And so this, right, 5 00:00:12,459 --> 00:00:15,560 this equation describes our graph. 6 00:00:15,642 --> 00:00:18,563 So this would be the number of radioactive nucleis 7 00:00:18,634 --> 00:00:21,915 at any time, T, is equal to N not, 8 00:00:21,915 --> 00:00:24,289 the initial number of nuclei, 9 00:00:24,383 --> 00:00:26,688 E to the negative lambda T. 10 00:00:28,877 --> 00:00:32,021 Lambda is equal to the decay constant. 11 00:00:32,063 --> 00:00:34,535 So this is just some constant number here. 12 00:00:34,852 --> 00:00:38,178 And you could also call this K, if you wanted to, 13 00:00:38,183 --> 00:00:39,240 you could call it K if you're thinking 14 00:00:39,240 --> 00:00:41,103 about the rate constant. 15 00:00:41,343 --> 00:00:42,727 But that's just some constant 16 00:00:42,727 --> 00:00:45,299 and now we're gonna multiply it by the time here. 17 00:00:45,775 --> 00:00:47,494 So let's say we wanted to, 18 00:00:47,676 --> 00:00:49,218 let's say we wanted to find 19 00:00:49,218 --> 00:00:52,631 what this point right here represents on our graph. 20 00:00:52,631 --> 00:00:54,765 Well that's when time is equal to zero. 21 00:00:55,062 --> 00:00:56,861 So let's plug in, let's plug in 22 00:00:56,861 --> 00:01:01,043 time is equal to zero into our equation. 23 00:01:01,485 --> 00:01:04,334 So this would be the number of radioactive nuclei 24 00:01:04,334 --> 00:01:07,060 when time is equal to zero, is equal to N not, 25 00:01:07,537 --> 00:01:10,484 times E to the negative lambda times T, 26 00:01:10,484 --> 00:01:11,460 which is zero. 27 00:01:11,519 --> 00:01:14,531 So that's that is equal to, 28 00:01:14,939 --> 00:01:17,731 this would be N not times E to the zero. 29 00:01:17,731 --> 00:01:19,389 Each of the zero is one. 30 00:01:19,653 --> 00:01:23,039 So the number of radioactive nuclei 31 00:01:23,039 --> 00:01:26,297 at time is equal to zero, is equal to N not. 32 00:01:26,486 --> 00:01:28,769 So this represents N not, 33 00:01:28,769 --> 00:01:32,411 the initial number of radioactive nuclei on our graph. 34 00:01:32,617 --> 00:01:35,451 And you could do this for any time T, right? 35 00:01:35,451 --> 00:01:36,599 You could just pick a time, 36 00:01:36,599 --> 00:01:38,646 let's say that's our time that we wanted, right, 37 00:01:38,646 --> 00:01:40,790 and go up to here and find what 38 00:01:40,790 --> 00:01:42,874 this value is on the graph. 39 00:01:42,874 --> 00:01:44,896 And so at any time T, right, 40 00:01:44,896 --> 00:01:48,121 this would represent the number of radioactive nuclei. 41 00:01:48,642 --> 00:01:50,529 Let's do it for half-life. 42 00:01:50,892 --> 00:01:53,625 So remember, when half-life-- 43 00:01:53,625 --> 00:01:56,637 when time is equal to a half-life, 44 00:01:56,719 --> 00:02:00,900 right, the number of radioactive nuclei, 45 00:02:01,068 --> 00:02:02,969 this would be, let's see one half-life, 46 00:02:02,969 --> 00:02:06,182 so this would be the initial divided by two. 47 00:02:06,184 --> 00:02:07,532 So half of it remains. 48 00:02:07,532 --> 00:02:09,990 So let's look at that on our graph, right, 49 00:02:09,990 --> 00:02:11,589 so if we take N not, we divide that by two, 50 00:02:11,589 --> 00:02:13,050 that's approximately here, 51 00:02:13,050 --> 00:02:15,027 so let's say this is N not over two. 52 00:02:15,322 --> 00:02:18,132 And we go over to, we go over to here, 53 00:02:18,326 --> 00:02:20,696 and we drop down to our time, 54 00:02:20,824 --> 00:02:24,363 so this should represent our half-life. 55 00:02:24,434 --> 00:02:25,990 That time should represent our half-life. 56 00:02:25,990 --> 00:02:27,557 So that's what it looks like graphically. 57 00:02:27,823 --> 00:02:30,357 Let's take these numbers, 58 00:02:31,126 --> 00:02:32,702 let's take the half-life 59 00:02:32,948 --> 00:02:36,462 and what the number of radioactive nuclei would be 60 00:02:36,462 --> 00:02:38,495 and let's plug it in to this equation. 61 00:02:38,914 --> 00:02:40,400 Alright, so let's go ahead and do that. 62 00:02:40,400 --> 00:02:41,743 Let's get some more room. 63 00:02:41,743 --> 00:02:44,833 So I'm just gonna rewrite that equation here. 64 00:02:44,870 --> 00:02:46,082 So we had 65 00:02:47,370 --> 00:02:49,659 the number of radioactive nuclei as a function of time 66 00:02:49,682 --> 00:02:53,595 is equal to the initial number of radioactive nuclei 67 00:02:53,608 --> 00:02:56,476 times E to the negative lambda T. 68 00:02:56,654 --> 00:02:58,360 So let's plug in those, 69 00:02:58,813 --> 00:02:59,781 let's plug in those. 70 00:02:59,781 --> 00:03:02,590 So when we're gonna talk about the half-life 71 00:03:02,616 --> 00:03:04,299 we're gonna plug that in here for the time. 72 00:03:04,550 --> 00:03:06,632 And then the number of radioactive nuclei 73 00:03:06,632 --> 00:03:08,450 would be N not divided by two. 74 00:03:08,567 --> 00:03:10,015 So let's plug those in. 75 00:03:10,243 --> 00:03:13,369 So we would have N not divided by two, 76 00:03:13,519 --> 00:03:17,407 is equal to N not E to the negative lambda 77 00:03:17,701 --> 00:03:19,846 times the half-life. 78 00:03:20,490 --> 00:03:23,964 Alright, well that cancels out the N not's. 79 00:03:24,484 --> 00:03:26,183 Alright so that gives us on the left side 80 00:03:26,183 --> 00:03:30,060 one half is equal to E to the negative lambda 81 00:03:30,540 --> 00:03:32,380 times the half-life. 82 00:03:32,924 --> 00:03:36,235 So next, let's get rid of the E. 83 00:03:36,444 --> 00:03:40,399 And we can do that by taking the natural log of both sides. 84 00:03:40,596 --> 00:03:43,899 Alright, so if I take the natural log of one half 85 00:03:44,240 --> 00:03:47,476 on the left side and I take the natural log of E 86 00:03:47,525 --> 00:03:51,222 to the negative lambda T one half. 87 00:03:51,945 --> 00:03:54,983 Alright, on the left side natural log of one half 88 00:03:54,983 --> 00:03:59,366 is equal to negative .693 89 00:03:59,505 --> 00:04:01,666 so this is just plug it in your calculator 90 00:04:01,666 --> 00:04:04,203 you'll get negative .693 91 00:04:04,403 --> 00:04:06,794 and the right, that takes care of this 92 00:04:07,317 --> 00:04:10,023 all that's equal to just this over here. 93 00:04:10,023 --> 00:04:11,392 Right, so now you would have-- 94 00:04:11,656 --> 00:04:15,779 this would negative lambda T one half. 95 00:04:16,189 --> 00:04:18,089 And so we don't have to worry about the negative signs, 96 00:04:18,089 --> 00:04:21,235 right, so this is just .693 97 00:04:21,411 --> 00:04:25,454 is equal to lambda times T one half. 98 00:04:25,761 --> 00:04:27,489 So we could solve for the half-life, 99 00:04:27,904 --> 00:04:30,376 right, so if we solve for T one half. 100 00:04:30,376 --> 00:04:32,108 So T one half would be 101 00:04:32,108 --> 00:04:35,502 equal to .693 102 00:04:35,877 --> 00:04:38,206 divided by lambda, 103 00:04:38,206 --> 00:04:40,193 divided by the decay constant. 104 00:04:40,264 --> 00:04:43,302 So this is one of those equations, right, 105 00:04:43,302 --> 00:04:45,323 that you see for half-life. 106 00:04:45,923 --> 00:04:49,489 So what if you wanted to go ahead and solve for lambda, 107 00:04:49,489 --> 00:04:50,586 the decay constant. 108 00:04:50,586 --> 00:04:52,724 Right, so that's obviously really simple. 109 00:04:52,724 --> 00:04:54,819 We just do the decay constant 110 00:04:54,819 --> 00:04:57,798 is equal to .693 111 00:04:59,234 --> 00:05:00,958 divided by the half-life. 112 00:05:01,096 --> 00:05:04,999 And so obviously I'm just rearranging this equation here. 113 00:05:04,999 --> 00:05:07,217 So you could solve for half-life 114 00:05:07,217 --> 00:05:09,843 or you could solve for the decay constant. 115 00:05:09,904 --> 00:05:11,894 You could go back and forth between the half-life 116 00:05:11,894 --> 00:05:13,185 and the decay constant. 117 00:05:13,199 --> 00:05:16,527 So if you know one, you can figure out the other one. 118 00:05:16,698 --> 00:05:18,588 Alright so that's thinking about 119 00:05:18,588 --> 00:05:20,793 the exponential decay graph. 120 00:05:20,793 --> 00:05:23,053 Let's talk about semi-log plots next, 121 00:05:23,053 --> 00:05:25,812 which is another way at looking at the data. 122 00:05:25,812 --> 00:05:27,685 And so let's get some room here. 123 00:05:27,685 --> 00:05:30,179 I'm going to rewrite our equation, right, 124 00:05:30,179 --> 00:05:32,517 so the number of radioactive nuclei 125 00:05:32,517 --> 00:05:35,756 is equal to the initial number 126 00:05:35,756 --> 00:05:39,622 times E to the negative lambda T. 127 00:05:40,755 --> 00:05:43,301 Alright so let's convert this into 128 00:05:44,417 --> 00:05:47,003 a linear, into a straight line. 129 00:05:47,003 --> 00:05:49,398 So what we have to do, we have divide by N not. 130 00:05:49,398 --> 00:05:51,925 We have N divided by N not. 131 00:05:52,189 --> 00:05:53,932 So we divide both sides by N not 132 00:05:53,932 --> 00:05:57,010 and we get E to the negative lambda T here. 133 00:05:57,351 --> 00:05:59,412 Right now, to get rid of this E 134 00:05:59,412 --> 00:06:01,064 once again we just take the natural log. 135 00:06:01,064 --> 00:06:03,101 So we take the natural log of both sides, 136 00:06:03,101 --> 00:06:05,452 so natural log of N over N not 137 00:06:05,452 --> 00:06:10,340 is equal to the natural log of E to the negative lambda T. 138 00:06:10,682 --> 00:06:14,156 Alright so on the left side we have a log property 139 00:06:14,349 --> 00:06:16,563 so natural log of N over N not 140 00:06:16,667 --> 00:06:18,746 is equal to natural log of N 141 00:06:18,746 --> 00:06:22,030 minus natural log of N not. 142 00:06:22,303 --> 00:06:25,574 And then on the right, right, this goes away 143 00:06:25,574 --> 00:06:26,980 and we're left with this. 144 00:06:26,980 --> 00:06:28,995 So negative lambda T, 145 00:06:29,112 --> 00:06:31,452 so we have negative lambda T 146 00:06:31,452 --> 00:06:32,888 on the right side here. 147 00:06:33,040 --> 00:06:34,365 And if we just rearrange this, 148 00:06:34,365 --> 00:06:37,403 right, so let's just do natural log of N 149 00:06:37,403 --> 00:06:40,373 is equal to negative lambda T, 150 00:06:40,373 --> 00:06:45,037 so we're gonna add LN of N not to this side. 151 00:06:45,264 --> 00:06:49,180 And now we have, now we have a very interesting, 152 00:06:49,810 --> 00:06:50,912 very interesting form. 153 00:06:50,912 --> 00:06:54,165 If you look at it closely you'll see this is the same thing 154 00:06:54,165 --> 00:06:58,203 as Y is equal to MX + B, 155 00:06:58,203 --> 00:07:00,753 which is the equation for a straight line. 156 00:07:00,988 --> 00:07:04,845 So Y, Y would be equal to natural log of N. 157 00:07:05,243 --> 00:07:09,514 M would be equal to negative lambda, 158 00:07:09,740 --> 00:07:12,822 alright so we're talking about T, right, 159 00:07:12,822 --> 00:07:14,015 as being X here. 160 00:07:14,015 --> 00:07:18,178 And then natural log of N not is equal to B. 161 00:07:18,178 --> 00:07:21,353 So Y is equal to MX + B if you remember the equation, 162 00:07:21,814 --> 00:07:24,359 it's the graph of a straight line, right, 163 00:07:25,204 --> 00:07:28,203 where M is the slope. 164 00:07:28,604 --> 00:07:31,041 Alright so, we could say the slope of this 165 00:07:31,496 --> 00:07:34,787 is equal to negative lambda. 166 00:07:35,320 --> 00:07:38,351 And remember that this is 167 00:07:38,351 --> 00:07:40,723 your vertical intercept, right? 168 00:07:40,723 --> 00:07:43,771 So B is your vertical intercept 169 00:07:43,771 --> 00:07:47,154 so if we graph this our vertical intercept 170 00:07:47,181 --> 00:07:51,406 should be natural log of N not. 171 00:07:51,511 --> 00:07:54,029 So let's go ahead and sketch this out really quickly. 172 00:07:54,086 --> 00:07:57,101 I won't be too concerned with details here 173 00:07:57,101 --> 00:07:58,960 but if we're graphing it. 174 00:07:58,960 --> 00:08:01,094 So if you think about this is being your Y axis, 175 00:08:01,094 --> 00:08:02,264 I'll just put this in parentheses 176 00:08:02,264 --> 00:08:03,434 cuz that's not really what we're doing. 177 00:08:03,434 --> 00:08:05,650 And this being your X axis, right, 178 00:08:05,650 --> 00:08:06,929 let's look at those again. 179 00:08:06,932 --> 00:08:08,917 So for my Y, 180 00:08:09,122 --> 00:08:12,021 I would be graphing natural log of N. 181 00:08:12,081 --> 00:08:13,487 So let me go ahead and use a different color 182 00:08:13,487 --> 00:08:14,601 so we can see it here. 183 00:08:14,601 --> 00:08:18,084 So natural of N on the Y axis. 184 00:08:18,943 --> 00:08:22,078 On the X axis here, that would be time. 185 00:08:22,362 --> 00:08:24,845 So we have time over here. 186 00:08:26,188 --> 00:08:28,247 Alright and we know that the vertical intercept 187 00:08:28,247 --> 00:08:31,876 is going to be natural log of N not. 188 00:08:31,925 --> 00:08:34,636 So this vertical intercept is going to be here 189 00:08:34,636 --> 00:08:36,837 natural log of N not. 190 00:08:36,837 --> 00:08:39,323 And we could prove that really quickly, right, 191 00:08:40,261 --> 00:08:44,168 we could say when time is equal to zero, 192 00:08:44,543 --> 00:08:46,770 right, so when time is equal to zero, 193 00:08:46,770 --> 00:08:48,966 let's plug that in, we could have natural log of N, 194 00:08:48,966 --> 00:08:52,729 which is equal to negative lambda times zero 195 00:08:52,792 --> 00:08:55,477 plus natural log of N not. 196 00:08:55,888 --> 00:08:59,589 Right, so this would go away and you can see 197 00:08:59,589 --> 00:09:02,252 that we would have natural log of N not 198 00:09:02,252 --> 00:09:04,848 would be equal to this point here. 199 00:09:04,848 --> 00:09:06,598 So that's our vertical intercept. 200 00:09:06,794 --> 00:09:10,447 And we know this is the graph of a straight line. 201 00:09:11,427 --> 00:09:13,861 And we're gonna have a negative slope here, 202 00:09:13,861 --> 00:09:15,842 so if I go ahead and draw this in 203 00:09:15,842 --> 00:09:16,788 it would look something like, 204 00:09:16,788 --> 00:09:19,224 oh just pretend like that's a straight line. 205 00:09:19,224 --> 00:09:20,304 I didn't do a very good job 206 00:09:20,304 --> 00:09:22,588 but you can use your imagination there. 207 00:09:22,588 --> 00:09:24,998 And the slope of this, right, 208 00:09:24,998 --> 00:09:27,501 so the slope, remember what slope is. 209 00:09:27,518 --> 00:09:30,996 That's change in Y over change in X. 210 00:09:32,998 --> 00:09:35,496 That would be the change in this axis 211 00:09:35,496 --> 00:09:36,917 over the change in this axis. 212 00:09:36,917 --> 00:09:39,055 That's equal to, right, 213 00:09:39,055 --> 00:09:41,882 that is equal to negative lambda, 214 00:09:41,942 --> 00:09:43,895 what we talked about over here. 215 00:09:43,978 --> 00:09:46,565 And so if you do a semi-log plot, right, 216 00:09:46,565 --> 00:09:49,647 so it's semi-log because we have this natural log 217 00:09:49,647 --> 00:09:51,213 over here versus here. 218 00:09:51,213 --> 00:09:52,966 It tells us some information, right, 219 00:09:52,966 --> 00:09:54,676 so it's another way to look at the data. 220 00:09:55,106 --> 00:09:57,549 You could find the slope of this straight line, 221 00:09:57,549 --> 00:10:01,478 right, take the negative of it and get your decay constant. 222 00:10:01,478 --> 00:10:04,008 And then from your decay constant 223 00:10:04,008 --> 00:10:05,922 you could get your half-life. 224 00:10:07,122 --> 00:10:08,985 And so once again sometimes you'll see 225 00:10:09,094 --> 00:10:11,467 semi-log plots done as just 226 00:10:11,467 --> 00:00:00,000 a different way at looking at the data.