1 00:00:00,000 --> 00:00:00,580 2 00:00:00,580 --> 00:00:03,200 SAL: Two videos ago we learned about half-lives. 3 00:00:03,200 --> 00:00:05,750 And we saw that they're good if we are trying to figure out 4 00:00:05,750 --> 00:00:09,020 how much of a compound we have left after one half-life, or 5 00:00:09,020 --> 00:00:10,830 two half-lives, or three half-lives. 6 00:00:10,830 --> 00:00:13,250 We can just take 1/2 of the compound at every period. 7 00:00:13,250 --> 00:00:15,940 But it's not as useful if we're trying to figure out how 8 00:00:15,940 --> 00:00:20,960 much of a compound we have after 1/2 of a half-life, or 9 00:00:20,960 --> 00:00:24,610 after one day, or 10 seconds, or 10 billion years. 10 00:00:24,610 --> 00:00:28,030 And to address that issue in the last video, I proved that 11 00:00:28,030 --> 00:00:30,720 it involved a little bit of sophisticated math. 12 00:00:30,720 --> 00:00:32,490 And if you haven't taken calculus, you can really just 13 00:00:32,490 --> 00:00:33,220 skip that video. 14 00:00:33,220 --> 00:00:35,990 You don't have to watch it for an intro math class. 15 00:00:35,990 --> 00:00:38,180 But if you're curious, that's where we proved 16 00:00:38,180 --> 00:00:38,950 the following formula. 17 00:00:38,950 --> 00:00:41,050 That at any given point of time, if you have some 18 00:00:41,050 --> 00:00:46,220 decaying atom, some element, it can be described as the 19 00:00:46,220 --> 00:00:49,220 amount of element you have at any period of time is equal to 20 00:00:49,220 --> 00:00:53,080 the amount you started off with, times e to some 21 00:00:53,080 --> 00:00:55,280 constant-- in the last video I use lambda. 22 00:00:55,280 --> 00:00:59,450 I could use k this time-- minus k times t. 23 00:00:59,450 --> 00:01:01,670 And then for a particular element with a particular 24 00:01:01,670 --> 00:01:04,980 half-life you can just solve for the k, and then apply it 25 00:01:04,980 --> 00:01:05,660 to your problem. 26 00:01:05,660 --> 00:01:08,030 So let's do that in this video, just so that all of 27 00:01:08,030 --> 00:01:12,110 these variables can become a little bit more concrete. 28 00:01:12,110 --> 00:01:14,790 So let's figure out the general formula for carbon. 29 00:01:14,790 --> 00:01:17,660 Carbon-14, that's the one that we addressed in the half-life. 30 00:01:17,660 --> 00:01:24,530 We saw that carbon-14 has a half-life of 5,730 years. 31 00:01:24,530 --> 00:01:28,200 So let's see if we can somehow take this information and 32 00:01:28,200 --> 00:01:31,390 apply it to this equation. 33 00:01:31,390 --> 00:01:35,500 So this tells us that after one half-life-- so 34 00:01:35,500 --> 00:01:39,360 t is equal to 5,730. 35 00:01:39,360 --> 00:01:45,930 N of 5,730 is equal to the amount we start off with. 36 00:01:45,930 --> 00:01:48,320 So we're starting off with, well, we're starting off with 37 00:01:48,320 --> 00:01:52,590 N sub 0 times e to the minus-- wherever you see the t you put 38 00:01:52,590 --> 00:02:01,180 the minus 5,730-- so minus k, times 5,730. 39 00:02:01,180 --> 00:02:02,860 That's how many years have gone by. 40 00:02:02,860 --> 00:02:06,660 And half-life tells us that after 5,730 years we'll have 41 00:02:06,660 --> 00:02:08,690 1/2 of our initial sample left. 42 00:02:08,690 --> 00:02:12,050 So we'll have 1/2 of our initial sample left. 43 00:02:12,050 --> 00:02:14,580 So if we try to solve this equation for 44 00:02:14,580 --> 00:02:16,000 k, what do we get? 45 00:02:16,000 --> 00:02:18,180 Divide both sides by N naught. 46 00:02:18,180 --> 00:02:22,110 Get rid of that variable, and then we're left with e to the 47 00:02:22,110 --> 00:02:27,620 minus 5,730k-- I'm just switching these two around-- 48 00:02:27,620 --> 00:02:29,075 is equal to 1/2. 49 00:02:29,075 --> 00:02:32,060 If we take the natural log of both sides, what do we get? 50 00:02:32,060 --> 00:02:34,702 51 00:02:34,702 --> 00:02:38,220 The natural log of e to anything, the natural log of e 52 00:02:38,220 --> 00:02:39,540 to the a is just a. 53 00:02:39,540 --> 00:02:47,170 So the natural log of this is minus 5,730k is equal to the 54 00:02:47,170 --> 00:02:49,030 natural log of 1/2. 55 00:02:49,030 --> 00:02:50,950 I just took the natural log of both sides. 56 00:02:50,950 --> 00:02:55,940 The natural log and natural log of both sides of that. 57 00:02:55,940 --> 00:03:01,400 And so to solve for k, we could just say, k is equal to 58 00:03:01,400 --> 00:03:08,170 the natural log of 1/2 over minus 5,730, which we did in 59 00:03:08,170 --> 00:03:09,030 the previous video. 60 00:03:09,030 --> 00:03:12,860 But let's see if we can do that again here, to avoid-- 61 00:03:12,860 --> 00:03:15,560 for those who might have skipped it. 62 00:03:15,560 --> 00:03:21,590 So if you have 1/2, 0.5, take the natural log, and then you 63 00:03:21,590 --> 00:03:29,580 divide it by 5,730, it's a negative 5,730, you get 1.2 64 00:03:29,580 --> 00:03:33,380 times 10 to the negative 4. 65 00:03:33,380 --> 00:03:38,830 So it equals 1.2 times 10 to the minus 4. 66 00:03:38,830 --> 00:03:42,250 So now we have the general formula for carbon-14, given 67 00:03:42,250 --> 00:03:43,480 its half-life. 68 00:03:43,480 --> 00:03:47,990 At any given point in time, after our starting point-- so 69 00:03:47,990 --> 00:03:52,940 this is for, let's call this for carbon-14, for c-14-- the 70 00:03:52,940 --> 00:03:55,560 amount of carbon-14 we're going to have left is going to 71 00:03:55,560 --> 00:04:00,690 be the amount that we started with times e to the minus k. k 72 00:04:00,690 --> 00:04:02,070 we just solved for. 73 00:04:02,070 --> 00:04:06,930 1.2 times 10 to the minus 4, times the amount of time that 74 00:04:06,930 --> 00:04:08,360 has passed by. 75 00:04:08,360 --> 00:04:11,700 This is our formula for carbon, for carbon-14. 76 00:04:11,700 --> 00:04:13,650 If we were doing this for some other element, we would use 77 00:04:13,650 --> 00:04:16,610 that element's half-life to figure out how much we're 78 00:04:16,610 --> 00:04:18,660 going to have at any given period of time to figure out 79 00:04:18,660 --> 00:04:20,170 the k value. 80 00:04:20,170 --> 00:04:21,880 So let's use this to solve a problem. 81 00:04:21,880 --> 00:04:25,680 Let's say that I start off with, I don't know, say I 82 00:04:25,680 --> 00:04:34,530 start off with 300 grams of carbon, carbon-14. 83 00:04:34,530 --> 00:04:39,150 And I want to know, how much do I have after, I don't know, 84 00:04:39,150 --> 00:04:43,250 after 2000 years? 85 00:04:43,250 --> 00:04:44,510 How much do I have? 86 00:04:44,510 --> 00:04:46,940 Well I just plug into the formula. 87 00:04:46,940 --> 00:04:53,900 N of 2000 is equal to the amount that I started off 88 00:04:53,900 --> 00:05:02,670 with, 300 grams, times e to the minus 1.2 times 10 to the 89 00:05:02,670 --> 00:05:07,990 minus 4, times t, is times 2000, times 2000. 90 00:05:07,990 --> 00:05:10,330 So what is that? 91 00:05:10,330 --> 00:05:13,230 So I already have that 1.2 times 10 the minus 4 there. 92 00:05:13,230 --> 00:05:18,920 So let me say, times 2000 equals-- and of course, this 93 00:05:18,920 --> 00:05:20,790 throws a negative out there, so let me put the negative 94 00:05:20,790 --> 00:05:21,730 number out there. 95 00:05:21,730 --> 00:05:23,060 So there's a negative. 96 00:05:23,060 --> 00:05:25,130 And I have to raise e to this power. 97 00:05:25,130 --> 00:05:27,430 So it's 0.241. 98 00:05:27,430 --> 00:05:30,790 So this is equal to N of 2000. 99 00:05:30,790 --> 00:05:34,110 The amount of the substance I can expect after 2000 years is 100 00:05:34,110 --> 00:05:42,630 equal to 300 times e to the minus 0.2419. 101 00:05:42,630 --> 00:05:48,370 And let's see, my calculator doesn't have an e to the 102 00:05:48,370 --> 00:05:50,660 power, so Let me just take e. 103 00:05:50,660 --> 00:05:51,900 I need to get a better calculator. 104 00:05:51,900 --> 00:05:53,840 I should get my scientific calculator back. 105 00:05:53,840 --> 00:05:57,830 But e is, let's say 2.71-- I can keep adding digits but 106 00:05:57,830 --> 00:06:05,130 I'll just do 2.71-- to the 0.24 negative, which is equal 107 00:06:05,130 --> 00:06:13,830 to 0.78 times the amount that I started off with, times 300, 108 00:06:13,830 --> 00:06:21,550 which is equal to 236 grams. So this is equal to 236 grams. 109 00:06:21,550 --> 00:06:24,480 So just like that, using this exponential decay formula, I 110 00:06:24,480 --> 00:06:27,140 was able to figure out how much of the carbon I have 111 00:06:27,140 --> 00:06:30,890 after kind of an unusual period of time, a 112 00:06:30,890 --> 00:06:32,540 non-half-life period of time. 113 00:06:32,540 --> 00:06:35,080 Let's do another one like this. 114 00:06:35,080 --> 00:06:36,170 Let's go the other way around. 115 00:06:36,170 --> 00:06:39,430 Let's say, I'm trying to figure out. 116 00:06:39,430 --> 00:06:48,980 Let's say I start off with 400 grams of c-14. 117 00:06:48,980 --> 00:06:53,440 And I want to know how long-- so I want to know a certain 118 00:06:53,440 --> 00:06:55,790 amount of time-- does it take for me to get to 119 00:06:55,790 --> 00:06:59,380 350 grams of c-14? 120 00:06:59,380 --> 00:07:03,040 So, you just say that 350 grams is how much 121 00:07:03,040 --> 00:07:03,870 I'm ending up with. 122 00:07:03,870 --> 00:07:07,570 It's equal to the amount that I started off with, 400 grams, 123 00:07:07,570 --> 00:07:10,430 times e to the minus k. 124 00:07:10,430 --> 00:07:15,160 That's minus 1.2 times 10 to the minus 4, times time. 125 00:07:15,160 --> 00:07:18,190 And now we solve for time. 126 00:07:18,190 --> 00:07:19,620 How do we do that? 127 00:07:19,620 --> 00:07:22,840 Well we could divide both sides by 400. 128 00:07:22,840 --> 00:07:24,960 What's 350 divided by 400? 129 00:07:24,960 --> 00:07:27,180 350 by 400. 130 00:07:27,180 --> 00:07:29,340 It's 7/8. 131 00:07:29,340 --> 00:07:31,230 So 0.875. 132 00:07:31,230 --> 00:07:37,780 So you get 0.875 is equal to e to the minus 1.2 times 10 to 133 00:07:37,780 --> 00:07:40,100 the minus 4t. 134 00:07:40,100 --> 00:07:41,890 You take the natural log of both sides. 135 00:07:41,890 --> 00:07:46,460 You get the natural log of 0.875 is equal to-- the 136 00:07:46,460 --> 00:07:49,100 natural log of e to anything is just the anything-- so it's 137 00:07:49,100 --> 00:07:54,990 equal to minus 1.2 times 10 to the minus 4t. 138 00:07:54,990 --> 00:08:00,280 And so t is equal to this divided by 1.2 times 10 139 00:08:00,280 --> 00:08:01,130 to the minus 4. 140 00:08:01,130 --> 00:08:07,560 So the natural log, 0.875 divided by minus 1.2 times 10 141 00:08:07,560 --> 00:08:10,650 to the minus 4, is equal to the amount of time it would 142 00:08:10,650 --> 00:08:13,620 take us to get from 400 grams to 350. 143 00:08:13,620 --> 00:08:14,535 [PHONE RINGS] 144 00:08:14,535 --> 00:08:17,930 My cell phone is ringing, let me turn that off. 145 00:08:17,930 --> 00:08:19,350 To 350. 146 00:08:19,350 --> 00:08:22,120 So let me do the math. 147 00:08:22,120 --> 00:08:26,030 So if you have 0.875, and we want to take the natural log 148 00:08:26,030 --> 00:08:37,120 of it, and divide it by minus 1.-- So divided by 1.2e 4 149 00:08:37,120 --> 00:08:38,480 negative, 10 to the negative 4. 150 00:08:38,480 --> 00:08:41,169 This is all a negative number. 151 00:08:41,169 --> 00:08:43,250 Oh, I'll just divide it by this, and then just take the 152 00:08:43,250 --> 00:08:44,290 negative of that. 153 00:08:44,290 --> 00:08:47,170 Equals that and then I have to take a negative. 154 00:08:47,170 --> 00:08:55,410 So this is equal to 1,112 years to get from 400 to 350 155 00:08:55,410 --> 00:08:57,440 grams of my substance. 156 00:08:57,440 --> 00:09:00,970 This might seem a little complicated, but if there's 157 00:09:00,970 --> 00:09:02,890 one thing you just have to do, is you just have to remember 158 00:09:02,890 --> 00:09:04,570 this formula. 159 00:09:04,570 --> 00:09:06,160 And if you want to know where it came from, watch the 160 00:09:06,160 --> 00:09:07,230 previous video. 161 00:09:07,230 --> 00:09:12,370 For any particular element you solve for this k value. 162 00:09:12,370 --> 00:09:15,000 And then you just substitute what you know, and then solve 163 00:09:15,000 --> 00:09:16,230 for what you don't know. 164 00:09:16,230 --> 00:00:00,000 I'll do a couple more of these in the next video.