1 00:00:01,510 --> 00:00:03,027 - [Voiceover] Let's say you wanted to calculate the mass 2 00:00:03,027 --> 00:00:05,771 of a helium four nucleus, well first we need 3 00:00:05,771 --> 00:00:07,247 to figure out what's in the nucleus, 4 00:00:07,247 --> 00:00:09,113 so, with an atomic number of two, 5 00:00:09,113 --> 00:00:11,817 we know there are two protons in the nucleus. 6 00:00:11,817 --> 00:00:14,533 And, subtract the atomic number from the mass number, 7 00:00:14,533 --> 00:00:17,558 four minus two gives us two neutrons. 8 00:00:17,558 --> 00:00:20,644 And so, if we know the mass of a proton, 9 00:00:20,644 --> 00:00:21,922 and the mass of a neutron, 10 00:00:21,922 --> 00:00:24,475 we could easily calculate the expected mass 11 00:00:24,475 --> 00:00:27,529 of a helium four nucleus, and the mass 12 00:00:27,529 --> 00:00:30,812 of a proton in amus, atomic mass units, 13 00:00:30,812 --> 00:00:35,812 is equal to 1.00727647 and we have two protons, 14 00:00:38,928 --> 00:00:41,487 so we need to multiply this number by two. 15 00:00:41,487 --> 00:00:43,867 So let's go ahead and get out the calculator, 16 00:00:43,867 --> 00:00:48,867 and let's do that, so 1.00727647 times two 17 00:00:52,455 --> 00:00:57,455 gives us 2.01455294 so this is equal to 18 00:00:58,367 --> 00:01:03,367 2.01455294, remember these are amus, atomic mass units. 19 00:01:08,549 --> 00:01:11,068 A neutron, one neutron, has a mass 20 00:01:11,068 --> 00:01:16,068 in amus of 1.00866490 and we have two neutrons, 21 00:01:19,876 --> 00:01:21,919 so we have to multiply this number by two. 22 00:01:21,919 --> 00:01:23,791 So let's go ahead and do that. 23 00:01:23,791 --> 00:01:28,791 So we have 1.00866490 times two, 24 00:01:31,763 --> 00:01:35,857 and this gives us 2.0173298. 25 00:01:35,857 --> 00:01:40,857 So 2.0173298 amus, so the mass 26 00:01:45,496 --> 00:01:47,923 of a helium nucleus, if we add those two 27 00:01:47,923 --> 00:01:51,295 numbers together, we should get that mass. 28 00:01:51,295 --> 00:01:54,276 So let's do that math, so this number plus 29 00:01:54,276 --> 00:01:59,276 our first number 2.01455294 gives us 4.03188274. 30 00:02:06,770 --> 00:02:11,770 So, 4.031882748 amus, so this is the predicted mass 31 00:02:18,834 --> 00:02:21,429 of the helium four nucleus. 32 00:02:21,429 --> 00:02:22,806 So let me go ahead and write this, 33 00:02:22,806 --> 00:02:26,637 this is the predicted mass. 34 00:02:26,637 --> 00:02:29,086 The actual mass of a helium four nucleus 35 00:02:29,086 --> 00:02:34,086 has been measured to be 4.00150608 amus. 36 00:02:40,789 --> 00:02:42,330 Let me go ahead and write this, 37 00:02:42,330 --> 00:02:47,287 this is the actual mass, so the actual mass. 38 00:02:47,287 --> 00:02:48,897 All right, there's a difference there. 39 00:02:48,897 --> 00:02:50,544 They're not the same number. 40 00:02:50,544 --> 00:02:54,106 The predicted number is higher than the actual mass. 41 00:02:54,106 --> 00:02:57,689 So let's calculate the difference between those two numbers. 42 00:02:57,689 --> 00:03:00,147 So if we subtract the actual 43 00:03:00,147 --> 00:03:02,607 from the predicted, we can see 44 00:03:02,607 --> 00:03:04,169 the difference between those numbers. 45 00:03:04,169 --> 00:03:06,017 So let's go ahead and do that. 46 00:03:06,017 --> 00:03:08,100 All right, so we have the predicted, 47 00:03:08,100 --> 00:03:10,344 and then we're going to subtract the actual from that. 48 00:03:10,344 --> 00:03:15,344 So 4.00150608 is going to give us .03037666 amus. 49 00:03:24,223 --> 00:03:25,938 So let's go ahead and write that. 50 00:03:25,938 --> 00:03:30,938 So, 0.03037666 amus, so this is the difference 51 00:03:34,913 --> 00:03:39,913 between those two numbers, and we call this the mass defect. 52 00:03:40,051 --> 00:03:41,577 So let me go ahead and write that down here. 53 00:03:41,577 --> 00:03:44,610 This is called the mass defect. 54 00:03:44,610 --> 00:03:47,943 The difference between the predicted mass 55 00:03:47,943 --> 00:03:51,647 of the nucleus and the actual mass of the nucleus. 56 00:03:51,647 --> 00:03:55,559 And it looks like we lost some mass here, 57 00:03:55,559 --> 00:03:57,505 and really what's happened is the mass, 58 00:03:57,505 --> 00:04:01,000 this mass right here, the mass defect, 59 00:04:01,000 --> 00:04:04,927 was converted into energy when the nucleus was formed. 60 00:04:04,927 --> 00:04:06,159 So that's pretty interesting, 61 00:04:06,159 --> 00:04:09,437 and we can calculate how much energy, 62 00:04:09,437 --> 00:04:12,269 according to Einstein's famous equation, 63 00:04:12,269 --> 00:04:15,194 which relates energy and mass. 64 00:04:15,194 --> 00:04:17,709 So this is the one that most people know. 65 00:04:17,709 --> 00:04:20,899 It's E is equal to MC squared. 66 00:04:20,899 --> 00:04:24,413 So E is equal to MC squared, where E refers 67 00:04:24,413 --> 00:04:28,277 to the energy in Joules, M is the mass 68 00:04:28,277 --> 00:04:32,451 in kilograms, and C is the speed of light, 69 00:04:32,451 --> 00:04:34,755 which is in meters per second, 70 00:04:34,755 --> 00:04:36,223 and so you'd be squaring that, 71 00:04:36,223 --> 00:04:38,966 so it would be meters squared over seconds squared. 72 00:04:38,966 --> 00:04:42,789 All right, so let's calculate the mass 73 00:04:42,789 --> 00:04:44,283 that we're dealing with here. 74 00:04:44,283 --> 00:04:46,189 So, using Einstein's equation, 75 00:04:46,189 --> 00:04:49,306 we see we need kilograms, and we've calculated 76 00:04:49,306 --> 00:04:52,356 the mass in amus and so the first thing we need 77 00:04:52,356 --> 00:04:56,628 to do, is convert the amus into kilograms. 78 00:04:56,628 --> 00:04:58,900 And I briefly mentioned in an earlier video, 79 00:04:58,900 --> 00:05:02,308 the conversion factor between amus and kilograms, 80 00:05:02,308 --> 00:05:04,611 so amus is just a different measure of mass. 81 00:05:04,611 --> 00:05:06,869 Let's get some more room down here. 82 00:05:06,869 --> 00:05:09,236 All right, so one amu is equal 83 00:05:09,236 --> 00:05:14,236 to 1.66054 times 10 to the negative 27 kilograms. 84 00:05:14,302 --> 00:05:16,255 And so the first thing we need to do, 85 00:05:16,255 --> 00:05:19,286 is convert that number, so let's go ahead and write it down. 86 00:05:19,286 --> 00:05:24,286 So 0.03037666 amus, so mathematically, 87 00:05:27,229 --> 00:05:31,514 how would I convert the amus into kilograms? 88 00:05:31,514 --> 00:05:34,692 Well, I need to cancel out the amu units. 89 00:05:34,692 --> 00:05:36,842 So my conversion factor is going 90 00:05:36,842 --> 00:05:41,842 to be 1.66054 times 10 to the negative 27, 91 00:05:44,293 --> 00:05:46,091 that's how many kilograms we have 92 00:05:46,091 --> 00:05:50,407 for every one amu, so I can put that in here. 93 00:05:50,407 --> 00:05:52,067 So there's my conversion factor. 94 00:05:52,067 --> 00:05:53,901 And notice what happens when we do this, 95 00:05:53,901 --> 00:05:55,578 our units for amus cancel, and this 96 00:05:55,578 --> 00:05:57,272 is going to give us kilograms. 97 00:05:57,272 --> 00:06:00,120 So let's go ahead and do this math. 98 00:06:00,120 --> 00:06:02,081 So what is this equal to? 99 00:06:02,081 --> 00:06:04,753 All right, so we get out the calculator, 100 00:06:04,753 --> 00:06:08,213 and we take this number, and we multiply it 101 00:06:08,213 --> 00:06:10,920 by, let's use some parentheses here, 102 00:06:10,920 --> 00:06:15,920 1.66054 times 10 to the negative 27, 103 00:06:23,721 --> 00:06:25,738 so let's see what this gives us here, 104 00:06:25,738 --> 00:06:30,738 so this gives us 5.04417 times 10 to the negative 29, 105 00:06:32,762 --> 00:06:34,331 so let's round it like that. 106 00:06:34,331 --> 00:06:38,209 So this is going to give us, let's put it right here, 107 00:06:38,209 --> 00:06:43,209 5.04417 times 10 to the negative 29 kilograms. 108 00:06:47,301 --> 00:06:50,476 All right, so now we have the mass 109 00:06:50,476 --> 00:06:53,393 in kilograms, so let's get some more room, 110 00:06:53,393 --> 00:06:55,753 and let's go ahead and calculate the energy. 111 00:06:55,753 --> 00:06:58,841 All right, so this is the mass that was lost 112 00:06:58,841 --> 00:07:01,861 when the nucleus was formed, so let's figure out 113 00:07:01,861 --> 00:07:03,946 how much energy was given off. 114 00:07:03,946 --> 00:07:07,158 So the total energy, energy is equal 115 00:07:07,158 --> 00:07:10,202 to that mass, so let's go ahead and plug that in. 116 00:07:10,202 --> 00:07:15,202 5.04417 times 10 to the negative 29, 117 00:07:16,645 --> 00:07:19,471 times the speed of light, which is approximately 118 00:07:19,471 --> 00:07:21,997 three times 10 to the eighth meters per second, 119 00:07:21,997 --> 00:07:23,692 we'll use a more exact number, 120 00:07:23,692 --> 00:07:28,692 2.99792 times 10 to the eighth, 121 00:07:29,582 --> 00:07:32,883 and we need to square that number. 122 00:07:32,883 --> 00:07:35,836 All right, so let's do our last calculation here. 123 00:07:35,836 --> 00:07:37,879 So let's start with the speed of light. 124 00:07:37,879 --> 00:07:42,879 2.99792 times 10 to the eighth, 125 00:07:47,182 --> 00:07:49,810 and then we need to square that number. 126 00:07:49,810 --> 00:07:51,277 So we get this number, and we're going 127 00:07:51,277 --> 00:07:54,333 to multiply that by our mass, 128 00:07:54,333 --> 00:07:59,333 5.04417 times 10 to the negative 29th, 129 00:08:03,452 --> 00:08:05,259 and so let's see what that gives us. 130 00:08:05,259 --> 00:08:10,181 4.3346 times 10 to the negative 12, 131 00:08:10,181 --> 00:08:11,295 so let's write that down. 132 00:08:11,295 --> 00:08:16,295 4 point, I think there was a five in there, 4.5334. 133 00:08:16,815 --> 00:08:21,815 So, 4.53346 times 10 to the negative 12, 134 00:08:25,881 --> 00:08:27,870 let me just double check that real fast here. 135 00:08:27,870 --> 00:08:32,241 So 4.55346 times 10 to the negative 12 is our answer. 136 00:08:32,241 --> 00:08:34,178 And the units should be Joules, 137 00:08:34,178 --> 00:08:37,397 so that's how much energy is given off. 138 00:08:37,397 --> 00:08:39,235 So here's our final calculation, 139 00:08:39,235 --> 00:08:41,212 so it took us awhile to get there. 140 00:08:41,212 --> 00:08:43,821 So remember, this is the energy that's 141 00:08:43,821 --> 00:08:46,584 released when the nucleus is formed. 142 00:08:46,584 --> 00:08:48,349 So let me go ahead and get some more space, 143 00:08:48,349 --> 00:08:49,937 and let me write this down. 144 00:08:49,937 --> 00:08:54,937 The energy released when the nucleus is formed. 145 00:09:00,884 --> 00:09:03,569 The energy released when the nucleus is formed. 146 00:09:03,569 --> 00:09:07,306 So let's draw a picture of what's happening. 147 00:09:07,306 --> 00:09:10,780 So we were talking about the helium nucleus, 148 00:09:10,780 --> 00:09:14,584 which had two protons, let me go ahead and draw those 149 00:09:14,584 --> 00:09:18,348 two protons in here, and two neutrons, 150 00:09:18,348 --> 00:09:21,552 so let me use neutrons here like that. 151 00:09:21,552 --> 00:09:25,579 And these came together, to form our nucleus. 152 00:09:25,579 --> 00:09:27,624 All right, so these two things come together. 153 00:09:27,624 --> 00:09:32,178 So we have our two positive charges in our nucleus. 154 00:09:32,178 --> 00:09:35,876 And then we have our neutrons as well. 155 00:09:35,876 --> 00:09:40,753 And this is supposed to represent our nucleus, 156 00:09:40,753 --> 00:09:42,863 our helium four nucleus here. 157 00:09:42,863 --> 00:09:45,727 So, energy is released when the nucleus is formed. 158 00:09:45,727 --> 00:09:47,946 So we could also put in this energy. 159 00:09:47,946 --> 00:09:49,677 So this energy is given off, 160 00:09:49,677 --> 00:09:51,693 so that's the number we just calculated. 161 00:09:51,693 --> 00:09:54,740 We spent several minutes getting this number, 162 00:09:54,740 --> 00:09:56,345 and this is the energy that's released 163 00:09:56,345 --> 00:09:58,779 when the nucleus is formed here. 164 00:09:58,779 --> 00:10:00,710 And so this is just a nice little picture 165 00:10:00,710 --> 00:10:02,455 to think about what's happening. 166 00:10:02,455 --> 00:10:03,797 So whatever this nucleus formed, 167 00:10:03,797 --> 00:10:05,099 energy was given off. 168 00:10:05,099 --> 00:10:08,777 The nucleus is stable because energy is given off here. 169 00:10:08,777 --> 00:10:10,922 And we can also think about going the opposite direction. 170 00:10:10,922 --> 00:10:13,100 So if we started with the nucleus, 171 00:10:13,100 --> 00:10:14,699 and you wanted to break it up 172 00:10:14,699 --> 00:10:17,290 into the individual components, 173 00:10:17,290 --> 00:10:19,080 so if you took this nucleus here, 174 00:10:19,080 --> 00:10:21,357 and you applied some energy, 175 00:10:21,357 --> 00:10:23,356 you could break it up and turn it back 176 00:10:23,356 --> 00:10:26,133 into protons and neutrons, and that energy 177 00:10:26,133 --> 00:10:27,565 that you would have to apply, 178 00:10:27,565 --> 00:10:30,528 is also equal to this energy. 179 00:10:30,528 --> 00:10:33,949 So this is also called the nuclear binding energy. 180 00:10:33,949 --> 00:10:36,222 So let me go ahead and write that. 181 00:10:36,222 --> 00:10:41,222 Nuclear binding energy, so again, 182 00:10:42,969 --> 00:10:45,244 that is the term for the energy 183 00:10:45,244 --> 00:10:47,591 that we just calculated here. 184 00:10:47,591 --> 00:10:49,165 So you can think about it two different ways, 185 00:10:49,165 --> 00:10:52,417 it's the energy that's released when the nucleus is formed, 186 00:10:52,417 --> 00:10:55,686 and that's also the amount of energy that's needed 187 00:10:55,686 --> 00:10:58,970 to break the nucleus apart. 188 00:10:58,970 --> 00:11:02,162 And so the nucleus is stable in this case. 189 00:11:02,162 --> 00:11:04,522 So we have a stable nucleus right? 190 00:11:04,522 --> 00:11:06,867 This is a stable nucleus, but that's a little bit weird 191 00:11:06,867 --> 00:11:08,714 because we have these positive charges, 192 00:11:08,714 --> 00:11:11,187 and we know that positive charges repel. 193 00:11:11,187 --> 00:11:13,343 So these two positive charges here 194 00:11:13,343 --> 00:11:14,613 are repelling each other right? 195 00:11:14,613 --> 00:11:16,695 We know that like charges repel, 196 00:11:16,695 --> 00:11:19,153 and so there must be some other force 197 00:11:19,153 --> 00:11:22,375 that's holding our nucleus together. 198 00:11:22,375 --> 00:11:24,430 And that's called the nuclear strong force, 199 00:11:24,430 --> 00:00:00,000 and we'll talk more about that in the next video.