1 00:00:01,716 --> 00:00:03,700 - In the Bohr model of the hydrogen atom 2 00:00:03,700 --> 00:00:05,837 we have one proton in the nucleus. 3 00:00:05,837 --> 00:00:07,740 So I draw in a positive charge here 4 00:00:07,740 --> 00:00:11,421 and a negatively charged electron orbiting the nucleus, 5 00:00:11,421 --> 00:00:13,998 so kind of like the planets orbiting the sun. 6 00:00:13,998 --> 00:00:16,494 Even though the Bohr model is not reality 7 00:00:16,494 --> 00:00:20,314 it is useful for a concept of the atom. 8 00:00:20,314 --> 00:00:21,777 It's useful to calculate, 9 00:00:21,777 --> 00:00:25,596 say for example we can calculate the radius of this circle. 10 00:00:25,596 --> 00:00:27,651 We're actually gonna do that in this video. 11 00:00:27,651 --> 00:00:30,127 It's worth going in to some of the details. 12 00:00:30,127 --> 00:00:31,401 But I should warn you that this is 13 00:00:31,401 --> 00:00:33,386 a lot of physics in this video as well. 14 00:00:33,386 --> 00:00:36,080 If you don't like physics you can jump to the next video 15 00:00:36,080 --> 00:00:38,425 where I show you the result of what we're going to calculate 16 00:00:38,425 --> 00:00:40,201 in this video. 17 00:00:40,201 --> 00:00:41,618 Going back to the electron here, 18 00:00:41,618 --> 00:00:43,719 let's say the electron is going around 19 00:00:43,719 --> 00:00:45,298 counter-clockwise. 20 00:00:45,298 --> 00:00:47,886 The velocity of that electron at this point 21 00:00:47,886 --> 00:00:50,127 is tangent to the circle. 22 00:00:50,127 --> 00:00:52,925 That's the direction of the velocity vector. 23 00:00:52,925 --> 00:00:55,839 The electron has mass, m let's say. 24 00:00:55,839 --> 00:00:58,727 The electron is going to feel a force. 25 00:00:58,727 --> 00:01:01,811 It's going to be attracted to the nucleus. 26 00:01:01,811 --> 00:01:04,295 Opposite charges attract. 27 00:01:04,295 --> 00:01:07,418 This negatively charged electron is gonna feel a force 28 00:01:07,418 --> 00:01:09,937 towards the center of the circle. 29 00:01:09,937 --> 00:01:12,109 That's a centripetal force. 30 00:01:12,109 --> 00:01:15,057 In this case we're talking about the electric force. 31 00:01:15,057 --> 00:01:18,145 This is the electric force that's causing the 32 00:01:18,145 --> 00:01:21,332 electron to move in a circle. 33 00:01:21,332 --> 00:01:24,594 We can find the electric force by using Coulomb's law. 34 00:01:24,594 --> 00:01:26,555 Over here in the left this is Coulomb's law, 35 00:01:26,555 --> 00:01:28,495 the electric force is equal to 36 00:01:28,495 --> 00:01:30,813 K, which is a constant, 37 00:01:30,813 --> 00:01:33,402 times q1, which is one of the charges. 38 00:01:33,402 --> 00:01:36,722 Let's just say that q1 is the charge on the proton. 39 00:01:36,722 --> 00:01:39,415 Times the other charge, q2, which we'll say 40 00:01:39,415 --> 00:01:41,517 is the charge on the electron. 41 00:01:41,517 --> 00:01:46,517 Divided by the distance between those two charges squared. 42 00:01:46,938 --> 00:01:48,134 This is Coulomb's law. 43 00:01:48,134 --> 00:01:50,943 Let's go ahead and plug in what we know so far. 44 00:01:50,943 --> 00:01:53,601 K is some constant which we'll get to later. 45 00:01:53,601 --> 00:01:55,981 q1 I said was the charge on the proton, 46 00:01:55,981 --> 00:01:59,162 and the charge in the proton we'll say is e for now, 47 00:01:59,162 --> 00:02:00,602 so elemental charge. 48 00:02:00,602 --> 00:02:03,099 q2 I said was the charge in the electron, 49 00:02:03,099 --> 00:02:06,372 and the electron has the same magnitude of charge 50 00:02:06,372 --> 00:02:08,183 as the proton but it's negative. 51 00:02:08,183 --> 00:02:11,457 So we put in a negative e here. 52 00:02:11,457 --> 00:02:16,240 Divided by the distance between the two charges squared. 53 00:02:16,240 --> 00:02:19,793 Force is equal to mass times acceleration 54 00:02:19,793 --> 00:02:21,426 using Newton's second law. 55 00:02:21,426 --> 00:02:25,377 m is the mass of the electron. 56 00:02:25,377 --> 00:02:27,468 This will be the centripetal acceleration 57 00:02:27,468 --> 00:02:30,486 since we're talking about a centripetal force. 58 00:02:30,486 --> 00:02:32,622 We know that the centripetal acceleration 59 00:02:32,622 --> 00:02:34,539 is equal to V squared over r. 60 00:02:34,539 --> 00:02:35,989 We can go ahead and plug in 61 00:02:35,989 --> 00:02:39,530 m times v squared over r. 62 00:02:39,530 --> 00:02:42,500 Immediately we can cancel out one of the r's. 63 00:02:42,500 --> 00:02:44,852 Since we only care about the magnitude of the force 64 00:02:44,852 --> 00:02:46,988 we know the direction of the electric force, 65 00:02:46,988 --> 00:02:48,904 we don't really care about this negative sign, 66 00:02:48,904 --> 00:02:50,645 so we can just say we only care about the magnitude 67 00:02:50,645 --> 00:02:52,468 of the electric force here. 68 00:02:52,468 --> 00:02:54,105 We can go ahead and simplify a little bit. 69 00:02:54,105 --> 00:02:58,145 This would be ke squared over r on the left, 70 00:02:58,145 --> 00:03:02,487 and on the right this would be mv squared. 71 00:03:02,487 --> 00:03:04,762 Continuing with some more classical physics, 72 00:03:04,762 --> 00:03:07,200 next we're gonna talk about angular momentum 73 00:03:07,200 --> 00:03:08,884 which is a tricky concept. 74 00:03:08,884 --> 00:03:11,240 Angular momentum is capital L, 75 00:03:11,240 --> 00:03:16,151 and one equation for it is r cross p 76 00:03:16,151 --> 00:03:20,284 where r is a vector and p is the linear momentum. 77 00:03:20,284 --> 00:03:23,790 Linear momentum is equal to the mass times 78 00:03:23,790 --> 00:03:24,905 the velocity. 79 00:03:24,905 --> 00:03:27,475 We're talking about the linear momentum of the electrons, 80 00:03:27,475 --> 00:03:28,613 so the mass of the electron 81 00:03:28,613 --> 00:03:31,712 times the velocity of the electron. 82 00:03:31,712 --> 00:03:34,615 Let's go ahead and plug this in for angular momentum. 83 00:03:34,615 --> 00:03:37,296 We're gonna take the angular momentum about the center 84 00:03:37,296 --> 00:03:40,071 of our circle here. 85 00:03:40,071 --> 00:03:41,777 The angular momentum at the center, 86 00:03:41,777 --> 00:03:43,496 so r is a vector. 87 00:03:43,496 --> 00:03:45,215 Let me go ahead and draw r in. 88 00:03:45,215 --> 00:03:46,607 So r is a vector. 89 00:03:46,607 --> 00:03:49,212 It's the distance from the center to 90 00:03:49,212 --> 00:03:50,651 where our electron is. 91 00:03:50,651 --> 00:03:52,346 So we have r right there. 92 00:03:52,346 --> 00:03:53,692 This is the r vector. 93 00:03:53,692 --> 00:03:55,968 I put in r. 94 00:03:55,968 --> 00:03:57,950 Cross is the cross product. 95 00:03:57,950 --> 00:04:00,361 This would be times the linear momentum, 96 00:04:00,361 --> 00:04:02,776 so times p which is mv, 97 00:04:02,776 --> 00:04:05,283 times the sine of the angle between 98 00:04:05,283 --> 00:04:07,663 the two vectors. 99 00:04:07,663 --> 00:04:10,218 Let's think about their other vector here. 100 00:04:10,218 --> 00:04:12,541 The other vector is the momentum vector. 101 00:04:12,541 --> 00:04:14,142 We took care of the r vector. 102 00:04:14,142 --> 00:04:16,742 The momentum vector is in the same direction 103 00:04:16,742 --> 00:04:19,459 as the velocity. 104 00:04:19,459 --> 00:04:21,449 If this is the direction of the velocity 105 00:04:21,449 --> 00:04:24,711 that's also the direction of the linear momentum vector. 106 00:04:24,711 --> 00:04:27,184 The angle between those two vectors, 107 00:04:27,184 --> 00:04:29,498 the angle between those two vectors is obviously 108 00:04:29,498 --> 00:04:30,802 90 degrees. 109 00:04:30,802 --> 00:04:33,403 Sine of 90 is 1. 110 00:04:33,403 --> 00:04:35,709 We can just say the angular momentum is equal to 111 00:04:35,709 --> 00:04:38,647 rmv times 1. 112 00:04:38,647 --> 00:04:41,793 Niels Bohr thought that this angular momentum 113 00:04:41,793 --> 00:04:43,627 should be quantized. 114 00:04:43,627 --> 00:04:45,752 What he did was he set 115 00:04:45,752 --> 00:04:49,512 this angular momentum equal to some integer, 116 00:04:49,512 --> 00:04:52,612 so like 1, 2 or 3, or you can keep going. 117 00:04:52,612 --> 00:04:54,911 But let's just say an integer n, 118 00:04:54,911 --> 00:04:58,251 times h which is Planck's constant 119 00:04:58,251 --> 00:05:01,251 divided by 2 pi. 120 00:05:01,251 --> 00:05:03,863 This is what Bohr came up with. 121 00:05:03,863 --> 00:05:08,461 He took this and he solved for the velocity. 122 00:05:08,461 --> 00:05:09,784 Let's go ahead and do that. 123 00:05:09,784 --> 00:05:14,126 On the right we're just gonna solve for v. 124 00:05:14,126 --> 00:05:16,250 The velocity is equal to, 125 00:05:16,250 --> 00:05:18,967 this would just be n times h 126 00:05:18,967 --> 00:05:23,390 divided by 2 pi mr. 127 00:05:23,390 --> 00:05:24,563 We just solve for V, 128 00:05:24,563 --> 00:05:26,780 and then we're gonna take that. 129 00:05:26,780 --> 00:05:28,940 We just solve for V and we're gonna plug that 130 00:05:28,940 --> 00:05:32,468 into our other equation over here on the left. 131 00:05:32,468 --> 00:05:34,083 Let's go ahead and do that. 132 00:05:34,083 --> 00:05:37,740 We would have ke squared over r. 133 00:05:37,740 --> 00:05:41,780 On the right we would have m times all of that, 134 00:05:41,780 --> 00:05:45,924 n times h over 2 pi mr. 135 00:05:45,924 --> 00:05:48,269 Then we just square all of that. 136 00:05:48,269 --> 00:05:50,080 Let's go ahead and get some more room 137 00:05:50,080 --> 00:05:52,925 and let's continue with our algebra here. 138 00:05:52,925 --> 00:05:57,407 We have ke squared over r is equal to 139 00:05:57,407 --> 00:05:59,902 the mass times, 140 00:05:59,902 --> 00:06:01,899 so we square everything in parenthesis. 141 00:06:01,899 --> 00:06:04,302 n squared, h squared, 142 00:06:04,302 --> 00:06:08,621 4 pi squared, 143 00:06:08,621 --> 00:06:12,266 m squared, r squared. 144 00:06:12,266 --> 00:06:13,404 We can cancel a few things. 145 00:06:13,404 --> 00:06:15,559 We can cancel out one of these m's here. 146 00:06:15,559 --> 00:06:18,299 We can cancel out on of these r's. 147 00:06:18,299 --> 00:06:20,377 Now we would have on the left side 148 00:06:20,377 --> 00:06:22,959 ke squared is equal to 149 00:06:22,959 --> 00:06:25,791 n squared h squared 150 00:06:25,791 --> 00:06:30,191 over 4 pi squared. 151 00:06:30,191 --> 00:06:33,548 We would have one m left and one r left. 152 00:06:35,078 --> 00:06:37,818 The goal of all this is to solve for 153 00:06:37,818 --> 00:06:40,175 the radius of that circle. 154 00:06:40,175 --> 00:06:42,520 To solve for r we could start by multiplying 155 00:06:42,520 --> 00:06:45,098 both sides by 4 pi squared mr. 156 00:06:45,098 --> 00:06:46,549 We would get 157 00:06:46,549 --> 00:06:51,549 ke squared 4 pi squared mr 158 00:06:51,925 --> 00:06:53,539 on the left side. 159 00:06:53,539 --> 00:06:57,501 In the right side we would get m squared h squared. 160 00:06:57,501 --> 00:07:00,505 We're going to solve for r. 161 00:07:00,505 --> 00:07:02,200 Let's go ahead and do that. 162 00:07:02,200 --> 00:07:04,324 r would be equal to 163 00:07:04,324 --> 00:07:08,562 n squared h squared over, 164 00:07:08,562 --> 00:07:12,545 this would be over ke squared 165 00:07:12,545 --> 00:07:16,956 4 pi squared m. 166 00:07:19,189 --> 00:07:23,122 Now next we're going to take all of this stuff 167 00:07:23,122 --> 00:07:26,919 and we are going to plug in what those numbers are. 168 00:07:26,919 --> 00:07:29,461 For example h is Planck's constant, 169 00:07:29,461 --> 00:07:30,912 we know what that is. 170 00:07:30,912 --> 00:07:34,918 That's 6.626 times 171 00:07:34,918 --> 00:07:38,493 10 to the negative 34. 172 00:07:38,493 --> 00:07:41,837 We're going to be squaring that number. 173 00:07:41,837 --> 00:07:44,577 That's going to be over all of this. 174 00:07:44,577 --> 00:07:46,619 k, if you're taking physics, 175 00:07:46,619 --> 00:07:50,079 you know that k is equal to 9 times 10 to the ninth. 176 00:07:50,079 --> 00:07:51,400 It's a constant. 177 00:07:51,400 --> 00:07:53,465 e is elemental charge, 178 00:07:53,465 --> 00:07:56,287 the magnitude of charge on a proton or an electron 179 00:07:56,287 --> 00:08:01,024 is 1.6 times 10 to the negative 19 coulombs. 180 00:08:01,024 --> 00:08:02,685 We put that in there and that number 181 00:08:02,685 --> 00:08:04,843 needs to be squared as well. 182 00:08:04,843 --> 00:08:07,838 We have a 4 pi squared in there. 183 00:08:07,838 --> 00:08:10,764 Remember, m was the mass of the electron. 184 00:08:10,764 --> 00:08:12,157 You can look up the mass of the electron, 185 00:08:12,157 --> 00:08:17,046 it's 9.11 times 10 to the negative 31st kilograms. 186 00:08:18,405 --> 00:08:20,528 That's a lot of math. 187 00:08:20,528 --> 00:08:22,815 Rather than take out the calculator and show you, 188 00:08:22,815 --> 00:08:26,093 you can do that yourself. 189 00:08:26,093 --> 00:08:29,217 You'll see that that number comes out to be, 190 00:08:29,217 --> 00:08:30,302 this comes out to be, 191 00:08:30,302 --> 00:08:32,211 this is equal to, I'll put it down here, 192 00:08:32,211 --> 00:08:34,626 5.3 times 10 193 00:08:34,626 --> 00:08:36,345 to the negative 11. 194 00:08:36,345 --> 00:08:37,969 If you had time to do all the units 195 00:08:37,969 --> 00:08:40,245 you would get meters for this. 196 00:08:40,245 --> 00:08:42,068 Go ahead and do that calculation yourself 197 00:08:42,068 --> 00:08:43,751 and you would see that you get that number. 198 00:08:43,751 --> 00:08:45,144 That's a very important number. 199 00:08:45,144 --> 00:08:47,327 Let's plug that in 200 00:08:47,327 --> 00:08:48,871 to what we have so far in the left. 201 00:08:48,871 --> 00:08:52,609 The radius is equal to n squared times that number now. 202 00:08:52,609 --> 00:08:57,609 5.3 times 10 to the negative 11. 203 00:08:58,306 --> 00:09:01,662 Let's go ahead and plug in n is equal to 1, 204 00:09:01,662 --> 00:09:02,927 so an integer. 205 00:09:02,927 --> 00:09:06,260 This represents a ground state electron 206 00:09:06,260 --> 00:09:07,664 in hydrogen. 207 00:09:07,664 --> 00:09:12,157 If n is equal to 1, this would be r1 208 00:09:12,157 --> 00:09:15,825 is equal to 1 squared times this number. 209 00:09:15,825 --> 00:09:18,066 Obviously that's very simple math. 210 00:09:18,066 --> 00:09:21,743 We know that the radius, 211 00:09:21,743 --> 00:09:23,125 when n is equal to 1 212 00:09:23,125 --> 00:09:25,203 the radius is equal to this number, 213 00:09:25,203 --> 00:09:29,068 5.3 times 10 to the negative 11 meters. 214 00:09:29,068 --> 00:09:30,659 Let's go back up here. 215 00:09:30,659 --> 00:09:32,772 Let's go back up here to the picture 216 00:09:32,772 --> 00:09:34,386 so I can show you what we're talking about, 217 00:09:34,386 --> 00:09:36,545 why that's an important value. 218 00:09:36,545 --> 00:09:38,520 This is what we just calculated. 219 00:09:38,520 --> 00:09:42,357 We calculated this radius 220 00:09:42,357 --> 00:09:45,465 for a ground state electron in hydrogen. 221 00:09:45,465 --> 00:09:49,688 We calculated this distance and we called it r1. 222 00:09:49,688 --> 00:09:54,246 The idea of Niels Bohr 223 00:09:54,246 --> 00:09:56,684 by quantizing angular momentum 224 00:09:56,684 --> 00:09:59,900 that's going to limit your radii, 225 00:09:59,900 --> 00:10:01,525 the different radii that you could have. 226 00:10:01,525 --> 00:10:04,521 Let's go ahead and generalize this equation. 227 00:10:04,521 --> 00:10:09,362 We could say r, for any integer n, 228 00:10:09,362 --> 00:10:12,903 would be equal to n squared times this number, 229 00:10:12,903 --> 00:10:14,957 times r1. 230 00:10:14,957 --> 00:10:17,500 n squared times r1 which we just calculated 231 00:10:17,500 --> 00:10:21,435 to be 5.3 times 10 to the negative 11 meters. 232 00:10:21,435 --> 00:10:23,854 This is very important. 233 00:10:23,854 --> 00:10:28,441 r for any integer n is equal to n squared times r 1. 234 00:10:28,441 --> 00:10:31,934 This means only certain radii are allowed 235 00:10:31,934 --> 00:10:35,719 because Niels Bohr quantized the angular momentum. 236 00:10:35,719 --> 00:10:38,854 You have to have specific radii. 237 00:10:38,854 --> 00:10:42,197 We'll talk about the other radii in the next video. 238 00:10:42,197 --> 00:10:43,892 This video, after all that physics, 239 00:10:43,892 --> 00:10:45,935 we got this equation. 240 00:10:45,935 --> 00:10:48,419 We're gonna use that to go into more detail 241 00:10:48,419 --> 00:00:00,000 about the Bohr model radii.