1
00:00:01,475 --> 00:00:03,089
- If we continue with our Bohr model,

2
00:00:03,089 --> 00:00:04,574
the next thing we have to talk about

3
00:00:04,574 --> 00:00:06,374
are the different energy levels.

4
00:00:06,374 --> 00:00:09,335
And so we're gonna be talking
about energy in this video,

5
00:00:09,335 --> 00:00:11,915
and once again, there's a lot
of derivation using physics,

6
00:00:11,915 --> 00:00:13,614
so you can jump ahead to the next video

7
00:00:13,614 --> 00:00:16,041
to see what we come up with in this video,

8
00:00:16,041 --> 00:00:17,631
to see how it's applied.

9
00:00:17,938 --> 00:00:20,170
Alright, so we need to talk about energy,

10
00:00:20,170 --> 00:00:22,491
and first, we're going to try to find

11
00:00:22,491 --> 00:00:25,627
the kinetic energy of the electron,

12
00:00:25,627 --> 00:00:28,029
and we know that kinetic
energy is equal to:

13
00:00:28,029 --> 00:00:30,640
1/2 mv squared,

14
00:00:30,640 --> 00:00:33,264
where "m" is the mass of the electron,

15
00:00:33,264 --> 00:00:36,305
and "v" is the velocity.

16
00:00:36,305 --> 00:00:40,287
So, if our electron is
going this way around,

17
00:00:40,287 --> 00:00:42,122
if it's orbiting our nucleus,

18
00:00:42,122 --> 00:00:44,033
so this is our electron,
the negative charge,

19
00:00:44,033 --> 00:00:47,991
the velocity vector, it'd
be tangent at this point.

20
00:00:48,483 --> 00:00:52,071
And we know that this electron
is attracted to the nucleus.

21
00:00:52,071 --> 00:00:54,425
We have one proton in the nucleus

22
00:00:55,195 --> 00:00:58,510
for a hydrogen atom, using the Bohr model,

23
00:00:58,510 --> 00:01:01,817
and we know, we know, that if
we're doing the Bohr model,

24
00:01:01,817 --> 00:01:04,138
there's a certain radius associated

25
00:01:04,138 --> 00:01:05,997
with where that electron is.

26
00:01:05,997 --> 00:01:09,316
So we know the electron is
also attracted to the nucleus.

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00:01:09,316 --> 00:01:12,335
There's an electric force,
alright, so this electron

28
00:01:12,335 --> 00:01:15,412
is pulled to the nucleus,
this is an attractive force.

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00:01:15,412 --> 00:01:19,210
This is the electric force,
this is a centripetal force,

30
00:01:19,210 --> 00:01:20,997
the force that's holding that electron

31
00:01:20,997 --> 00:01:25,152
in a circular orbit
around the nucleus here.

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00:01:25,152 --> 00:01:29,134
And, once again, we talked
about the magnitude of this

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00:01:29,134 --> 00:01:31,258
electric force in an earlier video,

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00:01:31,258 --> 00:01:32,699
and we need it for this video, too.

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00:01:32,699 --> 00:01:34,290
We're gonna use it to come up with

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00:01:34,290 --> 00:01:37,377
the kinetic energy for that electron.

37
00:01:37,715 --> 00:01:40,523
So the electric force is
given by Coulomb's Law,

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00:01:40,523 --> 00:01:41,626
the magnitude of the electric force

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00:01:41,626 --> 00:01:43,566
is equal to K, which is a constant,

40
00:01:43,566 --> 00:01:47,282
"q1", which is, let's say
it's the charge on the proton,

41
00:01:47,282 --> 00:01:50,651
times "q2", charge on the electron,

42
00:01:50,651 --> 00:01:52,585
divided by "r squared",

43
00:01:52,585 --> 00:01:55,866
where "r" is the distance
between our two charges.

44
00:01:56,389 --> 00:01:58,519
We know that Newton's Second Law:

45
00:01:58,519 --> 00:02:01,319
force is equal to the mass
times the acceleration.

46
00:02:01,319 --> 00:02:03,335
We're talking about the electron here,

47
00:02:03,335 --> 00:02:05,436
so the mass of the electron times

48
00:02:05,436 --> 00:02:07,537
the acceleration of the electron.

49
00:02:07,537 --> 00:02:09,940
The electric force is a centripetal force,

50
00:02:09,940 --> 00:02:11,601
keeping it in circular motion,

51
00:02:11,601 --> 00:02:14,777
so we can say this is the
"centripetal acceleration".

52
00:02:14,946 --> 00:02:16,914
Alright, let's go ahead and
write down what we know.

53
00:02:17,268 --> 00:02:19,973
"K" is a constant, we'll
write that in here,

54
00:02:19,973 --> 00:02:24,973
"q1", "q1" is the charge on a proton,

55
00:02:25,975 --> 00:02:28,714
which we know is elemental charge,

56
00:02:28,714 --> 00:02:31,401
so it would be positive "e"...

57
00:02:32,570 --> 00:02:35,822
"q2" is the charge on the electron.

58
00:02:35,822 --> 00:02:38,605
The charge on the electron
is the same magnitude

59
00:02:38,605 --> 00:02:41,764
as the charge on the proton,
but it's a negative value.

60
00:02:41,764 --> 00:02:44,782
So we have negative "e", is
the charge on the electron,

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00:02:44,782 --> 00:02:46,966
divided by "r squared",

62
00:02:46,966 --> 00:02:49,763
is equal to the mass of the electron

63
00:02:49,763 --> 00:02:52,363
times the centripetal acceleration.

64
00:02:52,363 --> 00:02:54,246
So, centripetal acceleration

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00:02:54,246 --> 00:02:57,072
is equal to "v squared" over "r".

66
00:02:57,072 --> 00:02:59,613
So, we did this in a previous video.

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00:02:59,613 --> 00:03:01,555
We're gonna do the exact
same thing we did before.

68
00:03:01,555 --> 00:03:04,676
We only care about the
magnitude of the electric force

69
00:03:04,676 --> 00:03:06,120
because we already know the direction

70
00:03:06,120 --> 00:03:08,257
is always going to be towards the center,

71
00:03:08,257 --> 00:03:11,309
and therefore, we only care...

72
00:03:11,388 --> 00:03:13,572
we don't care about
this negative sign here.

73
00:03:13,572 --> 00:03:16,079
We can also cancel one of the "r"s.

74
00:03:16,079 --> 00:03:17,578
So if we don't care about...

75
00:03:17,578 --> 00:03:19,021
if we only care about the magnitude,

76
00:03:19,021 --> 00:03:20,004
on the left side, we get:

77
00:03:20,004 --> 00:03:25,004
Ke squared over r is equal to
mv squared, on the right side.

78
00:03:26,761 --> 00:03:29,767
And you can see, we're
almost to what we want.

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00:03:29,767 --> 00:03:31,346
Our goal was to try to find the expression

80
00:03:31,346 --> 00:03:34,387
for the kinetic energy,
that's 1/2 mv squared.

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00:03:34,387 --> 00:03:37,081
Here, we have mv squared, so if we

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00:03:37,081 --> 00:03:39,380
multiply both sides by 1/2, right,

83
00:03:39,380 --> 00:03:41,342
multiply both sides by 1/2,

84
00:03:41,342 --> 00:03:42,769
now we have an expression

85
00:03:42,769 --> 00:03:45,591
for the kinetic energy of the electron.

86
00:03:45,591 --> 00:03:48,658
So: 1/2 mv squared is equal
to the kinetic energy.

87
00:03:48,658 --> 00:03:51,376
So we know the kinetic energy is equal to:

88
00:03:51,376 --> 00:03:55,000
1/2 Ke squared over r

89
00:03:55,246 --> 00:03:58,694
Alright, so we will come
back to the kinetic energy.

90
00:03:58,694 --> 00:04:01,132
Next, we're gonna find
the potential energy.

91
00:04:01,132 --> 00:04:04,709
So the potential energy of that electron.

92
00:04:04,709 --> 00:04:07,890
And that potential energy is given

93
00:04:07,890 --> 00:04:09,874
by this equation in physics.

94
00:04:09,874 --> 00:04:12,507
So the electrical potential energy

95
00:04:12,507 --> 00:04:16,002
is equal to: "K", our same "K",

96
00:04:16,002 --> 00:04:19,345
times "q1", so the charge of one...

97
00:04:19,345 --> 00:04:21,512
so we'll say, once again,
that's the charge of the proton,

98
00:04:21,512 --> 00:04:24,079
times the charge of the electron,

99
00:04:24,079 --> 00:04:26,712
divided by the distance between them.

100
00:04:26,712 --> 00:04:28,974
So again, it's just physics.

101
00:04:29,682 --> 00:04:32,288
So let's plug in what we know.

102
00:04:32,734 --> 00:04:34,521
This would be equal to K.

103
00:04:34,521 --> 00:04:38,319
"q1", again, "q1" is the
charge on the proton,

104
00:04:38,319 --> 00:04:40,171
so that's positive "e",

105
00:04:40,787 --> 00:04:44,200
and "q2" is the charge on the electron,

106
00:04:44,200 --> 00:04:46,877
so that's negative "e",

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00:04:48,138 --> 00:04:49,422
negative "e",

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00:04:49,577 --> 00:04:51,111
divided by "r".

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00:04:51,111 --> 00:04:54,966
This time, we're going to
leave the negative sign in,

110
00:04:54,966 --> 00:04:57,044
and that's a consequence of how we define

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00:04:57,044 --> 00:04:59,238
electrical potential energy.

112
00:04:59,238 --> 00:05:02,593
So we get: negative Ke squared over r

113
00:05:02,593 --> 00:05:06,134
So we define the
electrical potential energy

114
00:05:06,134 --> 00:05:08,537
equal to zero at infinity.

115
00:05:08,537 --> 00:05:11,056
And so we need to keep
this negative sign in,

116
00:05:11,056 --> 00:05:13,031
because it's actually important.

117
00:05:13,416 --> 00:05:16,109
Alright, so now we have the
electrical potential energy,

118
00:05:16,109 --> 00:05:18,535
and we have the kinetic energy.

119
00:05:18,535 --> 00:05:22,843
And to find the total energy
associated with that electron,

120
00:05:23,797 --> 00:05:26,486
the total energy associated
with that electron,

121
00:05:26,486 --> 00:05:28,460
the total energy would be equal to:

122
00:05:28,460 --> 00:05:31,356
so, E-total is equal
to the kinetic energy,

123
00:05:31,956 --> 00:05:34,125
plus the potential energy.

124
00:05:34,125 --> 00:05:37,422
So this would be the
electrical potential energy.

125
00:05:37,422 --> 00:05:39,593
So let's plug in those values.

126
00:05:39,593 --> 00:05:41,626
We found the kinetic energy over here,

127
00:05:41,626 --> 00:05:44,550
1/2 Ke squared over r, so
we plug that into here,

128
00:05:44,550 --> 00:05:47,429
and then we also found the
electrical potential energy is:

129
00:05:47,429 --> 00:05:50,768
negative Ke squared over
r, so we plug that in,

130
00:05:50,768 --> 00:05:53,043
and now we can calculate the total energy.

131
00:05:53,043 --> 00:05:54,439
So we get some more room...

132
00:05:54,439 --> 00:05:57,620
The total energy is equal to:

133
00:05:57,620 --> 00:06:02,191
1/2 Ke squared over r,

134
00:06:02,191 --> 00:06:04,049
our expression for the kinetic energy,

135
00:06:04,049 --> 00:06:06,337
and then, this was plus,

136
00:06:06,337 --> 00:06:08,205
and then we have a negative value,

137
00:06:08,205 --> 00:06:12,524
so we just write: minus Ke squared over r

138
00:06:12,832 --> 00:06:14,566
So, if you think about the math,

139
00:06:14,874 --> 00:06:17,255
this is just like 1/2 minus one,

140
00:06:17,255 --> 00:06:20,612
and so that's going to
give you negative 1/2.

141
00:06:20,612 --> 00:06:23,163
1/2 - 1 = -1/2

142
00:06:23,163 --> 00:06:28,163
So "negative 1/2 Ke squared
over r" is our expression

143
00:06:30,091 --> 00:06:32,169
for the total energy.

144
00:06:32,169 --> 00:06:37,169
So this is the total energy
associated with our electron.

145
00:06:38,791 --> 00:06:41,228
Alright, let's find the total energy

146
00:06:41,228 --> 00:06:45,385
when the radius is equal to r1.

147
00:06:45,385 --> 00:06:47,823
What we talked about in the last video.

148
00:06:47,823 --> 00:06:52,823
The radius of the electron
in the ground state.

149
00:06:53,024 --> 00:06:55,938
And r1, when we did that math,

150
00:06:55,938 --> 00:07:00,686
we got: 5.3 times 10 to
the negative 11 meters.

151
00:07:00,686 --> 00:07:05,179
And so, we're going to be
plugging that value in for this r.

152
00:07:05,179 --> 00:07:07,187
So we can calculate the total energy

153
00:07:07,187 --> 00:07:10,821
associated with that energy level.

154
00:07:10,821 --> 00:07:13,921
And remember, we got this r1 value,

155
00:07:13,921 --> 00:07:16,661
we got this r1 value, by doing some math

156
00:07:16,661 --> 00:07:21,661
and saying, n = 1, and plugging
that into our equation.

157
00:07:23,059 --> 00:07:25,374
The radius for any integer, n,

158
00:07:25,374 --> 00:07:29,740
is equal to n squared times r1.

159
00:07:29,740 --> 00:07:32,239
So when n = 1, we plugged it into here

160
00:07:32,239 --> 00:07:35,063
and we got our radius.

161
00:07:35,494 --> 00:07:36,608
So let's go ahead and plug that in.

162
00:07:36,608 --> 00:07:38,256
Let's do the math, actually.

163
00:07:38,256 --> 00:07:40,627
So, we're going to get the total energy

164
00:07:41,320 --> 00:07:43,024
for the first energy level,

165
00:07:43,024 --> 00:07:48,024
so when n = 1, it's equal
to negative 1/2 times K,

166
00:07:48,306 --> 00:07:50,943
which is nine times 10 to the 9th,

167
00:07:50,943 --> 00:07:53,112
times the elemental charge.

168
00:07:53,112 --> 00:07:55,043
Alright, so we just took care of K,

169
00:07:55,043 --> 00:07:59,722
E is the magnitude of charge
on a proton or an electron,

170
00:07:59,722 --> 00:08:04,505
which is equal to 1.6 times 10
to the negative 19 Coulombs,

171
00:08:04,505 --> 00:08:06,246
we're going to square that,

172
00:08:06,246 --> 00:08:09,207
and then put that over the radius,

173
00:08:09,207 --> 00:08:14,164
which was 5.3 times 10 to
the negative 11 meters.

174
00:08:14,164 --> 00:08:17,426
And to save time, I
won't do that math here,

175
00:08:17,426 --> 00:08:19,958
but if you do that calculation,

176
00:08:20,312 --> 00:08:23,334
if you do that calculation,
the energy associated

177
00:08:23,334 --> 00:08:26,527
with the ground state
electron of a hydrogen atom,

178
00:08:26,527 --> 00:08:31,527
is equal to: negative 2.17
times 10 to the negative 18

179
00:08:33,514 --> 00:08:35,111
and the units would be joules.

180
00:08:35,111 --> 00:08:36,868
So if you took the time
to do all those units,

181
00:08:36,869 --> 00:08:38,862
you would get joules here.

182
00:08:39,231 --> 00:08:42,259
So that's the lowest energy
state, the ground state.

183
00:08:42,259 --> 00:08:44,883
The energy is negative,
and I'll talk more about

184
00:08:44,883 --> 00:08:48,089
what the negative sign
means in the next video.

185
00:08:48,473 --> 00:08:52,244
Alright, so we could
generalize this energy.

186
00:08:52,244 --> 00:08:55,541
We could say, here we did it for n = 1,

187
00:08:55,541 --> 00:08:59,465
but we could say that:
E at any integer "n",

188
00:08:59,465 --> 00:09:02,576
is equal to, then put an "r sub n" here.

189
00:09:02,576 --> 00:09:03,982
Let me just re-write that equation.

190
00:09:03,982 --> 00:09:06,108
So we could generalize this and say:

191
00:09:07,307 --> 00:09:10,581
the energy at any energy level is equal to

192
00:09:10,581 --> 00:09:15,581
negative 1/2 Ke squared, r n.

193
00:09:17,445 --> 00:09:21,015
Okay, so we could now take
this equation, right here,

194
00:09:21,015 --> 00:09:22,799
the one we talked about and actually

195
00:09:22,799 --> 00:09:24,680
derived in the earlier video,

196
00:09:24,680 --> 00:09:27,942
and plug all of this in for our "n".

197
00:09:27,942 --> 00:09:30,743
So we're gonna plug all of that into here.

198
00:09:30,743 --> 00:09:34,201
So let's get some more
room, and continue...

199
00:09:34,201 --> 00:09:38,485
So the energy at an energy level "n",

200
00:09:38,485 --> 00:09:43,485
is equal to negative 1/2
Ke squared, over, right?

201
00:09:44,986 --> 00:09:47,981
So we're gonna plug in
"n squared r1" here.

202
00:09:47,981 --> 00:09:51,939
So this would be: n squared r1

203
00:09:52,277 --> 00:09:54,488
We can re-write that.

204
00:09:54,488 --> 00:09:56,404
This is the same thing as:

205
00:09:56,404 --> 00:10:01,404
negative 1/2 Ke squared over
r1 times one over n squared.

206
00:10:06,284 --> 00:10:09,883
So I just re-wrote this in a certain way

207
00:10:09,883 --> 00:10:11,682
because I know what all
of this is equal to.

208
00:10:11,682 --> 00:10:16,165
I know what negative 1/2 Ke
squared over r1 is equal to.

209
00:10:16,165 --> 00:10:18,160
We just did the math for that.

210
00:10:18,160 --> 00:10:23,160
Alright, so this is negative
1/2 Ke squared over r1.

211
00:10:23,679 --> 00:10:25,415
And so we got this number:

212
00:10:25,415 --> 00:10:29,086
this is the energy associated
with the first energy level.

213
00:10:29,409 --> 00:10:31,566
And so we can go ahead and plug that in.

214
00:10:31,566 --> 00:10:32,797
We can plug in this number.

215
00:10:32,797 --> 00:10:36,257
We can take this number and
plug it in for all of this.

216
00:10:36,257 --> 00:10:38,485
So that's what all of that is equal to.

217
00:10:38,485 --> 00:10:41,481
So, here's another way
to write our energy.

218
00:10:41,481 --> 00:10:43,396
So, energy is equal to:

219
00:10:43,396 --> 00:10:48,396
negative 2.17 times 10 to the negative 18

220
00:10:49,538 --> 00:10:51,001
and then this would be:

221
00:10:51,001 --> 00:10:53,160
times one over n squared.

222
00:10:53,160 --> 00:10:57,199
So we can just put it
over n squared like that.

223
00:10:57,753 --> 00:11:00,619
And then we could write it
in a slightly different way.

224
00:11:00,619 --> 00:11:02,349
Since that's equal to E1,

225
00:11:02,349 --> 00:11:04,630
we could just make it
look even shorter here.

226
00:11:04,630 --> 00:11:05,776
We could just say that:

227
00:11:05,776 --> 00:11:08,005
The energy at energy level n...

228
00:11:08,005 --> 00:11:12,892
So: the energy at energy
level n is equal to the energy

229
00:11:12,892 --> 00:11:17,108
associated with the first energy
level divided by n squared.

230
00:11:17,108 --> 00:11:20,835
Either one of these is fine.

231
00:11:20,835 --> 00:11:22,855
So we could write it like this,

232
00:11:22,855 --> 00:11:25,351
or we could write it like
this, it doesn't really matter

233
00:11:25,351 --> 00:11:28,114
which one you use, but
we're gonna be using

234
00:11:28,114 --> 00:11:30,412
these equations, or this equation,

235
00:11:30,412 --> 00:11:31,968
it's really the same equation,

236
00:11:31,968 --> 00:11:33,970
in the next video, and
we're gonna come up with

237
00:11:34,370 --> 00:11:36,939
the different energies,
the different energies

238
00:11:36,939 --> 00:11:38,402
at different energy levels.

239
00:11:38,402 --> 00:11:40,140
So we're gonna change what "n" is

240
00:11:40,140 --> 00:11:42,370
and come up with a different energy.

241
00:11:42,370 --> 00:11:44,202
So energy is quantized.

242
00:11:44,525 --> 00:11:45,835
Now, this is really important to think

243
00:11:45,835 --> 00:11:48,610
about this idea of energy being quantized.

244
00:11:48,610 --> 00:11:51,373
And this is one reason why the Bohr model

245
00:11:51,373 --> 00:11:53,533
is nice to look at, because it

246
00:11:53,533 --> 00:11:55,472
gives us these quantized energy levels,

247
00:11:55,472 --> 00:11:57,108
which actually explains some things,

248
00:11:57,108 --> 00:11:58,814
as we'll see in later videos.

249
00:11:58,814 --> 00:12:00,731
So the next video, we'll
continue with energy,

250
00:12:00,731 --> 00:12:02,785
and we'll take these
equations we just derived,

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and we'll talk some more about

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the Bohr model of the hydrogen atom.