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If these bonds were all completely of equal distance apart, whether is was a lone pair or bonding electrons, 5° the angles would be 109 . 5 degrees.
如果不管它是孤对，还是成键，它们等距分开的话，键角是109。
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You put a quarter in and you look through and you can see all through New York
你只要在望远镜里投入两角五分钱，整个纽约的景色就一览无遗了。
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But because there's this lone pair here, it's pushing down on the other bonds, 5° so we end up with an angle of less than 109 . 5 degrees.
但因为这里有孤对，它会把其它键向下压，所以键角会小于109。
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He sees weird kinds of beasts and animals that had like--they're bodies of lambs, but they've got horns and they're bleeding all over the place.
看见了奇怪的动物与野兽,有着羔羊的身体，却长着角，浑身是血。
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I'm going to give you the data set of the scattering angles of the atoms.
我会把,这些原子散射角的数据发给你。
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Isn't the reason I think unicorns are logically coherent is because I can imagine them so easily?
我认为,独角兽在逻辑上不矛盾的原因,不正是因为我能够轻易的想象它们吗
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Why do you like to measure angle in the peculiar way?
为什么要用这种特殊的方式衡量一个角呢
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But right opposite that magnificent site, the golden horn is there.
就是在那块宝地对面,金角湾就在那啊
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Now we're going to--notice there's no subscript on this P and you'll see why in a second.
我们要,注意这个P没有角标,你们很快会知道这是为什么
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They want to get as far away from each other possible, the ideal angle is 120. But what we have here is a four-membered ring, so what angle does 90° that have to be, that bond? 90 degrees.
它们想要尽量远离彼此，最理想的是形成120°键角,但现在是个四元环,所以这键角应该多大？
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You can go ahead and use that equation, or you could figure it out every time, because if you know the total number of nodes, and you know the angular node number, then you know how many nodes you're going to have left.
你们可以直接用这个方程，或者每次都自己算出来，因为如果你们知道了总的节点数，又知道角向节点数，就知道剩下的节点数是多少。
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Angular nodes, we're not going to have any of those, we'll have zero, l equals 0, so we have zero angular nodes.
角向节点，当然,是没有的，０个，ｌ等于０，所以是０个角向节点。
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It doesn't mean I think unicorns are physically possible.
但并不代表我认为独角兽真的可能存在
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Three. Good, so everyone that recognized that probably got the right answer of three angular nodes here.
三，很好，那么知道这一点的同学,应该都得到了正确结果,也就是三个角向节点。
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No matter where you specify your electron is in terms of those two angles, it doesn't matter the angular part of your wave function is going to be the same.
不论你将，这两个角度,取成什么值，波函数的角向部分,都是，相同的。
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So you can see there's this radial part here, and you have the angular part, you can combine the two parts to get the total wave function.
你们可以看到，这是径向部分，这是角向部分，把这两部分结合到一起,就是总的波函数。
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The bond, it turns out, 5° is 104 . 5 degrees, that h o h bond.
这个键即，ＨＯＨ键角，是104。
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So, you remember from last time radial nodes are values of r at which the wave function and wave function squared are zero, so the difference is now we're just talking about the angular part of the wave function.
你们记得上次说径向节点在,波函数和波函数的平方,等于零的ｒ的处，现在的区别是我们讨论的是,角向波函数。
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So again, if we think about that shape of that carbon atom, it's going to be trigonal planar, 120° it's going to have bond angles of 120 degrees, because we have this set up of having three hybrid orbitals.
如果我们考虑碳原子的形状，它是平面三角形，键角是，因为我们有这三个杂化轨道。
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So, in a settling, what is the bond angle here?
这里的键角是多少？
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What do you know the bond angle should be?
你们觉得键角是多少？
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It seems to follow that unicorns are logically possible.
可以推测,独角兽在逻辑上是可能的
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I could imagine a world with unicorns.
我可以想象一个有独角兽的世界
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And as a reminder, hopefully I don't need to remind any of you, but exam 1 is on Wednesday, so rather than our clicker question being on something from last class, which is exam 2 material, let's just make sure everyone remembers some small topic from exam 1 material, which is the idea of angular nodes.
而作为一个提醒--我希望大家都不需要提醒，但是这周三就要进行第一次考试了，因此我们的这个选择题,不是关于上节课内容的，那是第二次考试的内容，这里只是想确认大家都记得,第一次考试所要求的一些小问题，也就是关于角向节点的概念。
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It doesn't mean unicorns exist.
但这并不意味着独角兽就存在
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I have yet to show you the solution to a wave function for the hydrogen atom, so let me do that here, and then we'll build back up to probability densities, and it turns out that if we're talking about any wave function, we can actually break it up into two components, which are called the radial wave function and angular wave function.
我还没有给你们看过,氢原子波函数的解，让我现在给你们看一下，然后再来说,概率密度，实际上,对于任何一个波函数来说，我们可以把它,分解为两部分，分别叫做径向波函数,和角向波函数。
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If I pick an angle of 60 degrees, these are some numbers like half and root 3 over 2.
要是我选择一个 60°的角,上面的系数就会是二分之一和二分之根号三
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So, let me get a little bit more specific about what we mean by nodal plane and where the idea of nodal plane comes from, and nodal planes arise from any place you have angular nodes.
关于节面的意义,或者节面概念的起源，让我们讲的更具体一点，节面起源于角向节点。
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You know from the m quantum number there are three.
你可以从角量子数上看出是3个。
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I can imagine a world with unicorns.
我可以想象一个有独角兽的世界
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