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PROFESSOR: OK.
 

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As you're settling into your
seats, why don't we take

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10 more seconds on the
clicker question here.

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All right, so this is a
question that you saw on your

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problem-set, so this is how
many electrons would we

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expect to see in a single
atom in the 2 p state.

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So, let's see what
you said here.

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Six.
 

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And the correct answer
is, in fact, six.

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And most of you got that, about
75% of you got that right.

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So, let's consider some people
got it wrong, however, and

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let's see where that wrong
answer might have come from, or

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actually, more importantly,
let's see how we can all

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get to the correct answer.
 

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So if we say that we have a 2 p
orbital here, that means that

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we can have how many different
complete orbitals have a 2 for

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an n, and a p as its
l value? three.

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So, we can have the 2 p x,
2 p y, and 2 p z orbitals.

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Each of these orbitals can
have two electrons in them,

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so we get two electrons
here, here, and here.

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So, we end up with a total
of six electrons that are

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possible that have that
2 p orbital value.

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So this is a question that,
hopefully, if we see another

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one like this we'll get a 100%
on, because you've already seen

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this in your problem-set as
much as you're going to see it,

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and you're seen it in class as
much as you're going to see it.

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So if you're still having
trouble with this, this is

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something you want to bring
up in your recitation.

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And the idea behind this, of
course, is that we know that

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every electron has to have
its own distinct set of

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four quantum numbers.
 

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So that means that if we have
three orbitals, we can only

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have six electrons in those
complete three orbitals.

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All right, so today we're
going to fully have our

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discussion focused on
multi-electron atoms.

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We started talking about these
on Wednesday, and what we're

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going to start with is
considering specifically

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the wave functions for
multi-electron atoms.

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So, the wave functions for
multi-electron atoms.

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Then we'll move on to talking
about the binding energies, and

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we'll specifically talk about
how that differs from the

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binding energies we saw
of hydrogen atoms.

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We talked about that quite in
depth, but there are some

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differences now that we have
more than one electron

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in the atom.
 

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Then something that you
probably have a lot of

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experience with is talking
about electron configuration

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and writing out the
electron configuration.

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But we'll go over that,
particularly some exceptions,

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when we're filling in electron
configurations, and how we

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would go about doing that for
positive ions, which

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follow a little bit of a
different procedure.

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And if we have time today,
we'll start in on the

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photo-electron spectroscopy, if
not, that's where we'll start

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when we come back on Wednesday.
 

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So, what we saw just on
Wednesday, in particular, but

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also as we have been discussing
the Schrodinger equation for

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the hydrogen atom, is that this
equation can be used to

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correctly predict the atomic
structure of hydrogen, and also

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all of the energy levels of the
different orbitals in hydrogen,

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which matched up with what we
observed, for example, when we

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looked at the hydrogen
atom emission spectra.

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And what we can do is we can
also use the Schrodinger

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equation to make these accurate
predictions for any other atom

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that we want to talk about
in the periodic table.

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The one problem that we run
into is as we go to more and

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more atoms on the table, as
we add on electrons, the

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Schrodinger equation is going
to get more complicated.

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So here I've written for the
hydrogen atom that deceptively

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simple form of the Schrodinger
equation, where we don't

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actually write out the
Hamiltonian operator, but you

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remember that's a series of
second derivatives, so we have

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a differential equation that
were actually dealing with.

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If you think about what happens
when we go from hydrogen to

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helium, now instead of one
electron, so three position

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variables, we have to describe
two electrons, so now we have

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six position variables that we
need to plug into our

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Schrodinger equation.
 

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So similarly, as we now move up
only one more atom in the

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table, so to an atomic number
of three or lithium, now we're

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going from six variables all
the way to nine variables.

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So you can see that we're
starting to have a very

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complicated equation, and it
turns out that it's

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mathematically impossible to
even solve the exact

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Schrodinger equation as
we move up to higher

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numbers of electrons.
 

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So, what we say here is we need
to take a step back here and

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come up with an approximation
that's going to allow us to

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think about using the
Schrodinger equation when we're

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not just talking about hydrogen
or one electron, but

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when we have these
multi-electron atoms.

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The most straightforward way to
do this is to make what's

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called a one electron orbital
approximation, and when you do

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you get out what are called
Hartree orbitals, and what this

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means is that instead of
considering the wave function

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as a function, for example, for
helium as six different

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variables, what we do is we
break it up and treat each

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electron has a separate wave
function and say that our

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assumption is that the total
wave function is equal to

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the product of the two
individual wave functions.

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So, for example, for helium, we
can break it up into wave

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function for it the r, theta,
and phi value for electron one,

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and multiply that by the
wave function for the r,

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theta, and phi value for
electron number two.

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So essentially what we're
saying is we have a wave

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function for electron one,
and a wave function

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for electron two.
 

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We know how to write that in
terms of the state numbers, so

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it would be 1, 0, 0, because
we're talking about

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the ground state.
 

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We're always talking about the
ground state unless we specify

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that we're talking about
an excited state.

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And we have the spin quantum
number as plus 1/2 for

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electron one, and minus
1/2 for the electron two.

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It's arbitrary which one I
assigned to which, but we know

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that we have to have each of
those two magnetic spin quantum

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numbers in order to have
the distinct four letter

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description of an electron.
 

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We know that it's not enough
just to describe the orbital by

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three quantum numbers, we need
that fourth number to fully

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describe an electron.
 

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And when we describe this in
terms of talking about

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chemistry terminology, we would
call the first one the 1 s, and

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1 is in parentheses because
we're talking about the first

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electron there, and we would
multiply it by the wave

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function for the second one,
which is also 1 s, but now we

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are talking about that
second electron.

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We can do the exact same thing
when we talk about lithium, but

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now instead of breaking it up
into two wave functions, we're

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breaking it up into three
wave functions because

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we have three electrons.
 

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So, the first again is
the 1 s 1 electron.

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We then have the 1 s 2
electron, and what is our

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third electron going to be?
 

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Yeah.
 

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So it's going to be
the 2 s 1 electron.

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So we can do this essentially
for any atom we want, we just

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have more and more wave
functions that we're breaking

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it up to as we get to
more and more electrons.

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And we can also write this in
an even simpler form, which is

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what's called electron
configuration, and this is just

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a shorthand notation for these
electron wave functions.

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So, for example, again we see
hydrogen is 1 s 1, helium we

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say is 1 s 2, or 1 s squared,
so instead of writing out the

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1 s 1 and the 1 s 2, we just
combine it as 1 s squared,

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lithium is 1 s 2, 2 s 1.
 

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So writing out electron
configurations I realize is

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something that a lot of you had
experience with in high school,

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you're probably -- many of you
are very comfortable doing it,

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especially for the more
straightforward atoms.

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But what's neat to kind of
think about is if you think

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about what a question might
have been in high school, which

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is please write the electron
configuration for lithium, now

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we can also answer what sounds
like a much more impressive,

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and a much more complicated
question, which would be write

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the shorthand notation for the
one electron orbital

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approximation to solve the
Schrodinger equation

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for lithium.
 

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So essentially, that is
the exact same thing.

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The electronic configuration,
all it is is the shorthand

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notation for that one electron
approximation for the

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Schrodinger equation
for lithium.

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So, if you're at hanging your
exam from high school on the

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fridge and you want to make it
look more impressive, you could

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just rewrite the question as
that, and essentially you're

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answering the same thing.
 

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But now, hopefully, we
understand where that comes

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from, why it is that we use
the shorthand notation.

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So, let's write this one
electron orbital approximation

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for berylium, that sounds like
a pretty complicated question,

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but hopefully we know that
it's not at all, it's just

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1 s 2, and then 2 s 2.
 

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And we can go on and
on down the table.

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So, for example, for boron, now
we're dealing with 1 s 2, then

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2 s 2, and now we have to move
into the p orbital

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so we go to 2 p 1.
 

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So that's a little bit
of an introduction into

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electron configuration.
 

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We'll get into some spots where
it gets a little trickier, a

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little bit more complicated
later in class.

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But that's an idea of what it
actually means to talk about

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electron configuration.
 

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So now that we can do this, we
can compare and think about, we

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know how to consider wave
functions for individual

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electrons in multi-electron
atoms using those Hartree

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orbitals or the one electron
wave approximations.

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So let's compare what some
of the similarities and

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differences are between
hydrogen atom orbitals, which

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we spent a lot of time
studying, and now these one

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electron orbital approximations
for these multi-electron atoms.

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So, as an example, let's take
argon, I've written up the

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electron configuration here,
and let's think about what some

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of the similarities might be
between wave functions in argon

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and wave functions
for hydrogen.

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So the first is that the
orbitals are similar in shape.

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So for example, if you know how
to draw an s orbital for a

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hydrogen atom, then you already
know how to draw the shape of

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an s orbital or a p
orbital for argon.

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Similarly, if we were to look
at the radial probability

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distributions, what we would
find is that there's an

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identical nodal structure.
 

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So, for example, if we look at
the 2 s orbital of argon, it's

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going to have the same amount
of nodes and the same type of

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nodes that the 2 s orbital
for hydrogen has.

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So how many nodes does the 2
s orbital for hydrogen have?

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It has one node, right, because
if we're talking about nodes

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it's just n minus 1 is total
nodes, so you would just say 2

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minus 1 equals 1 node
for the 2 s orbital.

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And how many of those nodes
are angular nodes? zero.

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L equals 0, so we have zero
angular nodes, that means that

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they're all radial nodes.
 

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So what we end up with is one
radial node for the 2 s orbital

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of hydrogen, and we can apply
that for argon or any other

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multi-electron atom here, we
also have one radial node for

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the 2 s orbital of argon.
 

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But there's also some
differences that we need to

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keep in mind, and that will
be the focus of a lot

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of the lecture today.
 

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One of the main difference is
is that when you're talking

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about multi-electron orbitals,
they're actually smaller than

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the corresponding orbital
for the hydrogen atom.

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We can think about
why that would be.

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Let's consider again an s
orbital for argon, so let's

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say we're looking at the
1 s orbital for argon.

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What is the pull from
the nucleus from argon

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going to be equal to?
 

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What is the charge
of the nucleus?

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Does anyone know, it's a
quick addition problem here.

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Yeah, so it's 18.
 

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So z equals 18, so the nucleus
is going to be pulling at the

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electron with a Coulombic
attraction that has a charge of

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plus 18, if we're talking about
the 1 s electron or the

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1 s orbital in argon.
 

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It turns out, and we're going
to get the idea of shielding,

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so it's not going to actually
feel that full plus 18, but

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it'll feel a whole lot more
than it will just feel in terms

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of a hydrogen atom where we
only have a nuclear

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charge of one.
 

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So because we're feeling a
stronger attractive force from

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the nucleus, we're actually
pulling that electron in

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closer, which means that the
probability squared of where

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the electron is going to be is
actually a smaller radius.

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So when we talk about the size
of multi-electron orbitals,

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they're actually going to be
smaller because they're being

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pulled in closer to the nucleus
because of that stronger

254
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attraction because of the
higher charge of the nucleus

255
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in a multi-electron atom
compared to a hydrogen atom.

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The other main difference that
we're really going to get to

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today is that in multi-electron
atoms, orbital energies depend

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not just on the shell, which is
what we saw before, not just on

259
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the value of n, but also on the
angular momentum

260
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quantum number.
 

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So they also depend on
the sub-shell or l.

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And we'll really get to see a
picture of that, and I'll be

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repeating that again and again
today, because this is

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something I really want
everyone to get firmly

265
00:13:17 --> 00:13:18
into their heads.
 

266
00:13:18 --> 00:13:21
So, let's now take a
look at the energies.

267
00:13:21 --> 00:13:23
We looked at the wave
functions, we know the other

268
00:13:23 --> 00:13:26
part of solving the Schrodinger
equation is to solve for the

269
00:13:26 --> 00:13:29
binding energy of electrons to
the nucleus, so let's

270
00:13:29 --> 00:13:31
take a look at those.
 

271
00:13:31 --> 00:13:33
And there again is another
difference between

272
00:13:33 --> 00:13:35
multi-electron atom and
the hydrogen atoms.

273
00:13:35 --> 00:13:37
So when we talk about orbitals
in multi-electron atoms,

274
00:13:37 --> 00:13:41
they're actually lower in
energy than the corresponding

275
00:13:41 --> 00:13:43
h atom orbitals.
 

276
00:13:43 --> 00:13:45
And when we say lower in
energy, of course, what we

277
00:13:45 --> 00:13:48
mean is more negative.
 

278
00:13:48 --> 00:13:51
Right, because when we think of
an energy diagram, that lowest

279
00:13:51 --> 00:13:54
spot there is going to have the
lowest value of the binding

280
00:13:54 --> 00:14:00
energy or the most negative
value of binding.

281
00:14:00 --> 00:14:04
So, let's take a look here at
an example of an energy diagram

282
00:14:04 --> 00:14:07
for the hydrogen atom, and we
can also look at a energy

283
00:14:07 --> 00:14:10
diagram for a multi-electron
atom, and this is just a

284
00:14:10 --> 00:14:13
generic one here, so I haven't
actually listed energy numbers,

285
00:14:13 --> 00:14:15
but I want you to
see the trend.

286
00:14:15 --> 00:14:19
So for example, if you look at
the 1 s orbital here, you can

287
00:14:19 --> 00:14:23
see that actually it is
lower in the case of the

288
00:14:23 --> 00:14:26
multi-electron atom than it is
for the hydrogen atom.

289
00:14:26 --> 00:14:28
You see the same thing
regardless of which orbital

290
00:14:28 --> 00:14:30
you're looking at.
 

291
00:14:30 --> 00:14:33
For example, for the 2 s, again
what you see is that the

292
00:14:33 --> 00:14:37
multi-electron atom, its 2 s
orbital is lower in energy

293
00:14:37 --> 00:14:39
than it is for the hydrogen.
 

294
00:14:39 --> 00:14:41
The same thing we
see for the 2 p.

295
00:14:41 --> 00:14:45
Again the 2 p orbitals for the
multi-electron atom, lower in

296
00:14:45 --> 00:14:48
energy than for the
hydrogen atom.

297
00:14:48 --> 00:14:51
But there's something you'll
note here also when I point out

298
00:14:51 --> 00:14:54
the case of the 2 s versus the
2 p, which is what I mentioned

299
00:14:54 --> 00:14:57
that I would be saying again
and again, which is when we

300
00:14:57 --> 00:15:01
look at the hydrogen atom, the
energy of all of the n equals 2

301
00:15:01 --> 00:15:03
orbitals are exactly the same.
 

302
00:15:03 --> 00:15:05
That's what we call
degenerate orbitals,

303
00:15:05 --> 00:15:07
they're the same energy.
 

304
00:15:07 --> 00:15:09
But when we get to the
multi-electron atoms, we see

305
00:15:09 --> 00:15:12
that actually the p orbitals
are higher in energy

306
00:15:12 --> 00:15:14
than the s orbitals.
 

307
00:15:14 --> 00:15:17
So we'll see specifically why
it is that the s orbitals

308
00:15:17 --> 00:15:18
are lower in energy.
 

309
00:15:18 --> 00:15:21
We'll get to discussing that,
but what I want to point out

310
00:15:21 --> 00:15:24
here again is the fact that
instead of just being dependent

311
00:15:24 --> 00:15:29
on n, the energy level is
dependent on both n and l.

312
00:15:29 --> 00:15:33
And is no longer that sole
determining factor for

313
00:15:33 --> 00:15:38
energy, energy also depends
both on n and on l.

314
00:15:38 --> 00:15:41
And we can look at precisely
why that is by looking at the

315
00:15:41 --> 00:15:45
equations for the energy levels
for a hydrogen atom versus

316
00:15:45 --> 00:15:47
the multi-electron atom.
 

317
00:15:47 --> 00:15:50
So, for a hydrogen atom, and
actually for any one electron

318
00:15:50 --> 00:15:54
atom at all, this is our
energy or our binding energy.

319
00:15:54 --> 00:15:55
This is what came out of
solving the Schrodinger

320
00:15:55 --> 00:15:59
equation, we've seen this
several times before that the

321
00:15:59 --> 00:16:03
energy is equal to negative z
squared times the Rydberg

322
00:16:03 --> 00:16:05
constant over n squared.
 

323
00:16:05 --> 00:16:08
Remember the z squared, that's
just the atomic number or the

324
00:16:08 --> 00:16:13
charge on the nucleus, and we
can figure that out for any

325
00:16:13 --> 00:16:14
one electron atom at all.
 

326
00:16:14 --> 00:16:17
And an important thing to note
is in terms of what that

327
00:16:17 --> 00:16:20
physically means, so physically
the binding energy is just

328
00:16:20 --> 00:16:22
the negative of the
ionization energy.

329
00:16:22 --> 00:16:25
So if we can figure out the
binding energy, we can also

330
00:16:25 --> 00:16:28
figure out how much energy we
have to put into our atom in

331
00:16:28 --> 00:16:31
order to a eject or
ionize an electron.

332
00:16:31 --> 00:16:35
We can also look at the
energy equation now for

333
00:16:35 --> 00:16:37
a multi-electron atom.
 

334
00:16:37 --> 00:16:40
And the big difference is
right here in this term.

335
00:16:40 --> 00:16:43
So instead of being equal to
negative z squared, now we're

336
00:16:43 --> 00:16:48
equal to negative z
effective squared times r h

337
00:16:48 --> 00:16:49
all over n squared.
 

338
00:16:49 --> 00:16:53
So when we say z effective,
what we're talking about is

339
00:16:53 --> 00:16:56
instead of z, the charge on the
nucleus, we're talking about

340
00:16:56 --> 00:16:58
the effective charge
on a nucleus.

341
00:16:58 --> 00:17:02
So for an example, even if a
nucleus has a charge of 7, but

342
00:17:02 --> 00:17:05
the electron we're interested
in only feels the charge as if

343
00:17:05 --> 00:17:09
it were a 5, then what we would
say is that the z effective

344
00:17:09 --> 00:17:12
for the nucleus is 5
for that electron.

345
00:17:12 --> 00:17:15
And we'll talk about this more,
so if this is not completely

346
00:17:15 --> 00:17:17
intuitive, we'll see
why in a second.

347
00:17:17 --> 00:17:20
So the main idea here is z
effective is not z, so don't

348
00:17:20 --> 00:17:23
try to plug one in for the
other, they're absolutely

349
00:17:23 --> 00:17:26
different quantities in any
case when we're not talking

350
00:17:26 --> 00:17:29
about a 1 electron atom.
 

351
00:17:29 --> 00:17:32
And the point that I also want
to make is the way that they

352
00:17:32 --> 00:17:36
differ, z effective actually
differs from the total charge

353
00:17:36 --> 00:17:39
in the nucleus due to an
idea called shielding.

354
00:17:39 --> 00:17:43
So, shielding happens when you
have more than one electron in

355
00:17:43 --> 00:17:46
an atom, and the reason that
it's happening is because

356
00:17:46 --> 00:17:50
you're actually canceling out
some of that positive charge

357
00:17:50 --> 00:17:53
from the nucleus or that
attractive force with a

358
00:17:53 --> 00:17:56
repulsive force between
two electrons.

359
00:17:56 --> 00:17:59
So if you have some charge in
the nucleus, but you also have

360
00:17:59 --> 00:18:02
repulsion with another
electron, the net attractive

361
00:18:02 --> 00:18:05
charge that a given electron
going to feel is actually less

362
00:18:05 --> 00:18:08
than that total charge
in the nucleus.

363
00:18:08 --> 00:18:10
And shielding is a little bit
of a misnomer because it's not

364
00:18:10 --> 00:18:13
actually that's the electron's
blocking the charge from

365
00:18:13 --> 00:18:17
another electron, it's more
like you're canceling out a

366
00:18:17 --> 00:18:20
positive attractive force with
a negative repulsive force.

367
00:18:20 --> 00:18:23
But shielding is a good way to
think about it, and actually,

368
00:18:23 --> 00:18:26
that's what we'll use in this
class to sort of visualize

369
00:18:26 --> 00:18:29
what's happening when we have
many electrons in an atom and

370
00:18:29 --> 00:18:30
they're shielding each other.
 

371
00:18:30 --> 00:18:33
Shielding is the term that's
used, it brings up a certain

372
00:18:33 --> 00:18:35
image in our mind, and even
though that's not precisely

373
00:18:35 --> 00:18:38
what's going on, it's a very
good way to visualize

374
00:18:38 --> 00:18:39
what we're trying to
think about here.

375
00:18:39 --> 00:18:43
So let's take two cases of
shielding if we're talking

376
00:18:43 --> 00:18:46
about, for example, the
helium, a helium nucleus

377
00:18:46 --> 00:18:48
or a helium atom.
 

378
00:18:48 --> 00:18:52
So what is the charge
on a helium nucleus?

379
00:18:52 --> 00:18:54
What is z?
 

380
00:18:54 --> 00:18:56
Yup, so it's plus 2.
 

381
00:18:56 --> 00:19:00
So the charge is actually just
equal to z, we can write plus

382
00:19:00 --> 00:19:04
2, or you can write plus 2 e, e
just means the absolute value

383
00:19:04 --> 00:19:05
of the charge on an electron.
 

384
00:19:05 --> 00:19:08
When we plug it into equations
we just use the number,

385
00:19:08 --> 00:19:11
the e is assumed there.
 

386
00:19:11 --> 00:19:13
So, let's think of what we
could have if we have two

387
00:19:13 --> 00:19:16
electrons in a helium atom
that are shielded in

388
00:19:16 --> 00:19:17
two extreme ways.
 

389
00:19:17 --> 00:19:21
So, in the first extreme way,
let's consider that our first

390
00:19:21 --> 00:19:25
electron is at some distance
very far away from the nucleus,

391
00:19:25 --> 00:19:29
we'll call this electron one,
and our second electron is, in

392
00:19:29 --> 00:19:32
fact, much, much closer to the
nucleus, and let's think of the

393
00:19:32 --> 00:19:35
idea of shielding in more of
the classical sense where we're

394
00:19:35 --> 00:19:39
actually blocking some of
that positive charge.

395
00:19:39 --> 00:19:42
So if we have total and
complete shielding where that

396
00:19:42 --> 00:19:45
can actually negate a full
positive charge, because

397
00:19:45 --> 00:19:48
remember our nucleus is plus 2,
one of the electrons is minus

398
00:19:48 --> 00:19:51
1, so if it totally blocks it,
all we would have left from the

399
00:19:51 --> 00:19:54
nucleus is an effective
charge of plus 1.

400
00:19:54 --> 00:19:57
So in our first case, our first
extreme case, would be that the

401
00:19:57 --> 00:20:01
z effective that is felt by
electron number 1, is

402
00:20:01 --> 00:20:04
going to be plus 1.
 

403
00:20:04 --> 00:20:06
So, what we can do is figure
out what we would expect

404
00:20:06 --> 00:20:09
the binding energy of that
electron to be in the case

405
00:20:09 --> 00:20:10
of this total shielding.
 

406
00:20:10 --> 00:20:14
And remember again, the binding
energy physically is the

407
00:20:14 --> 00:20:17
negative of the ionization
energy, and that's actually how

408
00:20:17 --> 00:20:19
you can experimentally check
to see if this is

409
00:20:19 --> 00:20:20
actually correct.
 

410
00:20:20 --> 00:20:23
And that's going to be equal to
negative z effective squared

411
00:20:23 --> 00:20:25
times r h over n squared.
 

412
00:20:25 --> 00:20:28
So, let's plug in these values
and see what we would expect

413
00:20:28 --> 00:20:30
to see for the energy.
 

414
00:20:30 --> 00:20:34
So it would be negative 1
squared times r h all over 1

415
00:20:34 --> 00:20:38
squared, since our z effective
we're saying is 1, and n is

416
00:20:38 --> 00:20:41
also equal to 1, because we're
in the ground state here so

417
00:20:41 --> 00:20:46
we're talking about
a 1 s orbital.

418
00:20:46 --> 00:20:49
So if we have a look at what
the answer would be, this

419
00:20:49 --> 00:20:50
looks very familiar.
 

420
00:20:50 --> 00:20:53
We would expect our binding
energy to be a negative 2 .

421
00:20:53 --> 00:20:56
1 8 times 10 to the
negative 18 joules.

422
00:20:56 --> 00:20:59
This is actually what the
binding energy is for hydrogen

423
00:20:59 --> 00:21:02
atom, and in fact, that makes
sense because in our extreme

424
00:21:02 --> 00:21:04
case where we have total
shielding by the second

425
00:21:04 --> 00:21:07
electron of the electron of
interest, it's essentially

426
00:21:07 --> 00:21:11
seeing the same nuclear
force that an electron in a

427
00:21:11 --> 00:21:12
hydrogen atom would see.
 

428
00:21:12 --> 00:21:13
All right.
 

429
00:21:13 --> 00:21:17
Let's consider now the second
extreme case, or extreme case

430
00:21:17 --> 00:21:18
b, for our helium atom.
 

431
00:21:18 --> 00:21:22
Again we have the charge of the
nucleus on plus 2, but let's

432
00:21:22 --> 00:21:24
say this time the electron now
is going to be very, very

433
00:21:24 --> 00:21:26
close to the nucleus.
 

434
00:21:26 --> 00:21:29
And let's say our second
electron now is really far

435
00:21:29 --> 00:21:32
away, such that it's actually
not going to shield any of

436
00:21:32 --> 00:21:36
the nuclear charge at all
from that first electron.

437
00:21:36 --> 00:21:39
So what we end up saying is
that the z effective or the

438
00:21:39 --> 00:21:42
effective charge that that
first electron feels is

439
00:21:42 --> 00:21:45
now going to be plus 2.
 

440
00:21:45 --> 00:21:49
Again, we can just plug this
into our equation, so if we

441
00:21:49 --> 00:21:53
write in our numbers now saying
that z effective is equal to 2,

442
00:21:53 --> 00:21:58
we find that we get negative 2
squared r h, all divided again

443
00:21:58 --> 00:22:00
by 1 squared -- we're still
talking about a 1

444
00:22:00 --> 00:22:03
s orbital here.
 

445
00:22:03 --> 00:22:06
And if we do that calculation,
what we find out is that the

446
00:22:06 --> 00:22:10
binding energy, in this case
where we have no shielding,

447
00:22:10 --> 00:22:12
is negative 8 .
 

448
00:22:12 --> 00:22:17
7 2 times 10 to the
negative 18 joules.

449
00:22:17 --> 00:22:20
So, let's compare what we've
just seen as our two extremes.

450
00:22:20 --> 00:22:24
So in extreme case a, we saw
that z effective was 1.

451
00:22:24 --> 00:22:26
This is what we call
total shielding.

452
00:22:26 --> 00:22:29
The electron completely
canceled out it's equivalent of

453
00:22:29 --> 00:22:32
charge from the nucleus, such
that we only saw in

454
00:22:32 --> 00:22:33
a z effective of 1.
 

455
00:22:33 --> 00:22:38
In an extreme case b, we had
a z effective of 2, so

456
00:22:38 --> 00:22:41
essentially what we had
was no shielding at all.

457
00:22:41 --> 00:22:44
We said that that second
electron was so far out of the

458
00:22:44 --> 00:22:46
picture, that it had absolutely
no affect on what the charge

459
00:22:46 --> 00:22:49
was felt by that
first electron.

460
00:22:49 --> 00:22:52
So, we can actually think about
now, we know the extreme cases,

461
00:22:52 --> 00:22:56
but what is the reality, and
the reality is if we think

462
00:22:56 --> 00:22:58
about the ionization energy,
and we measure it

463
00:22:58 --> 00:23:00
experimentally, we
find that it's 3 .

464
00:23:00 --> 00:23:04
9 4 times 10 to the negative 18
joules, and what you can see is

465
00:23:04 --> 00:23:07
that falls right in the middle
between the two ionization

466
00:23:07 --> 00:23:10
energies that we would expect
for the extreme cases.

467
00:23:10 --> 00:23:13
And this is absolutely
confirming that what is

468
00:23:13 --> 00:23:16
happening is what we would
expect to happen, because we

469
00:23:16 --> 00:23:19
would expect the case of
reality is that, in fact, some

470
00:23:19 --> 00:23:21
shielding is going on, but it's
not going to be total

471
00:23:21 --> 00:23:24
shielding, but at the same time
it's not going to be

472
00:23:24 --> 00:23:26
no shielding at all.
 

473
00:23:26 --> 00:23:30
And if we experimentally know
what the ionization energy is,

474
00:23:30 --> 00:23:33
we actually have a way to find
out what the z effective

475
00:23:33 --> 00:23:34
will be equal to.
 

476
00:23:34 --> 00:23:38
And we can use this equation
here, this is just the equation

477
00:23:38 --> 00:23:41
for the ionization energy,
which is the same thing as

478
00:23:41 --> 00:23:44
saying the negative of the
binding energy that's equal to

479
00:23:44 --> 00:23:48
z effective squared
r h over n squared.

480
00:23:48 --> 00:23:51
So, what we can do instead of
talking about the ionization

481
00:23:51 --> 00:23:55
energy, because that's one of
our known quantities, is we

482
00:23:55 --> 00:23:58
can instead solve so that
we can find z effective.

483
00:23:58 --> 00:24:03
So, if we just rearrange this
equation, what we find is that

484
00:24:03 --> 00:24:10
z effective is equal to n
squared times the ionization

485
00:24:10 --> 00:24:16
energy, all over the
Rydberg constant and the

486
00:24:16 --> 00:24:17
square root of this.
 

487
00:24:17 --> 00:24:22
So the square root of n
squared r e over r h.

488
00:24:22 --> 00:24:27
So what's our value
for n here? one.

489
00:24:27 --> 00:24:29
Yup, that's right.
 

490
00:24:29 --> 00:24:35
And then what's our value
for ionization energy?

491
00:24:35 --> 00:24:35
Yup.
 

492
00:24:35 --> 00:24:37
So it's just that ionization
energy that we have

493
00:24:37 --> 00:24:40
experimentally measured, 3 .
 

494
00:24:40 --> 00:24:44
9 4 times 10 to the
negative 18 joules.

495
00:24:44 --> 00:24:48
We put all of this over the
Rydberg constant, which is 2 .

496
00:24:48 --> 00:24:53
1 8 times 10 to the negative
18 joules, and we want to

497
00:24:53 --> 00:24:59
raise this all to the 1/2.
 

498
00:24:59 --> 00:25:02
So what we end up seeing is
that the z effective is

499
00:25:02 --> 00:25:05
equal to positive 1 .
 

500
00:25:05 --> 00:25:07
3 4.
 

501
00:25:07 --> 00:25:11
So, this is what we find the
actual z effective is for an

502
00:25:11 --> 00:25:13
electron in the helium atom.
 

503
00:25:13 --> 00:25:17
Does this seem like a
reasonable number?

504
00:25:17 --> 00:25:17
Yeah?
 

505
00:25:17 --> 00:25:20
Who says yes, raise your hand
if this seems reasonable.

506
00:25:20 --> 00:25:23
Does anyone think this
seems not reasonable?

507
00:25:23 --> 00:25:23
OK.
 

508
00:25:23 --> 00:25:25
How can we check, for
example, if it does or if

509
00:25:25 --> 00:25:27
it doesn't seem reasonable.
 

510
00:25:27 --> 00:25:29
Well, the reason, the way that
we can check it is just to see

511
00:25:29 --> 00:25:32
if it's in between our
two extreme cases.

512
00:25:32 --> 00:25:34
We know that it has to be more
than 1, because even if we had

513
00:25:34 --> 00:25:36
total shielding, we would at
least feel is the

514
00:25:36 --> 00:25:38
effective of 1.
 

515
00:25:38 --> 00:25:41
We know that it has to be equal
to less than 2, because even if

516
00:25:41 --> 00:25:45
we had absolutely no shielding
at all, the highest z effective

517
00:25:45 --> 00:25:48
we could have is 2, so it makes
perfect sense that we have a z

518
00:25:48 --> 00:25:51
effective that falls somewhere
in the middle of those two.

519
00:25:51 --> 00:25:55
So, let's look at another
example of thinking about

520
00:25:55 --> 00:25:57
whether we get an answer
out that's reasonable.

521
00:25:57 --> 00:26:00
So we should be able to
calculate a z effective for any

522
00:26:00 --> 00:26:02
atom that we want to talk
about, as long as we know what

523
00:26:02 --> 00:26:04
that ionization energy is.
 

524
00:26:04 --> 00:26:07
And I'm not expecting you to do
that calculation here, because

525
00:26:07 --> 00:26:09
it involves the calculator,
among maybe a piece

526
00:26:09 --> 00:26:10
of paper as well.
 

527
00:26:10 --> 00:26:13
But what you should be able to
do is take a look at a list of

528
00:26:13 --> 00:26:17
answers for what we're saying z
effective might be, and

529
00:26:17 --> 00:26:20
determining which ones are
possible versus which

530
00:26:20 --> 00:26:21
ones are not possible.
 

531
00:26:21 --> 00:26:23
So, why don't you take a look
at this and tell me which are

532
00:26:23 --> 00:26:28
possible for a 2 s electron
in a lithium atom where z is

533
00:26:28 --> 00:26:48
going to be equal to three?
 

534
00:26:48 --> 00:27:04
Let's do 10 more
seconds on that.

535
00:27:04 --> 00:27:05
OK, great.
 

536
00:27:05 --> 00:27:06
So, the majority of
you got it right.

537
00:27:06 --> 00:27:09
There are some people that are
a little bit confused still on

538
00:27:09 --> 00:27:11
where this make sense, so,
let's just think about

539
00:27:11 --> 00:27:12
this a little bit more.
 

540
00:27:12 --> 00:27:17
So now we're saying that z is
equal to 3, so if, for example,

541
00:27:17 --> 00:27:21
we had total shielding by the
other two electrons, if they

542
00:27:21 --> 00:27:25
totally canceled out one unit
of positive charge each in the

543
00:27:25 --> 00:27:29
nucleus, what we would end up
with is we started with 3 and

544
00:27:29 --> 00:27:33
then we would subtract a charge
of 2, so we would end up with a

545
00:27:33 --> 00:27:36
plus 1 z effective
from the nucleus.

546
00:27:36 --> 00:27:40
So our minimum that we're going
to see is that the smallest we

547
00:27:40 --> 00:27:43
can have for a z effective
is going to be equal to 1.

548
00:27:43 --> 00:27:46
So any of the answers that
said a z effective of .

549
00:27:46 --> 00:27:47
3 9 or .
 

550
00:27:47 --> 00:27:50
8 7 are possible, they actually
aren't possible because even if

551
00:27:50 --> 00:27:53
we saw a total shielding, the
minimum z effective

552
00:27:53 --> 00:27:54
we would see is 1.
 

553
00:27:54 --> 00:27:58
And then I think it looks like
most people understood that

554
00:27:58 --> 00:27:59
four was not a possibility.
 

555
00:27:59 --> 00:28:02
Of course, if we saw no
shielding at all what we

556
00:28:02 --> 00:28:05
would end up with is
a z effective of 3.

557
00:28:05 --> 00:28:07
So again, when we check these,
what we want to see is that our

558
00:28:07 --> 00:28:10
z effective falls in between
the two extreme cases that we

559
00:28:10 --> 00:28:13
could envision for shielding.
 

560
00:28:13 --> 00:28:15
And again, just go back and
look at this and think about

561
00:28:15 --> 00:28:17
this, this should make sense if
you kind of look at those two

562
00:28:17 --> 00:28:21
extreme examples, so even if it
doesn't make entire sense in

563
00:28:21 --> 00:28:23
the 10 seconds you have to
answer a clicker question right

564
00:28:23 --> 00:28:26
now, make sure this weekend you
can go over it and be able to

565
00:28:26 --> 00:28:29
predict if you saw a list of
answers or if you calculate

566
00:28:29 --> 00:28:32
your own answer on the p-set,
whether or not it's right or

567
00:28:32 --> 00:28:35
it's wrong, you should be able
to qualitatively confirm

568
00:28:35 --> 00:28:37
whether you have a reasonable
or a not reasonable

569
00:28:37 --> 00:28:42
answer after you do
the calculation part.

570
00:28:42 --> 00:28:42
All right.
 

571
00:28:42 --> 00:28:44
So now that we have a general
idea of what we're talking

572
00:28:44 --> 00:28:48
about with shielding, we can
now go back and think about why

573
00:28:48 --> 00:28:50
it is that the orbitals are
ordered in the order

574
00:28:50 --> 00:28:51
that they are.
 

575
00:28:51 --> 00:28:54
We know that the orbitals
for multi-electron atoms

576
00:28:54 --> 00:28:56
depend both on n and on l.
 

577
00:28:56 --> 00:29:00
But we haven't yet addressed
why, for example, a 2 s orbital

578
00:29:00 --> 00:29:05
is lower in energy than the 2 p
orbital, or why, for example, a

579
00:29:05 --> 00:29:09
3 s orbital is lower in energy
than a 3 p, which in turn is

580
00:29:09 --> 00:29:11
lower than a 3 d orbital.
 

581
00:29:11 --> 00:29:15
So let's think about shielding
in trying to answer why, in

582
00:29:15 --> 00:29:20
fact, it's those s orbitals
that are the lowest in energy.

583
00:29:20 --> 00:29:22
And when we make these
comparisons, one thing I want

584
00:29:22 --> 00:29:26
to point out is that we need to
keep the constant principle

585
00:29:26 --> 00:29:29
quantum number constant, so
we're talking about a certain

586
00:29:29 --> 00:29:31
state, so we could talk about
the n equals 2 state, or

587
00:29:31 --> 00:29:33
the n equals 3 state.
 

588
00:29:33 --> 00:29:36
And when we're talking about
orbitals in the same state,

589
00:29:36 --> 00:29:39
what we find is that orbitals
that have lower values of l

590
00:29:39 --> 00:29:42
can actually penetrate
closer to the nucleus.

591
00:29:42 --> 00:29:45
This is an idea we introduced
on Wednesday when we were

592
00:29:45 --> 00:29:48
looking at the radial
probability distributions of p

593
00:29:48 --> 00:29:51
orbitals versus s orbitals
versus d orbitals.

594
00:29:51 --> 00:29:54
But now it's going to make more
sense because in that case we

595
00:29:54 --> 00:29:57
were just talking about single
electron atoms, and now we're

596
00:29:57 --> 00:30:00
talking about a case where we
actually can see shielding.

597
00:30:00 --> 00:30:03
So what is actually going to
matter is how closely that

598
00:30:03 --> 00:30:06
electron can penetrate to the
nucleus, and what I mean by

599
00:30:06 --> 00:30:10
penetrate to the nucleus is is
there probability density a

600
00:30:10 --> 00:30:13
decent amount that's very
close to the nucleus.

601
00:30:13 --> 00:30:16
So, if we superimpose, for
example, the 2 s radial

602
00:30:16 --> 00:30:21
probability distribution over
the 2 p, what we see is there's

603
00:30:21 --> 00:30:24
this little bit of probability
density in the 2 s, but it is

604
00:30:24 --> 00:30:27
significant, and that's closer
to the nucleus than

605
00:30:27 --> 00:30:29
it is for the 2 p.
 

606
00:30:29 --> 00:30:33
And remember, this is in
complete opposition to what we

607
00:30:33 --> 00:30:37
call the size of the orbitals,
because we know that the 2 p is

608
00:30:37 --> 00:30:39
actually a smaller orbital.
 

609
00:30:39 --> 00:30:41
For example, when we're talking
about radial probability

610
00:30:41 --> 00:30:45
distributions, the most
probable radius is closer into

611
00:30:45 --> 00:30:47
the nucleus than it is
for the s orbital.

612
00:30:47 --> 00:30:50
But what's important is not
where that most probable radius

613
00:30:50 --> 00:30:53
is when we're talking about the
z effective it feels, what's

614
00:30:53 --> 00:30:56
more important is how close the
electron actually can

615
00:30:56 --> 00:30:57
get the nucleus.
 

616
00:30:57 --> 00:31:01
And for the s electron, since
it can get closer, what we're

617
00:31:01 --> 00:31:04
going to see is that s
electrons are actually

618
00:31:04 --> 00:31:07
less shielded than the
corresponding p electrons.

619
00:31:07 --> 00:31:10
They're less shielded because
they're closer to the nucleus,

620
00:31:10 --> 00:31:15
they feel a greater
z effective.

621
00:31:15 --> 00:31:18
We can see the same thing when
we compare p electrons to d

622
00:31:18 --> 00:31:20
electrons, or p and d orbitals.
 

623
00:31:20 --> 00:31:24
I've drawn the 3 p and the 3 d
orbital here, and again, what

624
00:31:24 --> 00:31:27
you can see is that the p
electron are going to be able

625
00:31:27 --> 00:31:30
to penetrate closer to the
nucleus because of the fact

626
00:31:30 --> 00:31:33
that there's this bit of
probability density that's in

627
00:31:33 --> 00:31:36
significantly closer to
the nucleus than it is

628
00:31:36 --> 00:31:38
for the 3 d orbital.
 

629
00:31:38 --> 00:31:42
And if we go ahead and
superimpose the 3 s on top of

630
00:31:42 --> 00:31:46
the 3 p, you can see that the
3 s actually has some bit of

631
00:31:46 --> 00:31:49
probability density that gets
even closer to the nucleus

632
00:31:49 --> 00:31:51
than the 3 p did.
 

633
00:31:51 --> 00:31:53
So that's where that trend
comes from where the s orbital

634
00:31:53 --> 00:31:55
is lower than the d orbital,
which is lower than

635
00:31:55 --> 00:31:59
the d orbital.
 

636
00:31:59 --> 00:32:03
So now that we have this idea
of shielding and we can talk

637
00:32:03 --> 00:32:05
about the differences in the
radial probability

638
00:32:05 --> 00:32:09
distributions, we can consider
more completely why, for

639
00:32:09 --> 00:32:12
example, if we're talking about
lithium, we write the electron

640
00:32:12 --> 00:32:17
configuration as 1 s 2, 2 s 1,
and we don't instead jump

641
00:32:17 --> 00:32:20
from the 1 s 2 all the
way to a p orbital.

642
00:32:20 --> 00:32:23
So the most basic answer that
doesn't explain why is just to

643
00:32:23 --> 00:32:26
say well, the s orbital is
lower in energy than the p

644
00:32:26 --> 00:32:29
orbital, but we now have a more
complete answer, so we can

645
00:32:29 --> 00:32:32
actually describe why that is.
 

646
00:32:32 --> 00:32:37
And what we're actually talking
about again is the z effective.

647
00:32:37 --> 00:32:40
So that z effective felt
by the 2 p is going to be

648
00:32:40 --> 00:32:44
less than the z effective
felt by the 2 s.

649
00:32:44 --> 00:32:48
And another way to say this, I
think it's easiest to look at

650
00:32:48 --> 00:32:51
just the fact that there's some
probability density very close

651
00:32:51 --> 00:32:54
the nucleus, but what we can
actually do is average the z

652
00:32:54 --> 00:32:56
effective over this entire
radial probability

653
00:32:56 --> 00:33:00
distribution, and when we find
that, we find that it does turn

654
00:33:00 --> 00:33:04
out that the average of the z
effective over the 2 p is

655
00:33:04 --> 00:33:07
going to be less than
that of the 2 s.

656
00:33:07 --> 00:33:10
So we know that we can relate
to z effective to the actual

657
00:33:10 --> 00:33:14
energy level of each of those
orbitals, and we can do that

658
00:33:14 --> 00:33:17
using this equation here where
it's negative z effective

659
00:33:17 --> 00:33:19
squared r h over n squared,
we're going to see

660
00:33:19 --> 00:33:20
that again and again.
 

661
00:33:20 --> 00:33:24
And it turns out that if we
have a, for example, for s, a

662
00:33:24 --> 00:33:28
very large z effective or
larger z effective than for 2

663
00:33:28 --> 00:33:33
p, and we plug in a large value
here in the numerator, that

664
00:33:33 --> 00:33:35
means we're going to end up
with a very large

665
00:33:35 --> 00:33:36
negative number.
 

666
00:33:36 --> 00:33:39
So in other words a very low
energy is what we're going to

667
00:33:39 --> 00:33:42
have when we talk about the
orbitals -- the energy of the 2

668
00:33:42 --> 00:33:45
s orbital is going to be less
than the energy of

669
00:33:45 --> 00:33:46
the 2 p orbital.
 

670
00:33:46 --> 00:33:49
Another way to say that it's
going to be less, so you don't

671
00:33:49 --> 00:33:52
get confused with that the fact
this is in the numerator here,

672
00:33:52 --> 00:33:55
there is that negative sign so
it's less energy but it's a

673
00:33:55 --> 00:33:57
bigger negative number that
gives us that less

674
00:33:57 --> 00:34:01
energy there.
 

675
00:34:01 --> 00:34:04
All right, so let's go back to
electrons configurations now

676
00:34:04 --> 00:34:09
that we have an idea of why the
orbitals are listed in the

677
00:34:09 --> 00:34:12
energy that they are listed
under, why, for example, the

678
00:34:12 --> 00:34:14
2 s is lower than the 2 p.
 

679
00:34:14 --> 00:34:18
So now we can go back and think
about filling in these electron

680
00:34:18 --> 00:34:20
configurations for any atom.
 

681
00:34:20 --> 00:34:23
I think most and you are
familiar with the Aufbau or the

682
00:34:23 --> 00:34:27
building up principle, you
probably have seen it quite a

683
00:34:27 --> 00:34:29
bit in high school, and this is
the idea that we're filling up

684
00:34:29 --> 00:34:33
our energy states, again, which
depend on both n and l, one

685
00:34:33 --> 00:34:37
electron at a time starting
with that lowest energy and

686
00:34:37 --> 00:34:41
then working our way up into
higher and higher orbitals.

687
00:34:41 --> 00:34:44
And when we follow the Aufbau
principle, we have to

688
00:34:44 --> 00:34:45
follow two other rules.
 

689
00:34:45 --> 00:34:48
One is the Pauli exclusion
principal, we discussed

690
00:34:48 --> 00:34:49
this on Wednesday.
 

691
00:34:49 --> 00:34:52
So this is just the idea that
the most electrons that

692
00:34:52 --> 00:34:54
you can have in a single
orbital is two electrons.

693
00:34:54 --> 00:34:58
That makes sense because we
know that every single electron

694
00:34:58 --> 00:35:02
has to have its own distinct
set of four quantum numbers,

695
00:35:02 --> 00:35:06
the only way that we can do
that is to have a maximum of

696
00:35:06 --> 00:35:10
two spins in any single orbital
or two electrons per orbital.

697
00:35:10 --> 00:35:13
We also need to follow Hund's
rule, this is that a single

698
00:35:13 --> 00:35:16
electron enters each state
before it enters

699
00:35:16 --> 00:35:18
a second state.
 

700
00:35:18 --> 00:35:20
And by state we just mean
orbital, so if we're looking at

701
00:35:20 --> 00:35:24
the p orbitals here, that means
that a single electron goes in

702
00:35:24 --> 00:35:28
x, and then it will go in the z
orbital before a second one

703
00:35:28 --> 00:35:30
goes in the x orbital.
 

704
00:35:30 --> 00:35:32
This intuitively should make a
lot of sense, because we know

705
00:35:32 --> 00:35:36
we're trying to minimize
electron repulsions to keep

706
00:35:36 --> 00:35:39
things in as low an energy
state as possible, so it makes

707
00:35:39 --> 00:35:43
sense that we would put one
electron in each orbital first

708
00:35:43 --> 00:35:46
before we double up
in any orbital.

709
00:35:46 --> 00:35:49
And the third fact that we need
to keep in mind is that spins

710
00:35:49 --> 00:35:53
remain parallel prior to adding
a second electron in

711
00:35:53 --> 00:35:54
any of the orbitals.
 

712
00:35:54 --> 00:35:57
So by parallel we mean they're
either both spin up or they're

713
00:35:57 --> 00:36:00
both spin down -- remember
that's our spin quantum number,

714
00:36:00 --> 00:36:02
that fourth quantum number.
 

715
00:36:02 --> 00:36:04
And the reason for this comes
out of solving the relativistic

716
00:36:04 --> 00:36:07
version of the Schrodinger
equation, so unfortunately it's

717
00:36:07 --> 00:36:10
not as intuitive as knowing
that we want to fill separate

718
00:36:10 --> 00:36:14
before we double up a
degenerate orbital, but you

719
00:36:14 --> 00:36:17
just need to keep this in mind
and you need to just memorize

720
00:36:17 --> 00:36:21
the fact that you need to be
parallel before you double

721
00:36:21 --> 00:36:22
up in the orbital.
 

722
00:36:22 --> 00:36:24
So, we'll see how this
works in a second.

723
00:36:24 --> 00:36:29
So let's do this considering,
for example, what it would look

724
00:36:29 --> 00:36:31
like if we were to write out
the electron configuration for

725
00:36:31 --> 00:36:34
oxygen where z is going
to be equal to 8.

726
00:36:34 --> 00:36:37
So what we're doing is filling
in those eight electrons

727
00:36:37 --> 00:36:41
following the Aufbau principle,
so our first electron is going

728
00:36:41 --> 00:36:45
to go in the 1 s, and then we
have no other options for other

729
00:36:45 --> 00:36:48
orbitals that are at that same
energy, so we put the second

730
00:36:48 --> 00:36:50
electron in the 1 s as well.
 

731
00:36:50 --> 00:36:53
Then we go up to the 2 s, and
we have two electrons that

732
00:36:53 --> 00:36:55
we can fill in the 2 s.
 

733
00:36:55 --> 00:36:59
And now we get the p orbitals,
remember we want to fill up 1

734
00:36:59 --> 00:37:02
orbital at a time before we
double up, so we'll put one in

735
00:37:02 --> 00:37:08
the 2 p x, then one in the 2 p
z, and then one in the 2 p y.

736
00:37:08 --> 00:37:11
At this point, we have no other
choice but to double up before

737
00:37:11 --> 00:37:14
going to the next energy level,
so we'll put a second

738
00:37:14 --> 00:37:15
one in the 2 p x.
 

739
00:37:15 --> 00:37:18
And I arbitrarily chose to put
it in the 2 p x, we also could

740
00:37:18 --> 00:37:22
have put it in the 2 p y or the
2 p z, it doesn't matter where

741
00:37:22 --> 00:37:24
you double up, they're
all the same energy.

742
00:37:24 --> 00:37:27
So if we think about what we
would do to actually write out

743
00:37:27 --> 00:37:31
this configuration, we just
write the energy levels that we

744
00:37:31 --> 00:37:34
see here or the orbital
approximations.

745
00:37:34 --> 00:37:38
So if we're talking about
oxygen, we would say that it's

746
00:37:38 --> 00:37:45
1 s 2, then we have 2 s 2, and
then we have 2 p, and our

747
00:37:45 --> 00:37:49
total number of electrons in
the p orbitals are four.

748
00:37:49 --> 00:37:51
So it's OK to not specify.
 

749
00:37:51 --> 00:37:54
I want to point out, whether
you're in the p x, the p y, or

750
00:37:54 --> 00:37:58
the p z, unless a question
specifically asks you to

751
00:37:58 --> 00:38:02
specify the m sub l, which
occasionally will happen, but

752
00:38:02 --> 00:38:04
if it doesn't happen you
just write it like this.

753
00:38:04 --> 00:38:09
But if, in fact, you are asked
to specify the m sub l's, then

754
00:38:09 --> 00:38:11
we would have to write it out
more completely, which would be

755
00:38:11 --> 00:38:19
the 1 s 2, the 2 s 2, and then
we would say 2 p x 2,

756
00:38:19 --> 00:38:22
2 p z 1, and 2 p y 1.
 

757
00:38:22 --> 00:38:29
So again, in general, just go
ahead and write it out like

758
00:38:29 --> 00:38:32
this, but if we do ask you to
specify you should be able to

759
00:38:32 --> 00:38:35
know that the p orbital
separates into these three --

760
00:38:35 --> 00:38:42
the p sub-shell separates
into these three orbitals.

761
00:38:42 --> 00:38:46
So let's do a clicker question
on assigning electron

762
00:38:46 --> 00:38:49
configurations using
the Aufbau principle.

763
00:38:49 --> 00:38:52
So why don't you go ahead and
identify the correct electron

764
00:38:52 --> 00:38:55
configuration for carbon, and
I'll tell you that z

765
00:38:55 --> 00:38:57
is equal to 6 here.
 

766
00:38:57 --> 00:39:01
And in terms of doing this for
your homework, I actually want

767
00:39:01 --> 00:39:04
to mention that in the back
page of your notes I attached

768
00:39:04 --> 00:39:06
a periodic table that
does not have electron

769
00:39:06 --> 00:39:08
configurations on them.
 

770
00:39:08 --> 00:39:10
It's better to practice doing
electron configurations when

771
00:39:10 --> 00:39:13
you cannot actually see the
electron configurations.

772
00:39:13 --> 00:39:15
And this is the same periodic
table that you're going to get

773
00:39:15 --> 00:39:17
in your exams, so it's good
to practice doing your

774
00:39:17 --> 00:39:20
problem-sets with that periodic
table so you're not relying on

775
00:39:20 --> 00:39:23
having the double check right
there of seeing what the

776
00:39:23 --> 00:39:24
electron configuration is.
 

777
00:39:24 --> 00:39:39
So, let's do 10 seconds
on this problem here.

778
00:39:39 --> 00:39:40
OK, great.
 

779
00:39:40 --> 00:39:43
So this might be our best
clicker question yet.

780
00:39:43 --> 00:39:47
Most people were able to
identify the correct electron

781
00:39:47 --> 00:39:50
configuration here.
 

782
00:39:50 --> 00:39:52
Some people, the next most
popular answer with 5%, which

783
00:39:52 --> 00:39:56
is a nice low number, wanted
to put two in the 2 p x

784
00:39:56 --> 00:39:57
before they moved on.
 

785
00:39:57 --> 00:40:01
Remember we have to put one in
each degenerate orbital before

786
00:40:01 --> 00:40:04
we double up on any orbital, so
just keep that rule in mind

787
00:40:04 --> 00:40:07
that we would fill one in each
p orbital before we a

788
00:40:07 --> 00:40:07
to the second one.
 

789
00:40:07 --> 00:40:11
But it looks like you guys are
all experts here on doing these

790
00:40:11 --> 00:40:12
electron configurations.
 

791
00:40:12 --> 00:40:15
So, let's move on to
some more complicated

792
00:40:15 --> 00:40:17
electron configurations.
 

793
00:40:17 --> 00:40:19
So, for example, we can move
to the next periods in

794
00:40:19 --> 00:40:21
the periodic table.
 

795
00:40:21 --> 00:40:24
When we talk about a period,
we're just talking about that

796
00:40:24 --> 00:40:27
principle quantum number, so
period 2 means that we're

797
00:40:27 --> 00:40:31
talking about starting with the
2 s orbitals, period 3 starts

798
00:40:31 --> 00:40:35
with, what we're now filling
into the 3 s orbitals here.

799
00:40:35 --> 00:40:38
So if we're talking about the
third period, that starts with

800
00:40:38 --> 00:40:41
sodium and it goes all
the way up to argon.

801
00:40:41 --> 00:40:43
So if we write the electron
configuration for sodium, which

802
00:40:43 --> 00:40:46
you can try later -- hopefully
you would all get it correctly

803
00:40:46 --> 00:40:49
-- you see that this is the
electron configuration here, 1

804
00:40:49 --> 00:40:54
s 2, 2 s 2, 2 p 6, and now
we're going into that

805
00:40:54 --> 00:40:57
third shell, 3 s 1.
 

806
00:40:57 --> 00:41:00
And I want to point out the
difference between core

807
00:41:00 --> 00:41:02
electrons and valence
electrons here.

808
00:41:02 --> 00:41:05
If we look at this
configuration, what we say is

809
00:41:05 --> 00:41:08
all of the electrons in these
inner shells are what we

810
00:41:08 --> 00:41:10
call core electrons.
 

811
00:41:10 --> 00:41:14
The core electrons tend not to
be involved in much chemistry

812
00:41:14 --> 00:41:15
in bonding or in reactions.
 

813
00:41:15 --> 00:41:19
They're very deep and held very
tightly to the nucleus, so we

814
00:41:19 --> 00:41:23
can often lump them together,
and instead of writing them all

815
00:41:23 --> 00:41:27
out separately, we can just
write the equivalent noble gas

816
00:41:27 --> 00:41:28
that has that configuration.
 

817
00:41:28 --> 00:41:30
So, for example, for sodium,
we can instead write

818
00:41:30 --> 00:41:33
neon and then 3 s 1.
 

819
00:41:33 --> 00:41:37
So the 3 s 1, or any of the
other electrons that are in the

820
00:41:37 --> 00:41:39
outer-most shell, those are
what we call our valence

821
00:41:39 --> 00:41:43
electrons, and those are where
all the excitement happens.

822
00:41:43 --> 00:41:45
That's what we see are
involved in bonding.

823
00:41:45 --> 00:41:47
It makes sense, right, because
they're the furthest away from

824
00:41:47 --> 00:41:50
the nucleus, they're the ones
that are most willing to be

825
00:41:50 --> 00:41:54
involved in some chemistry or
in some bonding, or those are

826
00:41:54 --> 00:41:56
the orbitals that are most
likely to accept an electron

827
00:41:56 --> 00:41:59
from another atom, for example.
 

828
00:41:59 --> 00:42:01
So the valence electrons,
those are the exciting ones.

829
00:42:01 --> 00:42:04
We want to make sure we
have a full picture of

830
00:42:04 --> 00:42:05
what's going on there.
 

831
00:42:05 --> 00:42:08
So, no matter whether or not
you write out the full form

832
00:42:08 --> 00:42:11
here, or the noble gas
configuration where you write

833
00:42:11 --> 00:42:14
ne first or whatever the
corresponding noble gas is to

834
00:42:14 --> 00:42:17
the core electrons, we always
write out the valence

835
00:42:17 --> 00:42:18
electrons here.
 

836
00:42:18 --> 00:42:23
So for sodium, again, we can
write n e and then 3 s 1.

837
00:42:23 --> 00:42:26
We can go all the way down,
magnesium, aluminum, all the

838
00:42:26 --> 00:42:30
way to this noble gas, argon,
which would be n e and

839
00:42:30 --> 00:42:35
then 3 s 2, 3 p 6.
 

840
00:42:35 --> 00:42:37
Now we can think about the
fourth period, and the fourth

841
00:42:37 --> 00:42:41
period is where we start to run
into some exceptions, so this

842
00:42:41 --> 00:42:43
is where things get a teeny bit
more complicated, but you just

843
00:42:43 --> 00:42:46
need to remember the exceptions
and then you should be OK, no

844
00:42:46 --> 00:42:49
matter what you're
asked to write.

845
00:42:49 --> 00:42:52
So for the fourth period,
now we're into the 4 s

846
00:42:52 --> 00:42:54
1 for potassium here.
 

847
00:42:54 --> 00:42:58
And what we notice when we get
to the third element in and the

848
00:42:58 --> 00:43:02
fourth period is that we go 4
s 2 and then we're

849
00:43:02 --> 00:43:03
back to the 3 d's.
 

850
00:43:03 --> 00:43:07
So if you look at the energy
diagram, what we see is that

851
00:43:07 --> 00:43:10
the 4 s orbitals are actually
just a teeny bit lower in

852
00:43:10 --> 00:43:14
energy -- they're just ever so
slightly lower in energy

853
00:43:14 --> 00:43:15
than the 3 d orbitals.
 

854
00:43:15 --> 00:43:18
You can see that as you fill
up your periodic table,

855
00:43:18 --> 00:43:19
it's very clear.
 

856
00:43:19 --> 00:43:22
But also we'll tell you a
pneumonic device to keep

857
00:43:22 --> 00:43:24
that in mind, so you always
remember and get the

858
00:43:24 --> 00:43:26
orbital energy straight.
 

859
00:43:26 --> 00:43:29
But it just turns out that the
4 s is so low in energy that it

860
00:43:29 --> 00:43:32
actually surpasses the 3 d,
because we know the 3 d is

861
00:43:32 --> 00:43:35
going to be pretty high in
terms of the three shell, and

862
00:43:35 --> 00:43:38
the 4 s is going to be the
lowest in terms of the 4 shell,

863
00:43:38 --> 00:43:40
and it turns out that we need
to fill up the 4 s before

864
00:43:40 --> 00:43:42
we fill in the 3 d.
 

865
00:43:42 --> 00:43:46
And we can do that just going
along, 3 d 1, 2 3, and the

866
00:43:46 --> 00:43:50
problem comes when we get to
chromium here, which is instead

867
00:43:50 --> 00:43:52
of what we would expect,
we might expect to

868
00:43:52 --> 00:43:55
see 4 s 2, 3 d 4.
 

869
00:43:55 --> 00:44:01
What we see is that instead
it's 4 s 1, and 3 d 5.

870
00:44:01 --> 00:44:04
So this is the first exception
that you need to the

871
00:44:04 --> 00:44:05
Aufbau principle.
 

872
00:44:05 --> 00:44:09
The reason this an exception is
because it turns out that half

873
00:44:09 --> 00:44:11
filled d orbitals are more
stable than we could

874
00:44:11 --> 00:44:13
even predict.
 

875
00:44:13 --> 00:44:16
You wouldn't be expected to be
able to guess that this would

876
00:44:16 --> 00:44:19
happen, because using any kind
of simple theory, we would, in

877
00:44:19 --> 00:44:22
fact, predict that this would
not be the case, but what we

878
00:44:22 --> 00:44:25
find experimentally is that
it's more stable to have half

879
00:44:25 --> 00:44:32
filled d orbital than to
have a 4 s 2, and a 3 d 4.

880
00:44:32 --> 00:44:34
So you're going to need
to remember, so this

881
00:44:34 --> 00:44:36
is an exception, you
have to memorize.

882
00:44:36 --> 00:44:39
Another exception in the fourth
period is in copper here, we

883
00:44:39 --> 00:44:43
see that again, we have
4 s 1 instead of 4 s 2.

884
00:44:43 --> 00:44:49
This is 4 s 1, 3 d 10, we might
expect 4 s 2, 3 d 9, but again,

885
00:44:49 --> 00:44:53
this exception comes out of
experimental observation, which

886
00:44:53 --> 00:44:57
is the fact that full d
orbitals also are lower in

887
00:44:57 --> 00:45:00
energy then we could
theoretically predict using

888
00:45:00 --> 00:45:01
simple calculations.
 

889
00:45:01 --> 00:45:04
So again, you need to memorize
these two exceptions, and the

890
00:45:04 --> 00:45:09
exception in general is that
filled d 10, or half-filled d 5

891
00:45:09 --> 00:45:13
orbitals are lower in energy
than would be expected, so we

892
00:45:13 --> 00:45:16
got this flip-flip where if we
can get to that half filled

893
00:45:16 --> 00:45:20
orbital by only removing one s
electron, then we're going to

894
00:45:20 --> 00:45:22
do it, and the same with
the filled d orbital.

895
00:45:22 --> 00:45:27
And actually, when we get to
the fifth period of the

896
00:45:27 --> 00:45:31
periodic table, that again
takes place, so when you get to

897
00:45:31 --> 00:45:34
a half filled, or a filled d
orbital, again you want to do

898
00:45:34 --> 00:45:38
it, so those exceptions would
be with molybdenum and silver

899
00:45:38 --> 00:45:41
would be the corresponding
elements in the fifth period

900
00:45:41 --> 00:45:44
where you're going to see the
same case here where it's lower

901
00:45:44 --> 00:45:46
in energy to have the
half filled or the

902
00:45:46 --> 00:45:49
filled d orbitals.
 

903
00:45:49 --> 00:45:52
So here's the pneumonic I
mentioned for writing the

904
00:45:52 --> 00:45:53
electron configuration and
getting those orbital

905
00:45:53 --> 00:45:55
energies in the right order.
 

906
00:45:55 --> 00:45:58
All you do is just write out
all the orbitals, the 1 s, then

907
00:45:58 --> 00:46:02
the 2 s 2 p 3, 3 s 3 p d, just
write them in a straight line

908
00:46:02 --> 00:46:06
like this, and then if you draw
diagonals down them, what

909
00:46:06 --> 00:46:09
you'll get is the correct order
in terms of orbital energies.

910
00:46:09 --> 00:46:12
So if we go down the diagonal,
we start with 1 s, then we get

911
00:46:12 --> 00:46:17
2 s, then 2 p and 3 s, then 3
p, and 4 s, and then that's

912
00:46:17 --> 00:46:20
how we see here that 4 s is
actually lower in energy than 3

913
00:46:20 --> 00:46:23
d, then 4 p, 5 s and so on.
 

914
00:46:23 --> 00:46:26
So if you want to on an exam,
you can just write this down

915
00:46:26 --> 00:46:29
quickly at the beginning
and refer to it as you're

916
00:46:29 --> 00:46:32
filling up your electron
configurations, but also if you

917
00:46:32 --> 00:46:34
look at the periodic table it's
very clear as you try to fill

918
00:46:34 --> 00:46:38
it up that way that the same
order comes out of that.

919
00:46:38 --> 00:46:41
So, whichever works best for
you can do in terms of figuring

920
00:46:41 --> 00:46:44
out electron configurations.
 

921
00:46:44 --> 00:46:46
So the last thing I want to
mention today is how we

922
00:46:46 --> 00:46:50
can think about electron
configurations for ions.

923
00:46:50 --> 00:46:52
It turns out that it's going to
be a little bit different when

924
00:46:52 --> 00:46:54
we're talking about
positive ions here.

925
00:46:54 --> 00:46:57
We need to change our
rules just slightly.

926
00:46:57 --> 00:47:01
So what we know is that these 3
d orbitals are higher in energy

927
00:47:01 --> 00:47:03
than 4 s orbitals, so I've
written the energy of the

928
00:47:03 --> 00:47:07
orbital here for potassium
and for calcium.

929
00:47:07 --> 00:47:12
But what happens is that once a
d orbital is filled, I said the

930
00:47:12 --> 00:47:15
two are very close in energy,
and once a d orbital is filled,

931
00:47:15 --> 00:47:18
it actually drops to become
lower in energy than

932
00:47:18 --> 00:47:20
the 4 s orbital.
 

933
00:47:20 --> 00:47:24
So once we move past, we fill
the 4 s first, but once we fill

934
00:47:24 --> 00:47:28
in the d orbital, now that's
going to be lower in energy.

935
00:47:28 --> 00:47:30
So that doesn't make a
difference for us when we're

936
00:47:30 --> 00:47:34
talking about neutral atoms,
because we would fill up the 4

937
00:47:34 --> 00:47:38
s first, because that's lower
in energy until we fill it, and

938
00:47:38 --> 00:47:39
then we just keep going
with the d orbitals.

939
00:47:39 --> 00:47:42
So, for example, if we needed
to figure out the electron

940
00:47:42 --> 00:47:46
configuration for titanium, it
would just be argon then 4 s

941
00:47:46 --> 00:47:52
2, and then we would
fill in the 3 d 2.

942
00:47:52 --> 00:47:55
So, actually we don't have to
worry about this fact any time

943
00:47:55 --> 00:47:56
we're dealing with neutrals.
 

944
00:47:56 --> 00:47:59
The problem comes when instead
we're dealing with ions.

945
00:47:59 --> 00:48:02
So what I want to point out is
what we said now is that the 3

946
00:48:02 --> 00:48:05
d 2 is actually lower in
energy, so if we were to

947
00:48:05 --> 00:48:08
rewrite this in terms of what
the actual energy order is, we

948
00:48:08 --> 00:48:11
should instead write
it 3 d 2, 4 s 2.

949
00:48:11 --> 00:48:14
So you might ask in terms of
when you're writing electron

950
00:48:14 --> 00:48:16
configurations, which way
should you write it.

951
00:48:16 --> 00:48:18
And we'll absolutely
accept both answers

952
00:48:18 --> 00:48:18
for a neutral atom.
 

953
00:48:18 --> 00:48:18
They're both correct.
 

954
00:48:18 --> 00:48:18
In one case you decided to
order in terms of energy and in

955
00:48:18 --> 00:48:18
one case you decided to order
in terms of how it fills up.

956
00:48:18 --> 00:48:18
I don't care how you do it on
exams or on problem sets, but

957
00:48:18 --> 00:48:18
you do need to be aware that
the 3 d once filled is lower in

958
00:48:18 --> 00:48:18
energy than the 4 s, and the
reason you need to be aware of

959
00:48:18 --> 00:48:18
that is if you're asked for
the electron configuration

960
00:48:18 --> 00:48:18
now of the titanium ion.
 

961
00:48:18 --> 00:48:18
So, let's say we're asked
for the plus two ion.

962
00:48:18 --> 00:48:18
So a plus two ion means that
we're removing two electrons

963
00:48:18 --> 00:48:18
from the atom and the electrons
that we're going to remove

964
00:48:18 --> 00:48:18
are always going to be the
highest energy electrons.

965
00:48:18 --> 00:48:18
So it's good to write it like
this because this illustrates

966
00:48:18 --> 00:48:18
the fact that in fact the 4 s
electrons are the ones that

967
00:48:18 --> 00:48:18
are higher in energy.
 

968
00:48:18 --> 00:48:18
So the correct answer for
titanium plus two is going to

969
00:48:18 --> 00:48:18
be argon 3 d 2, whereas if we
did not rearrange our order

970
00:48:18 --> 00:48:18
here we might have been
tempted to write as 4

971
00:48:18 --> 00:48:18
s 2 so keep that in mind when
you're doing the positive

972
00:48:18 --> 00:48:18
ions of corresponding atoms.
 

973
00:48:18 --> 00:48:18
Alright, so we'll pick up with
photoelectron spectroscopy

974
00:48:18 --> 00:48:18
on Wednesday.
 

975
00:48:18 --> 00:48:19
Have a great weekend.
 

976
00:48:19 --> 00:49:24