Small x approximation for large Kc

- [Voiceover] In this video,
we're going to be talking about

using the small x approximation to solve

equilibrium problems when Kc is large.

And by large, I mean
that Kc is greater than

or approximately equal
to 10 to the fourth.

So we have another video
before this that talks about

using the small x approximation
for when Kc is small,

and I would say that's probably
a simpler place to start

if this is the first
time you're seeing it.

And in this video, we'll be talking about

the opposite situation.

As we talked about in the previous video,

the steps for solving this
kind of problem are fourfold.

So there are four steps.

The first step is to assume
that the reaction goes 100%

in the favored direction.

So the favored direction
when K is really large

is that means we're going
all the way to products.

We're assuming we have almost
no starting materials left.

The second step is to set up an ICE table.

And then to solve for x in our ICE table,

assuming that x is small,
hence small x approximation.

And then the last and possibly
the most important step

is to check our answer.

So make sure that the size
of x is actually small

compared to what we said
it was smaller than,

and also to make sure that
it gives us the right answer

when we plug it back in to calculate Kc.

In this video, we're going to
go through an example problem.

The example is the reaction of

NO gas reacting with

chlorine gas to make

2NOCl gas.

And this particular reaction has a K value

that is equal to 6.25 times 10

to the fourth.

So that is indeed on the
order of 10 to the fourth,

so we should be able to use
the small x approximation.

We're going to do it two ways.

So we're gonna set up our ICE table,

and we know the initial
concentrations are 2.0 molar

for NO, and 2.0 molar for Cl2.

And we have no NOCl at the beginning.

So if we are setting up an
ICE table as we normally did,

we would say, "Okay,
we'll, we're going to make

"some amount of NOCl."

So we'll just say minus 2x for our NO

since for every one Cl2 we use up,

we'll use up 2NO, which means
we'll have minus x for Cl2,

and we'll make 2x molar
concentration of our product.

And in the end, once we reach equilibrium,

we'll have to 2.0 molar minus 2x,

and we just get that by adding
the initial concentration

with the change.

In the same way, we get 2.0
molar minus x for chlorine gas.

And for a product, we just
get 2x at equilibrium.

Let's look back at our original
steps that we talked about

at the very beginning of this video.

We just made an ICE
table, so that's step two.

Did we assume that the
reaction goes to 100%

in our favored direction?

It turns out we didn't.

We actually skipped step
one to set up our ICE table

in what seems like a pretty natural way.

And it turns out that's not good.

So we'll see what happens
when we skip step one

and keep going.

So if we keep going, we can solve for x,

assuming x is small, and we'll get for

our equilibrium expression,

we get Kc is equal to 2x squared

divided by 2.0 molar

minus x, which our Cl2 concentration,

and multiply that by 2.0 molar minus 2x,

and that's all squared since
we have that stoichiometric

coefficient of two in front.

Just to point out, we already started

with a balanced reaction,
that's pretty important,

or else our Kc expression would be wrong.

So make sure everything is
balanced before you get started.

So we've written our expression for Kc,

and now we are going
to erroneously assume,

we're just going to be
like, "Oh, I don't care."

(chuckles)
"I'm just going to assume

"x is small.

"I don't care if that's a
good assumption at all."

So if x is small, what
we're really seeing is x

is a lot smaller than
two molar, here and here.

So we're really seeing 2x is
a lot smaller than two molar.

So we're seeing this is
approximately equal to

2x squared divided by

2.0 molar.

Since x is a lot smaller than two molar,

we'll just ignore it entirely,

and x is a lot smaller than 2x,

or sorry, x is a lot
smaller than 2.0 molar

even if you multiply x by two,

so this is going to be 2.0 molar squared.

And so if we multiply this out,
we get that this is equal to

x squared over

this two squared is
canceled out by this two,

so we actually get x
squared over two is equal to

our Kc, which is 6.25

times 10 to the fourth.

So that's what we get for x squared.

So if we multiply both sides by two,

we get that x squared is equal to

1.25 times 10 to the fifth,

which means x is equal to
the square root of that,

which is 354.

So this is where...

Well, okay.

So we've gone through steps two and three,

and now we're going to check our answer.

We're going to see if
our answer makes sense.

So we're saying that the
change in concentration here

is 354.

That clearly doesn't make sense
because it actually gives us

a negative concentration here.

So we're getting negative
concentrations for NO

and Cl2, so that's bad.

So that doesn't make sense.

The other thing is that
this already tells us

that our assumption was really bad.

We assumed to solve our equation for Kc.

We assumed that K is a
lot smaller than two,

or sorry, we assumed that x
is a lot smaller than two,

but then we got that x is 350,
which is clearly not smaller

than two.

So it looks like skipping
step one was bad.

So let's give this another try.

Try number two.

What was step one?

Step one says assume
the reaction goes 100%

in the favored direction.

And since Kc is really large
for this particular problem,

we're saying it's going
all the way to products.

What this basically means
is that we're gonna set up

our ICE table again,

except that this time,
our initial concentrations

are going to be assuming that

our reaction already went
all the way to products.

And we can figure that out,

we can figure out those
initial concentrations

using stoichiometry.

So in the beginning, we
started out with two molar NO

and two molar Cl2,

and we know that the NO and
Cl2 react in a 2:1 ratio.

So since we have the same
amount of both of these,

our limiting reactant

will be the NO.

And so that will get used up completely

when we get to equilibrium.

So that means if we assume
that it goes all the way to

products, we'll have zero molar NO left,

and will make 2.0 molar of our product.

And since we had our Cl2 in excess,

we'll use one molar of it
to react with two molar

of our NO, and we'll
have one molar left over.

So this is the special step
where we actually followed

step one and assumed

we got 100% of product.

And the reason why we made this assumption

was because we know
that K is really large.

So we know that at equilibrium,
we should have all product

or mostly product.

So now let's go through the other steps.

We assume that it's going to all product,

but we're not quite at equilibrium.

So if we're going to
equilibrium, that means we assume

that we'll see a little
bit of NO at equilibrium,

so that'll actually be plus 2x.

And we'll expect to see
a little bit more Cl2

at equilibrium.

So that'll be plus x because
of the stoichiometry,

and that would give us
minus 2x for the NOCl.

We expect a little of
this to get used up to go

in the reverse reaction.

So then if we add everything
together from the initial

and change, then we get 2x

for our NO concentration,

we get x for our Cl2 concentration,

and we get 2.0 minus 2x

for our product concentration.

So far, so good.

So now we're going to set up

our Kc expression just like we did before.

So Kc is still equal to 6.25

times 10 to the fourth,

and this is equal to the
concentration of NOCl squared.

So that is 2.0 minus 2x squared

divided by 2x squared,

which is our concentration
of NO squared times x.

Oh, oops, I made a mistake.

This is actually 1.0 plus x, sorry.

So this should be x plus 1.0 molar.

So that is our full expression for Kc

using our equilibrium concentrations,

and we have made zero approximations of R.

But now we're gonna
assume that we can assume,

(chuckles)
now we're gonna assume

that x is small.

We're gonna assume that assume.

So if x is small, what
we really mean is that x

is a lot smaller than one molar.

And we're also saying it's a
lot smaller than two molar.

So then we're saying that our
numerator is approximately

equal to 2.0 molar squared,

because we're saying that this is small.

I'm gonna make this an
approximately equal sign.

The 2x squared stays the same.

Since we're assuming x is
a lot more than 1.0 molar,

this just becomes 1.0 molar.

So now if we multiply this
all out, we get that four

divided by 4x squared

is equal to Kc,

and the fours cancel out.

So x squared is equal to one over Kc,

or 6.25 times 10 to the fourth.

And then if we take the square
root of both sides here,

we get that x is equal to, let's see,

we get x is equal to 4.0 times 10

to the minus three molar.

So this is where common sense is needed

to make sure that, well,

everything worked this time around.

So first of all, we can ask ourselves,

"Okay, is this x actually a
lot smaller than the numbers

"we said it was smaller than?"

So now we're comparing x to one,

which, it's about three
orders of magnitude smaller.

So that's good.

And we can also compare 2x to two.

And again, it's about three
orders of magnitude smaller.

So, so far, so good.

But our final test will
be to plug it back in

our Kc expression.

So if we plug in our
value of x, we get that Kc

is equal to 2.0 molar

minus two times 4.0

times 10 to the minus
three molar, all squared,

divided by two times 4.0

times 10 to the minus three molar,

and that's also squared, four,

and that's the NO
concentration at equilibrium.

And then the last part
is our Cl2 concentration,

which is 4.0 times 10
to the minus three molar

plus 1.0 molar.

And so if we multiply that all
out, what you get is that Kc

is equal to 6.23 times 10 to the fourth.

And so we can compare
that to the value of Kc

we started out our problem with,

which is, let's see.

It is 6.25 times 10 to the fourth.

And this is actually pretty good.

We did make an approximation,
so our answer isn't

exactly right.

But it's pretty close.

And so if we wanted to get it even closer,

there are other methods we could use.

But for most purposes, this is actually,

this tells us that our
approximation was good.

So we can see that when
Kc is really large,

what we need to do is assume that we have

100% product when we're
setting up our ICE table,

and that'll help us safely
assume that x is small.