Inert Effect on Chemical Equilibrium - Chemical Equilibrium Video Lecture | Physical Chemistry

We're going to look at the effect of adding
an inert on chemical equilibrium. The idea
is we have a system at constant temperature,
gas phase reaction, one mole reacting to two
moles, and then we add an inert at constant
pressure, or we add it at constant volume,
and we want to know what happens in each case.
So let's look first at the constant pressure
case. So we could look at this from the idea
of Le Chatelier's principle, that adding inert
is going to, at constant pressure, it's going
to lower partial pressures, the pressure of
A and the pressure of B because the total
pressure is constant, the inert has some pressure,
we've lowered those pressures, the system
is going to want to respond to make more moles,
and this is Le Chatelier's principle, more
moles is going to try and increase the pressure,
so we've lowered the pressure, this system
is going to go to the right to increase the
pressure, so that's looking at it from Le
Chatelier's principle, let's look at the actual
calculation to convince our self that is what's
happening. So the equilibrium constant is
the activity of B, we'll use the hat to indicate
it's in a mixture, squared, because of the
stoichiometric coefficient here, the activity
of A, first power, and I can write the activity
in terms of fugacity of B, fugacity of B at
standard conditions, that quantity squared,
fugacity of A in the mixture, fugacity of
pure A standard conditions. We assume ideal
gases, the ideas aren't going to change if
they're real but it's certainly easier to
visualize, then the fugacity of B is just
the partial pressure of B divided by the standard
state, pure component, which is one bar, that
quantity squared. and this is the pressure
of A over one bar also, which means this would
be the pressure of B squared, over the pressure
of A as long as we keep in mind we have to
write the pressures in units of bar. And so
let's continue in order to make it easier
to visualize this, let's write this in terms
of mole fractions, so the mole fraction of
B times the total pressure, that's the partial
pressure of B, mole fraction of A times the
total pressure, so we have mole fraction of
B squared, pressure and the mole fraction
of A. Well this is number of moles of B over
the total number of moles, and then likewise
number of moles of A over the total number
of moles. And so we end up with is number
of moles of B squared, over the total number
of moles, the number of moles of A and then
pressure, so now we can look at this and say
what happens at constant pressure if I change
the number of moles. So if I change the number
of moles, I haven't changed the equilibrium
constant, that's a constant because temperature
is constant, I've kept the pressure constant,
if I increase the total number of moles by
adding an inert then I must increase B, so
B must increase when N total increases, and
so again this tells us the reaction must go
to the right when I add an inert at constant
pressure, the same result that we predict
from Le Chatelier's principle. Okay now let's
look at the case for constant volume. So our
equilibrium expression is the same, if we
do this at constant volume but let's look
at what's going to happen, so I'm going to
write it this way, make it easier to visualize,
pressure over the total number of moles, if
I add an inert I'm going to increase the total
number of moles but I'm also going to increase
the pressure such that the pressure over total
number of moles is just R T over V from the
ideal gas law, R is constant, of course temperature
is constant, it's constant volume so this
term is constant, in other words we've increase
the total number of moles and we proportionally
increase the pressure so all these terms stay
constant, so there's no effect on equilibrium,
when we add an inert at constant volume, we've
kept the partial pressures of A and B the
same. Our equilibrium expression, remember,
the other way of writing our equilibrium expression
is the pressure of B squared over the pressure
of A, add an inert, constant volume, we increase
the total pressure, we don't change those
partial pressures, therefore there's no change,
the equilibrium is still satisfied. So the
situation is very different, constant volume
and constant pressure.