This is a problem that explores the concept of reactions that are elementary, versus elementary
as written, and we'll talk a little bit about non-elementary reactions. In this example
we're supposed to take this reaction, which is elementary as written, and write the rate
law for this reaction in terms of the forward rate constant and some form of an equilibrium
constant. Alright, if it's elementary as written, we can write the rate of the forward reaction
as being equal to a rate constant, k, times the reactant species that contribute to the
forward reaction, raised to their stoichiometric coefficients. So we get a concentration of
A to the first power, because we have stoichiometric coefficient of 1, an implied 1 in front of
the A, and we have a stoichiometric coefficient of 1/2 in front of B, and so we raise the
concentration of B to the 1/2 power. Alright, and this is just for the rate of the forward
reaction. We'll talk about the rate of the reverse reaction momentarily. One thing worth
noting here is that it is not correct to say that the rate of the forward reaction can
be determined from a different form of the stoichiometry, so one possibility is that
we could write this reaction as 2A plus B goes to 2C plus 4D, in order to remove the
fraction in front of B. And if we wrote it this way and assumed it was elementary as
written, then we would write that the rate of the forward reaction was the rate of A
squared times the concentration of B, because of the stoichiometry that's circled there,
but this is not the correct expression if the reaction is elementary as written in the
original expression that we have up at the top, and we can see this because for example
if we doubled the concentration of A, the top expression would double the rate, the
bottom expression would quadruple the rate, and so we don't have the same dependence on
the concentrations of A and B. So when we say a reaction is elementary as written, we
mean exactly as written, and we can't change the stoichiometry, for example by multiplying
through by a constant factor of 2, even though we can still conserve the mass balance and
the mole balance by doing that kind of multiplication. So we're always going to write an elementary
as written chemical expression based on exactly what is written there. It's also worth making
a distinction between a true elementary reaction and an elementary as written reaction. In
a true elementary reaction, the reaction expression that we write is actually describing an elementary
collision process. So let's take a different kind of reaction, let's say we have a reaction
of E and F, alright and they are colliding to produce some kind of joined molecule EF.
Now it's possible that for a simple reaction like this, that the rate of reaction is proportional
to the frequency with which molecules of E collide with molecules of F, and then we have
a true elementary reaction, that the expression that we write is describing a single collision
process that leads to reaction. In that case, we have a first order reaction of E and F.
Alright, that's not possible for the reaction of A and B that we have at the top, which
must be non elementary because we can't have a collision involving A and half a molecule
of B. Alright, so in an elementary reaction, the rate is proportional to the rate of collisions,
which is proportional to the concentrations of the species to their stoichiometric powers.
So in that case, there's a fundamental basis for the reaction rate expression that we write
using the stoichiometric coefficients to raise the concentrations to the appropriate power,
but sometimes the reaction is elementary as written, even if it's not truly elementary.
So it just happens to follow roughly elementary kinetics, and we'll see later in the course
why sometimes this just happens to be the case. Alright, so we've described the rate
expression for the forward reaction. now we need to describe it for the reverse reaction
as well, C plus 2D going back to A plus 1/2 B. And so I'll write that as an r_r, for reverse.
That's equal to a k reverse, concentration of C to the first power, concentration of
D to the second power based on the stoichiometry as written in that expression, with the implied
1 in front of the C. That allows us to write a rate for the net reaction as being equal
to the rate for the forward reaction minus the rate of the reverse reaction. that will
be the net reaction from reactants to products, and we just take our two expressions and sub
them in as is shown here. And so we could potentially stop here, but we're told in the
problem statement that we want to express this in terms of the forward rate constant
k_f, and then some equilibrium constant, and not this reverse rate constant. So let's take
a look at what we need to sub in for this reverse rate constant. Now, what's useful
to do whether you have an elementary rate expression or not is to look at an equilibrium
situation, if you want to bring in an equilibrium constant, and at equilibrium the net rate
of reaction is equal to 0 because we'll have a steady concentration of the reactants and
products. We won't have any net reaction occurring at equilibrium. Alright, and so therefore
if that net rate of reaction is equal to 0, the rate of the forward reaction is equal
to the rate of the reverse reaction, and we can simplify that expression to solve for
k_f divided by k_r, and I'll go ahead and write out that solution far to the right,
and I'll use a superscript "e" to denote that we have the concentrations at their equilibrium
condition. Okay, and this kind of ratio that we have here, that what we have on the left
is only a function of temperature, those are the rate constants, which as we'll see tend
to have an Arrhenius dependence on temperature, but only to depend on temperature. That ratio
is related to the concentrations of the products and reactants in a way that we expect based
on an equilibrium constant, and that's the concentration based equilibrium constant,
the products raised to their stoichiometric coefficients divided by the reactants raised
to their stoichiometric coefficients. That equilibrium constant is only a function of
temperature. And now we have this value, that k_c, is equal to k_f over k_r, so we can substitute
in for k_r in terms of k_c and k_f. And we'll just collect terms and write this out, and
we see that we get a k_c in the denominator of our expression involving the reactants,
and this is our final expression. One thing that we'll note here is that this expression
has the value that the net rate of reaction will go to zero as you approach equilibrium,
and so that's something that you actually expect to occur, and really should be the
case whether or not you have elementary or non elementary kinetics. You have these k's
that are a function of T, with concentrations and if your expression is thermodynamically
consistent, then the reaction rate should go to zero as you approach equilibrium. So
it's useful to do the kind of analysis that we showed here, to incorporate an equilibrium
constant, because if you don't, you will often find that you come up with an expression that
does not meet thermodynamic limitations. In other words, the rate does not go to zero
at the equilibrium conditions.