let's get some more practice calculating equilibrium concentrations. so let's look
at this equilibrium we have PCl5 in equilibrium with PCl3 and Cl2 now if
we are given the fact that the initial concentration of PCl5 is equal to one
point zero zero molar and then the initial concentrations of PCl3 and Cl2
are 0 so we have none of that we're just starting with PCl5 and letting it begin
to equilibrate and we know that the constant the equilibrium constant for
this system is zero point zero two one one what will be the equilibrium
concentrations for each of these three substances so if this seems confusing go
ahead and check out my tutorial on chemical equilibria and when you're
ready give it a try.
so let's put our equilibrium up here let's also find our
Kc expression so we have concentrations of the products on the top and
concentration of that lone reactant on the bottom and none of them will be
raised to any exponent because these are in a 1 to 1 to 1 ratio all of the
coefficients are 1 so that will be our equilibrium expression and now we're
going to need to make an icebox here so let's set this up with initial change in
equilibrium that's our ICE which is why it's an icebox and we have our PCl5, PCl3
and Cl2 so let's start by placing our initial concentrations these are the
ones that were given to us so that's one point zero zero for PCl5 and then we
have nothing at the start for the two products now let's enter our change
values so this is going to require the
stoichiometric coefficients we know that this is going to move in the forward
direction because all we have is reactant and we have no products so
we're gonna use up reactants and we're gonna make products so in what ratio
well for every one PCl5 that reacts we get one PCl3 and one Cl2 so however much
reacts let's just call it X we are going to lose X from PCl5 and we will gain X
for the two products if the coefficients were not all 1 let's say one of the
products had a coefficient of 2 then we would have + 2 X instead but here they
are all 1 so we just have minus X and then + X and plus X now to get the
equilibrium values we just take the initial and the change and add them
together so 1 point 0 0 minus X and then the others are 0 plus X so simply X so
let's place these in blue up top just so we have them when we get rid of the ice
box we can remember that our equilibrium values are 1 point 0 0 minus X and then
X and X because now we are ready to plug the equilibrium values into the
equilibrium constant expression so we have our kc value that is going to be 0
point 0 2 1 1 and then the two products are X and X and then 1 point 0 0
- X on the bottom for PCl5 so now all we have to do is some algebra so we will
multiply both sides by 1 point 0 0 - x and then on the right we can take x
times X and make that x squared and then we will simply distribute point 0 to 1 1
across that difference and we get this and then we move things around to get
this quadratic equation now when we have a quadratic equation and we want to
solve for x one method is to use the quadratic formula so I have a tutorial
all about this in my mathematics series if this seems unfamiliar
but as this is basic algebra we will just summarize what we're going to do
here there is the quadratic formula, A B and C are just the coefficients from our
equation so equals one because x squared is just one
x squared B is point 0 2 1 1 because that is the coefficient on X on the X
term and C is negative 0.0211 as that is the constant so when we plug that in and
do all of the simplification we get X as being equal to zero point one three five
and negative zero point one five six remember that a quadratic equation will
always have two solutions so we need to figure out which one makes sense well as
it turns out here only the positive value makes sense because if X were
negative that would mean that pcl5 s equilibrium concentration would be
greater than how it started and the equilibrium concentrations of the
products would be negative which doesn't make any sense so we get rid of that
negative value and we keep x equals zero point one three five and now we can find
the equilibrium concentrations PCl5 was one point zero zero minus x so that
becomes zero point eight six five molar and then PCl3 equals x and so does Cl2
so those will both be zero point one three five molar and that's how we use
an ice box to get equilibrium concentrations.