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## Le Châtelier's principle

Current time:0:00Total duration:7:11

# Worked example: Calculating the equilibrium total pressure after a change in volume

AP.Chem:

TRA‑8 (EU)

, TRA‑8.A (LO)

, TRA‑8.A.1 (EK)

## Video transcript

- [Instructor] Phosphorus
pentachloride will decompose into phosphorus trichloride
and chlorine gas. Kp for this reaction is
equal to .500 at 500 Kelvin. Let's say that this
reaction is at equilibrium and a reaction vessel that
has a volume of 2.0 liters and the equilibrium partial pressure, of PCl5 is equal to .980 atmospheres. The equilibrium partial pressure of PCl3 is equal to .700 atmospheres. And the equilibrium partial
pressure of chlorine gas is equal to .700 atmospheres. If we add up those
three partial pressures, we get the total pressure of
the gas mixture at equilibrium, which is equal to 2.38 atmospheres. And we're gonna call
this total pressure, P1. If we decrease the volume from two liters down to one liter, and we keep the temperature
constant at 500 Kelvin, we decrease the volume by a factor of two, which means we increase the
pressure by a factor of two. So, all of the partial
pressures of our gases double, and there's a new total pressure, which is twice the
original total pressure. And this new total pressure
is equal to 4.76 atmospheres. And from now on, we'll call
this total pressure, P2. So, when the volume was two liters, the reaction started out at equilibrium and by decreasing the volume to one liter, and by doubling the pressure, we've introduced a stress to the system. And so, at this moment in time, when these are the partial pressures, the reaction is not at equilibrium. Le Chatelier's principle
says the net reaction will move in the direction
that decreases the stress. So, if the stress is an
increase in the pressure, the net reaction will
move in the direction that decreases the pressure. Looking at the balanced equation, there's one mole of gas
on the reactant side, and there are two moles of
gas on the product side. So, if the net reaction goes from the products to the reactants, if the net reaction goes to the left, the net reaction is going to the side with the smaller number of moles of gas, which would decrease the
pressure and relieve the stress. The net reaction keeps moving to the left until equilibrium is reestablished. And when equilibrium is reestablished, there'll be a new total
pressure, which we'll call P3. So, our goal is to calculate P3, so we can compare it to P1 and P2. And we're gonna do this
in a quantitative way and in a more qualitative way. Let's use an ICE table to help us figure out the
final total pressure, P3. In an ICE table, I stands for the initial
partial pressure in this case, C is the change and E is the
equilibrium partial pressure. The initial partial pressure of PCl5, after the volume was reduced to one liter, was 1.96 atmospheres. And the partial pressures of PCl3 and Cl2 were both 1.40 atmospheres. We already used Le Chatelier's principle to realize the net reaction's
going to go to the left, which means we're going to decrease in the amount of our products, and we're going to increase
the amount of the reactants. So, if we're gonna increase
the amount of PCl5, we don't know how much
we're gonna increase, we're gonna call that x. But we know it's going to increase, so we write plus x under the
change part on the ICE table. And since our mole ratio of
PCl5 to PCl3 is one to one, if we're gaining x for PCl5, we must be losing x for PCl3. And the same goes for Cl2, since there's a coefficient of one. So, we write minus x in our ICE table. Therefore, the equilibrium
partial pressure of PCl5 would be 1.96 plus x. The equilibrium partial pressure of PCl3 would be 1.40 minus x. And the equilibrium
partial pressure of Cl2 would also be 1.40 minus x. Next, we can plug in the
equilibrium partial pressures into our Kp expression. So, we can plug in 1.40 minus x for the equilibrium
partial pressure of PCl3, 1.40 minus x for the equilibrium
partial pressure of Cl2, and 1.96 plus x for the equilibrium
partial pressure of PCl5. And we can also plug in the value for the equilibrium constant, Kp. Here's what our equilibrium
constant expression looks like with everything plugged in. And next, we would need to solve for x, which would involve the use
of a quadratic equation. And when you do all that math, you find that x is equal to .330. Now that we know x is equal to .33, we can solve for the
equilibrium partial pressures. 1.96 plus .33 is equal to 2.29. Therefore, the equilibrium
partial pressure of PCl5 is 2.29 atmospheres, for PCl3, it's 1.40 minus x, 1.40 minus 0.33 is equal to 1.07. Therefore, the equilibrium
partial pressure of PCl3 is equal to 1.07 atmospheres. And it's the same math for Cl2. So, the equilibrium
partial pressure of Cl2 is also 1.07 atmospheres. So, to find the total pressure, P3, we simply need to add up the
individual partial pressures. So, 2.29 plus 1.07 plus 1.07
is equal to 4.43 atmospheres. Therefore, P3 is equal
to 4.43 atmospheres. Doing the math helps us realize that x is not a very large number. And the reason why x is
not a very large number is because the Kp value is
equal to .500 for this reaction. And when K is close to one, there's an appreciable amount of both reactants and
products at equilibrium. And we can see that with our
equilibrium partial pressures. There's an appreciable
amount of both of them. And since there has to be a decent amount of both reactants and
products at equilibrium, we're not gonna see huge change from these initial partial pressures. So, there will definitely
be a shift to the left to decrease the pressure. And when these were the partial pressures, if you remember P2 was equal to 4.76, so there's gonna be a
decrease of pressure. So, there's gonna be a decrease, so the pressure is going
to go down from 4.76, but since there's not a large change, it's not gonna be a huge change. And that's why we saw P3 only
dropped to 4.43 atmospheres. So, if you go back to
the original problem, and our goal is to figure out P3 in relation to P1 and to P2, without doing all of that math, we could think to ourselves, okay, so, we decrease the
volume by a factor of two, which doubled the total pressure. But then, the net
reaction moves to the left to decrease the pressure. Since it's not gonna
move much to the left, it's not gonna decrease
the pressure by a lot. Therefore, the final pressure P3 is going to be a little less than 4.76, but greater than 2.38 atmospheres.

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