Consider the following reaction
CO2 (g) + H2 (g) ↔ CO (g) + H2O (g)
Calculate the equilibrium constant, Kc for the above system if 0.1908 moles of
CO2, 0.0908 moles of H2, 0.0092 moles of CO and 0.0092 moles of H2O were
present in a 2.00 L reaction vessel at equilibrium.
The general expression for the equilibrium constant, Kc, is:
Kc = ([CO][H2O])/([CO2][H2])
We can use the given concentrations to calculate the equilibrium concentrations of each species. The balanced equation tells us that for every 1 mole of CO2 and H2 that react, we form 1 mole of CO and H2O. Therefore, we can use the initial amounts to calculate the amounts at equilibrium:
CO2: 0.1908 - x
H2: 0.0908 - x
CO: 0.0092 + x
H2O: 0.0092 + x
Here, x is the amount (in moles) of CO and H2O that is formed at equilibrium. We can use the equilibrium constant expression to solve for x:
Kc = ([CO][H2O])/([CO2][H2])
Kc = ([0.0092 + x][0.0092 + x])/([0.1908 - x][0.0908 - x])
We can now use algebra to solve for x:
Kc([0.1908 - x][0.0908 - x]) = (0.0092 + x)(0.0092 + x)
Expand both sides:
Kc(0.01737 - 0.2824x + x^2) = 0.00008464 + 0.0184x + x^2
Simplify:
Kc(0.01737) - Kc(0.2824)x + Kc(x^2) = 0.00008464 + 0.0184x + x^2
Group like terms:
(Kc)x^2 + (-0.2824Kc - x)x + (0.01737Kc - 0.00008464) = 0
Now we can use the quadratic formula, where a = Kc, b = -0.2824Kc - 1, and c = 0.01737Kc - 0.00008464:
x = [-b ± sqrt(b^2 - 4ac)]/2a
Plugging in the values gives:
x = [-(-0.2824Kc - 1) ± sqrt((-0.2824Kc - 1)^2 - 4(Kc)(0.01737Kc - 0.00008464))]/2(Kc)
Simplifying:
x = (0.2824Kc + 1 ± sqrt((0.2824Kc + 1)^2 - 4(Kc)(0.01737Kc - 0.00008464)))/(2Kc)
We know that x represents the amount of CO and H2O formed at equilibrium, and we also know that the total amount of CO and H2O formed is equal to the initial amount of CO2 and H2 that reacted, which is 0.1908 + 0.0908 = 0.2816 moles. Therefore:
x = 0.2816 - 0.1908 - 0.0908 = 0
This tells us that the reaction goes to completion, meaning all of the CO2 and H2 have been converted to CO and H2O. We can use this information to simplify the equilibrium constant expression:
Kc = ([CO][H2O])/([CO2][H2]) = (0.0092)(0.0092)/(0.1908)(0.0908) = 0.00506
Therefore, the equilibrium constant, Kc, is 0.00506.