calculating the equilibrium concentrations of products and reactants

[Katherine Wu 1H]

A vial of SO2 (0.522 mol/L) and O2 (0.633 mol/L) react and reach equilibrium. Calculate the equilibrium concentrations of the products and reactants given that Kc= 5.66 x 10^-10 for this reaction: 2SO2 (g) + O2 (g) ⇌ 2SO3 (g)

concentration SO2 O2 SO3
initial 0.522 0.633 0
change -2X -X +2X
equilibrium 0.522-2X 0.633-X c+2X

I'm not really sure how to proceed after setting up Kc= ([SO3]^2)/([SO2]^2[O2]). Can someone show me how to solve the problem?

[Goyama_2A]

First off, since the initial concentration of SO3 is 0 and you are adding 2X, the equilibrium concentration should only be 2X rather than c+2X. You would then proceed using your equilibrium equations you have set up (0.522-2X, 0.633-X, and 2X) in filling out your equation for K. With these equilibrium, your K equation should come out to: K= (2X)^2/((0.522-2X)^2 x (0.633-X)). you would then solve that out to get 4X^2 in the numerator and some cubic function in the denominator. However, in the modules, Dr.Lavelle explained that cubics are very hard to solve, so we should turn to the K. If the K is very small (<10^-4) we can regard the Xs as not having extremely significant mathematical effects on K and then plug in 0 for the Xs in the cubic function that is currently in your denominator. From there, you can set your K function equal to your given K value, solve for X, and then solve out your equilibrium values.