Solving Equilibrium Problem Using Quadratic Formula

[1uigi]

I'm learning equilibirum in class, but I am stuck on this question. DOes anyone how how to do this problem? I use the quadratic formula but end end with answers A and B, but the correct answer is A. Can someone show me step by step how to do this problem? thanks

9. At a certain temperature, T, K for the reaction below is 7.5 liters/mole.
2NO2 <===> N2O4
If 2.0 moles of NO2 are placed in a 2.0-liter container and permitted to react at this temperature, what will be the concentration of N2O4 at equilibrium?

a) 0.39 moles/liter
b) 0.65 moles/liter
c) 0.82 moles/liter
d) 7.5 moles/liter
e) none of these

 

[ksinclair13]

You did the problem correctly. You end up having to solve the quadratic equation 30.x² - 31x + 7.5 = 0. You get the answers of x = .39 and x = .65.

Now, think about the problem. You are starting with 1.0 M NO₂, correct? You should notice that the equilibrium concentration of NO₂ will be 1.0 - 2x, and the equilibrium concentration of N₂O₄ will be x.

If x was .65, then 1.0 - 2x would yield a negative concentration, which is physically impossible. Therefore, the only correct answer can be [N₂O₄] = .39 M because 1.0 - 2(.39) gives you a positive concentration.

 

[Blueshift5]

To solve this equilibrium problem, we can use the following steps:

Step 1: Write the equilibrium expression for the given reaction:
K = [N2O4] / [NO2]^2

Step 2: Plug in the given values for the equilibrium constant and initial concentration of NO2:
7.5 = [N2O4] / (2.0)^2

Step 3: Rearrange the equation to solve for [N2O4]:
[N2O4] = 7.5 * (2.0)^2

Step 4: Solve for [N2O4]:
[N2O4] = 30 moles/liter

Step 5: Check the answer options and see that none of them match the calculated value of [N2O4]. This is because the given values for the equilibrium constant and initial concentration of NO2 are not in the correct units. We need to convert the units to match the answer options, which are in moles/liter.

Step 6: Convert the given equilibrium constant from liters/mole to moles/liter:
K = 7.5 liters/mole * (1 mole/1000 liters) = 0.0075 moles/liter

Step 7: Convert the given initial concentration of NO2 from liters to moles:
[NO2] = 2.0 liters * (1 mole/22.4 liters) = 0.0893 moles

Step 8: Plug in the converted values into the equilibrium expression:
0.0075 = [N2O4] / (0.0893)^2

Step 9: Solve for [N2O4]:
[N2O4] = 0.0075 * (0.0893)^2 = 0.000059 moles/liter

Step 10: Check the answer options and see that the closest one to our calculated value of [N2O4] is option A, 0.39 moles/liter.

Therefore, the correct answer is A) 0.39 moles/liter.