A 2.20 mol sample of NO 2 (g) is added to a 3.50 L vessel and heated to 500 K. N 2 O 4 (g) - ⇀ ↽ - 2 NO 2 (g) K c = 0.513 at 500 K Calculate the concentrations of NO 2 (g) and N 2 O 4 (g) at equilibrium.

[NO₂] = 0.434 M

[N₂O₄] = 0.0971 M

Explanation:

The equilibrum is: N₂O₄ (g) ⇆ 2NO₂ (g)

1 moles of nitrogen (IV) oxide is in equilibrium with 2 moles of nitrogen dioxide.

Initally we only have 2.20 moles of NO₂. So let's write the equilibrium again:

2NO₂ (g) ⇆ N₂O₄ (g)

Initially 2.20 mol -

React x x/2

X amount has reacted, and the half has been formed, according to stoichiometry.

Eq (2.20-x) / 3.50L (x/2) / 3.50L

We divide by the volume because we need molar concentrations. Let's make the Kc's expression:

Kc = [N₂O₄] / [NO₂]²

0.513 = ((x/2) / 3.50L) / [ (2.20-x) / 3.50L]

0.513 = ((x/2) / 3.50L) / [ (2.20-x) ² / 3.50L²]

0.513 = ((x/2) / 3.50L) / [2.20-x) ² / 3.50L²]

0.513 = ((x/2) / 3.50L) / (4.84 - 4.40x + x²) / 12.25)

0.513 / 12.25 (4.84 - 4.40x + x²) = x/2 / 3.50

0.203 - 0.184x + 0.0419x² = x/2 / 3.50

3.50 (0.203 - 0.184x + 0.0419x²) = x/2

7 (0.203 - 0.184x + 0.0419x²) - x = 0

1.421 - 2.288x + 0.2933x² = 0 → Quadratic formula

a = 0.2933; b = - 2.288; c = 1.421

(-b + - √ (b²-4ac)) / (2a)

x₁ = 7.12

x₂ = 0.68 → We consider this value, so we can have a (+) concentration.

Concentrations in the equilibrium are:

[NO₂] = (2.20-0.68) / 3.50 = 0.434 M

[N₂O₄] = (0.68/2) / 3.50 = 0.0971 M