decomposition - Can we combine two reactions and then calculate the equilibrium concentrations?

At a certain temperature, $\ce{N2O5}$ dissociates as: \begin{align} \ce{N2O5 (g) &<=> N2O3 (g) + O2 (g)} & K_1 &= 4.5 \end{align} At the same time, $\ce{N2O3}$ also dissociates as: \begin{align} \ce{N2O3 (g) &<=> N2O (g) + O2 (g)} & K_2 &= \mathrm{?} \end{align}

When 4 moles of $\ce{N2O5}$ are heated in a flask of $\pu{1 L}$ volume, it has been observed that the concentration of $\ce{O2}$ is $\pu{4.5 M}$ at equilibrium. What are the equilibrium concentrations of the other products?

The combined reaction:

$$\ce{ N2O5(g) <=> N2O(g) + 2O2(g)}$$ $$K_\text{combined} = K_1 \times K_2 = 4.5 K_2$$

There are 4 moles of $\ce{N2O5}$ initially. If $x$ moles of $\ce{N2O5}$ dissociate to give $x$ moles of $\ce{N2O}$ and $2x$ moles of $\ce{O2}$, then plugging in the values in the $K_c$ expression for this reaction we get $$4.5 K_2 = \frac{4x^3}{4-x},$$ where $$x = \frac{4.5}{2 \pu{mol}}.$$

Solving for $K_2$, I get $K_2=\frac{81}{14}$, which is incorrect.

Is it allowed to first combine the reactions and then use the combined reaction?

Answer

Different from Zhe's answer, I will use the approach of a combined ICE table using the model discussed in How can you use ICE tables to solve multiple coupled equilibria?. This results in a system with two unknowns that, in my opinion, gives a bit more insight into the problem as we are solving it. $x$ are changes due to dissociation of $\ce{N2O5}$ (first reaction), and $y$ are changes due to dissociation of $\ce{N2O3}$ (second reaction).

$$ \begin{array}{|c|c|c|c|c|} \hline &[\ce{N2O5}] & [\ce{N2O3}] & [\ce{N2O}]&[\ce{O2}] \\ \hline I & 4 & 0 & 0 & 0 \\ \hline C & -x & +x-y & +y & +x+y \\ \hline E & 4-x & +x-y & +y & 4.5 \\ \hline \end{array} $$

I have two unknowns, $x$ and $y$. Unless one of them is much bigger than the other (in which case I can first neglect the smaller one and come back to it later), I have to solve a system of two equations for $x$ and $y$ simultaneously. The first equation is already buried in the ICE table (subscript $eq$ is for equilibrium state):

$$ x + y = \ce{[O2]}_{eq} = 4.5 \ \ \ \ \text{or solved for y: }\ \ \ \ y = 4.5 - x$$

The second equation is via the equilibrium constant $K_1$:

$$ K_1 = 4.5 = \frac{[\ce{N2O3}]_{eq} [\ce{O2}]_{eq}}{[\ce{N2O5}]_{eq}} = \frac{[\ce{N2O3}]_{eq} \times 4.5}{[\ce{N2O5}]_{eq}}$$

Canceling 4.5 and rearranging gives $ [\ce{N2O5}]_{eq} = [\ce{N2O3}]_{eq}$, and substituting from the ICE table gives a second equation for x and y:

$$4-x = x - y\ \ \ \ \ \ \ \ \text{(or simplified } 2x = 4 + y)$$

Substituting the first equation into the second, we get:

$$ 2x = 4 + y = 4 + 4.5 - x$$ $$ x = 8.5/3 = 17/6$$

Substituting the value for $x$ back into the first equation:

$$ y = 4.5 - x = 4.5 - 17/6 = 10/6 = 5/3$$

Now we can tabulate the equilibrium concentrations and (to compare with Zhe's answer) the equilibrium constant $K_2$:

$$ c(\ce{N2O5})_{eq} = 7/6 M = 1.167 M$$ $$ c(\ce{N2O3})_{eq} = 7/6 M = 1.167 M$$ $$ c(\ce{N2O})_{eq} = 5/3 M = 1.667 M$$ $$ K_2 = \frac{[\ce{N2O}]_{eq} [\ce{O2}]_{eq}}{[\ce{N2O3}]_{eq}} = \frac{5/3 \times 4.5}{7/6} = \frac{45}{7}$$

We can check that the equilibrium concentrations add up to the initial concentration of 4 M, and that we obtain the correct value for $K_1$ when substituting our answers into the equilibrium expression.

Is it allowed to first combine the reactions and then use the combined reaction?

Yes, but we have to consider all species simultaneously because none of them are minor species. When calculating, for example, hydronium and hydroxide concentrations in a solution of a weak acid, we can set aside the autodissociation of water because hydroxide is a minor species, and adjusting the hydroxide concentration after calculating the weak acid equilibrium is fine because it does not change the hydronium concentration much, hydroxide being the minor species. Here, we can't do that and have to take the more comprehensive approach of considering the two equilibria simultaneously.