# Does stoichiometry matter in extent of reaction calculation?

I was recently introduced to the extent of reaction and learned that it can be calculated using this equation

$$\xi = \frac{n_{A,out}-n_{A,in}}{\nu_A}.$$

If I'm using this formula to calculate the extent of reaction, does it then depend on how I write the stoichometry? For example, reaction $$\ce{2CO + O2 -> 2CO2}$$ could also be written as $$\ce{CO + 1/2 O2 -> CO2}$$ which seems to result in a different value for the extent of reaction according to the formula above.

[OP] If i'm using this formula to calculate the extent of reaction is it then dependent on how i write the stoichometry?

No. The formula is general and, for the $$j$$th species in a chemical reaction, the amount of $$j$$ is $$n_j = n_{j,0} + \nu_j \xi \tag{1}$$ where $$n_{j,0}$$ is the initial amount of species $$j$$.

[OP] Which to me this seems to result in a different value for the extent of reaction according to this formula.

Yes. But the results will be the same. I will illustrate with the case you have given, considering that: (1) there is no carbon dioxide initially, and (2) carbon monoxide is the limiting reactant.

\begin{align} \ce{2CO(g) + O2(g) -> 2CO2(g)} \quad & \quad \ce{CO(g) + 1/2O2(g) -> CO2(g)} \tag{R1/R2} \end{align} By Eq. (1) and making a difference between $$\xi_1$$ for $$\text{R1}$$ and $$\xi_2$$ for $$\text{R2}$$ \begin{align} n_\ce{CO} = n_\ce{CO,0} - 2\xi_1 \quad & \quad n_\ce{CO} = n_\ce{CO,0} - \xi_2 \tag{2,3} \\ n_\ce{O2} = n_\ce{O2,0} - \xi_1 \quad & \quad n_\ce{O2} = n_\ce{O2,0} - \frac{1}{2}\xi_2 \tag{4,5} \\ n_\ce{CO2} = 2\xi_1 \quad & \quad n_\ce{CO2} = \xi_2 \tag{6,7} \\ \end{align} Consider as an example where the limiting reactant is depleted, i.e., $$n_\ce{CO} = 0$$. By Eqs. (2) and (3) we get that the final extents of reaction are $$\xi_{1,\text{max}} = \frac{n_\ce{CO,0}}{2} \quad \xi_{2,\text{max}} = n_\ce{CO,0} \tag{8,9}$$ But when you go to evaluate the final amounts in Eqs. (4-7) we get the same result \begin{align} n_\ce{O2} = n_\ce{O2,0} - \frac{n_\ce{CO,0}}{2} \quad & \quad n_\ce{O2} = n_\ce{O2,0} - \frac{n_\ce{CO,0}}{2} \tag{10,11} \\ n_\ce{CO2} = n_\ce{CO,0} \quad & \quad n_\ce{CO2} = n_\ce{CO,0} \tag{12,13} \\ \end{align} Thus, as we can see we are at the same point, however both extents have different bounds $$0 \leq \xi_1 \leq \frac{n_\ce{CO,0}}{2} \quad 0 \leq \xi_2 \leq n_\ce{CO,0} \tag{14,15}$$ if the reactions are irreversible. In my humble opinion, as a good practice, always have a stoichiometric number equal to $$1$$ for the limiting reactant. It keeps tracks of mistakes.

Does stoichiometry matter in extent of reaction calculation?

Yes, it does. There are several quantities that depend on the choice of a common multiplicative factor for the chemical reaction equation. The reaction quotient $$Q$$, the equilibrium constants $$K$$, and the enthalpy, entropy and Gibbs energy of reaction are other examples.

On the other hand, the measurable quantities do not depend on it. No matter which multiplicative factor you choose, e.g. the equilibrium concentrations or how much of one species is needed to react with a given amount of another species does not change.

If you look at a specific calculation done based on two net equations that differ by a common factor, you will see that the common factor cancels out for measurable quantities, even if the extent of reaction and many other quantities will have different values for the two distinct net equations.