Question 4.17: Calculating the Extent of Reaction. (a) Determine the stoich......
Calculating the Extent of Reaction
(a) Determine the stoichiometric numbers for the following reaction, which occurs to a limited extent at 25 °C: 4 KO_{2}(s) + 2 CO_{2}(g) → 2 K_{2}CO_{3}(s) + 3 O_{2}(g). (b) In an experiment carried out at 25 °C, a reaction mixture containing 2.478 mol KO_{2} and 0.979 mol CO_{2} yields 1.128 mol O_{2}. Calculate the extent of reaction, as well as the final amounts of all reactants and products.
Analyze
(a) For a given reactant or product, the stoichiometric number (\mathcal{v}) is set equal to the stoichiometric coefficient with a + or − sign included (a + sign for a product and a − sign for a reactant). (b) The extent of reaction (\xi) is defined by equation (4.10) and can be used to relate the changes in amount to each other. The relationships among the changes in amount are conveniently summarized by constructing a table with three rows immediately below the chemical equation. The first row specifies the initial amounts. The second row specifies the changes in amounts, each of which is expressed as the product of a stoichiometric number (\mathcal{v}) and the extent of reaction (\xi). The third row specifies the final amounts.
\xi = \frac{\Delta n_{k} }{\mathcal{v}_{k}} (4.10)
Solve
(a) The stoichiometric numbers are
\mathcal{v}_{KO_{2}} = −4 \mathcal{v}_{CO_{2}} = −2 \mathcal{v}_{K_{2}CO_{3}} = +2 \mathcal{v}_{O_{2}} = +3
(b) First, we construct a table with three rows below the chemical equation for the reaction. The rows specify the initial amounts, the changes in amount and the final amount, respectively.
Because the final amount of O_{2} is given as 1.128 mol, we have 3 \xi = 1.128 mol and \xi = \frac{1.128 mol}{3} = 0.376 mol
We can calculate the final amounts by using this value of \xi.
n_{KO_{2},final} = 2.478 mol − 4\xi = 2.478 mol − 4 × 0.376 mol = 0.974 mol
n_{CO_{2},final} = 0.979 mol − 2\xi = 0.979 mol − 2 × 0.376 mol = 0.227 mol
n_{K_{2}CO_{3},final} = 2\xi = 2 × 0.376 mol = 0.752 mol
n_{O_{2},final} = 3\xi = 3 × 0.376 mol = 1.128 mol
Analyze
Note that the unit of is mol. We will use the extent of reaction throughout the text. For example, in Chapters 7 and 13, we will express the energy consumed or produced by a reaction in terms of the extent of reaction. In Chapters 15 to 18, a primary goal is to determine the extent of reaction for reactions that occur to a limited extent only. In these chapters, the tabular approach illustrated above will be particularly useful.