Question

Manganate ions, \(\mathrm{MnO}_{4}^{2-}\), react at \(2.0 \mathrm{~mol} \cdot \mathrm{L}^{-1} \cdot \mathrm{min}^{-1}\) in acidic solution to form permanganate ions and manganese(IV) oxide: \(3 \mathrm{MnO}_{4}{ }^{2-}(\mathrm{aq})+4 \mathrm{H}^{+}(\mathrm{aq}) \rightarrow 2 \mathrm{MnO}_{4}{ }^{-}(\mathrm{aq})+\mathrm{MnO}_{2}(\mathrm{~s})+\) \(2 \mathrm{H}_{2} \mathrm{O}(\mathrm{l})\). (a) What is the rate of formation of permanganate ions? (b) What is the rate of reaction of \(\mathrm{H}^{+}(\mathrm{aq})\) ? (c) What is the unique rate of the reaction?

Short answer

The rate of formation of permanganate ions is approximately 1.33 mol L^-1 min^-1, the rate of reaction of H+ is approximately 2.67 mol L^-1 min^-1, and the unique rate of the reaction is approximately 0.67 mol L^-1 min^-1.

Step-by-step solution

Determine the Rate of Formation of Permanganate Ions

The stoichiometry of the reaction indicates that 3 moles of manganate ions form 2 moles of permanganate ions. Therefore, the rate of formation of permanganate ions is 2/3 times the rate of disappearance of manganate ions. We can calculate this using the rate of reaction of manganate ions which is given as 2.0 mol L^-1 min^-1.

Calculate the Rate of Reaction of H+

Using the stoichiometric ratios from the balanced chemical equation, we see that 4 moles of H+ react for every 3 moles of manganate ions that react. The rate of reaction of H+ can be found by multiplying the rate of disappearance of manganate ions by the ratio of 4/3.

Determine the Unique Rate of the Reaction

The unique rate of reaction is the rate of disappearance of reactants or the rate of formation of products divided by their stoichiometric coefficients. Since the stoichiometric coefficient for manganate ions is 3, the unique rate of the reaction can be found by dividing the rate of disappearance of manganate ions by 3.

Key concepts

Reaction Stoichiometry

Understanding reaction stoichiometry is essential when examining chemical reactions. It involves the quantitative relationship between reactants and products in a chemical equation. In the case of the reaction between manganate ions and hydrogen ions, the equation provided shows that 3 moles of manganate ions react with 4 moles of hydrogen ions to produce 2 moles of permanganate ions, some manganese(IV) oxide, and water.

Stoichiometry allows us to calculate the rates at which products are formed based on how reactants are consumed. By looking at the balanced chemical equation, one can determine the molar ratio of each reactant to each product. Here, for every 3 moles of manganate ions that react, 2 moles of permanganate ions are formed. This ratio helps us calculate the rate of formation of permanganate ions by applying the given rate of disappearance of manganate ions.

Unique Rate of Reaction

The unique rate of a reaction is a standardized measure of how fast the reaction occurs. It is independent of which reactant or product you are looking at and is calculated by taking the rate of change in concentration of one component and dividing it by its stoichiometric coefficient in the balanced equation.

For instance, if the rate of disappearance of manganate ions is given, to find the unique rate, we divide this rate by the stoichiometric coefficient of manganate ions, which is 3 in our reaction. This gives a single rate that characterizes the reaction, making it simpler to compare with other reactions or to express the rate of consumption or formation of different components of the reaction.

Rate of Formation

The rate of formation pertains to how quickly products are created in a chemical reaction. This can be calculated using reaction stoichiometry and the rates of disappearance of the reactants. In our example, once the rate at which manganate ions disappear is known, we can predict the rate at which permanganate ions are formed.

To calculate the rate of formation of permanganate ions from the given data, we utilize the stoichiometric ratio. This means mathematically expressing the rate as a fraction, 2/3 in this case, of the rate at which manganate ions are consumed. It is important to remember that the rates of formation and disappearance are intrinsically linked through the balanced chemical equation. So, by understanding one rate and the stoichiometry of the reaction, the other can be deduced.