Problem 4 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

For the reaction: \(2 \mathrm{~N}_{2} \mathrm{O}_{5}(\mathrm{~g}) \rightarrow 4 \mathrm{NO}_{2}(\mathrm{~g})\) \(+\mathrm{O}_{2}(\mathrm{~g})\), the concentration of \(\mathrm{NO}_{2}\) increases by \(2.4 \times 10^{-2} \mathrm{M}\) in \(6 \mathrm{~s}\). What will be the average rate of appearance of \(\mathrm{NO}_{2}\) and the average rate of disappearance of \(\mathrm{N}_{2} \mathrm{O}_{5} ?\) (a) \(2 \times 10^{-3} \mathrm{Ms}^{-1}, 4 \times 10^{-3} \mathrm{Ms}^{-1}\) (b) \(2 \times 10^{-3} \mathrm{Ms}^{-1}, 1 \times 10^{-3} \mathrm{Ms}^{-1}\) (c) \(2 \times 10 \mathrm{Ms}^{-1}, 2 \times 10^{-3} \mathrm{Ms}^{-1}\) (d) \(4 \times 10^{-3} \mathrm{Ms}^{-1}, 2 \times 10^{-3} \mathrm{Ms}^{-1}\)

Step-by-Step Solution

Answer

The average rate of appearance of NO2 is 4 x 10^-3 Ms^-1 and the average rate of disappearance of N2O5 is 2 x 10^-3 Ms^-1, which corresponds to choice (d).

Step 1: Identify Change in Concentration of NO2

Determine the change in concentration of NO2, which is given as an increase of 2.4 x 10^-2 M.

Step 2: Determine Stoichiometric Ratio

Identify the stoichiometric ratio between N2O5 and NO2 from the balanced chemical equation, which is 2:4 or 1:2.

Step 3: Calculate Average Rate of Disappearance of N2O5

Apply the stoichiometric ratio to the average rate of appearance of NO2 to find the average rate of disappearance of N2O5. Since the ratio is 1:2, the rate of disappearance of N2O5 is half the rate of appearance of NO2.

Step 4: Select the Correct Answer

Based on the calculations, match the calculated rates with the given options to select the correct answer.

Key Concepts

Rate of Reaction

Understanding the rate of a chemical reaction is crucial for interpreting how quickly reactants are converted into products. Simply put, the rate of reaction measures the speed at which a chemical reaction occurs. It can be defined as the change in concentration of a reactant or product per unit time. For gas-phase reactions, this is commonly expressed in molarity per second (M/s). Factors that can affect the rate include the concentration of reactants, temperature, catalysts, and the surface area of solid reactants.

Let's apply this to an example. If the concentration of a product in a reaction increases over time, the rate of the reaction can be calculated by dividing this increase by the time interval over which the change occurred. For instance, if nitrogen dioxide, NO2, increases by a certain molarity over a six-second timeframe, the rate of appearance of NO2 would be that molarity change divided by six seconds. Remember, the rate of appearance of a product and the rate of disappearance of a reactant are simply the reaction rate observed from different perspectives.

Stoichiometry

Stoichiometry is the quantitative relationship between reactants and products in a chemical reaction, based on the balanced chemical equation. For each reaction, the coefficients in the balanced equation tell us in what ratio the molecules react and are formed. This ratio allows us to predict the amounts of products formed from given reactants and vice versa.

Using our example of the decomposition of dinitrogen pentoxide, knowing the balanced equation is 2 N2O5 (g) → 4 NO2 (g) + O2 (g), we can see that the stoichiometric ratio is 1:2 between N2O5 and NO2. This means for every molecule of N2O5 that reacts, two molecules of NO2 are produced. Hence, if you know the rate of formation of NO2, you can easily determine the rate of consumption of N2O5 by applying this stoichiometric ratio. This relationship is fundamental to solving many chemical kinetics problems.

Average Rate of Reaction

The average rate of reaction is an approximation of the rate over a particular time period. It is calculated by dividing the change in concentration by the change in time. Unlike instantaneous rate, which tells us the rate at a specific moment, the average rate gives us a broader view of the reaction's speed across an interval.

In the case of our reaction, to find the average rate of appearance of NO2, we observe that the concentration increases by 2.4 x 10^-2 M in 6 seconds. By dividing this change by the time interval, we arrive at 4 x 10^-3 M/s. For the average rate of disappearance of N2O5, we know it's related by the stoichiometric ratio to the rate of appearance of NO2, as mentioned earlier. Since the rate of disappearance is half the rate of appearance for NO2 due to the 1:2 ratio, we calculate it as 2 x 10^-3 M/s. This approach helps us to connect the dots between stoichiometry and reaction rates.