Question

Consider the reaction: $$ 2 \mathrm{~N}_{2} \mathrm{O}(g) \longrightarrow 2 \mathrm{~N}_{2}(g)+\mathrm{O}_{2}(g) $$ a. Express the rate of the reaction in terms of the change in concentration of each of the reactants and products. b. In the first \(15.0 \mathrm{~s}\) of the reaction, \(0.015 \mathrm{~mol}\) of \(\mathrm{O}_{2}\) is produced in a reaction vessel with a volume of \(0.500 \mathrm{~L}\). What is the average rate of the reaction during this time interval? c. Predict the rate of change in the concentration of \(\mathrm{N}_{2} \mathrm{O}\) during this time interval. In other words, what is \(\Delta\left[\mathrm{N}_{2} \mathrm{O}\right] / \Delta t ?\)

Short answer

a. Reaction rate = \( -\frac{1}{2} \frac{\Delta [\mathrm{N}_2\mathrm{O}]}{\Delta t} = \frac{1}{2} \frac{\Delta [\mathrm{N}_2]}{\Delta t} = \frac{\Delta [\mathrm{O}_2]}{\Delta t} \); b. Average rate = 0.002 M/s; c. \( \frac{\Delta [\mathrm{N}_2\mathrm{O}]}{\Delta t} = -0.004 \ M/s \).

Step-by-step solution

Express the rate of reaction

For a given balanced chemical reaction, the rate of the reaction can be expressed in terms of the change of concentration of the reactants or products over time. The general form for the rate of reaction is rate = \( -\frac{1}{a} \frac{\Delta[\text{reactant}]}{\Delta t} = \frac{1}{b} \frac{\Delta[\text{product}]}{\Delta t} \) where a and b are the stoichiometric coefficients of the reactant and product respectively in the balanced equation.

Define the rate of reaction in terms of each species

Based on the balanced equation, the rate of the reaction in terms of the reactant N2O can be expressed as: rate = \( -\frac{1}{2} \frac{\Delta [\mathrm{N}_2\mathrm{O}]}{\Delta t} \). The negative sign indicates the concentration of the reactant decreases over time. Similarly, the rates for the products N2 and O2 are: rate = \( \frac{1}{2} \frac{\Delta [\mathrm{N}_2]}{\Delta t} \) and rate = \( \frac{\Delta [\mathrm{O}_2]}{\Delta t} \) since their stoichiometric coefficients are 2 and 1 respectively.

Calculate the average rate of reaction

Using the data for O2, the average rate of the reaction over the given time interval can be calculated with the formula: average rate = \( \frac{\Delta [\mathrm{O}_2]}{\Delta t} \). Given that \( \Delta [\mathrm{O}_2] = 0.015 \ mol \) and \( \Delta t = 15.0 \ s \) and the volume of the reaction vessel is 0.500 L, we first convert the amount of O2 produced to molarity (moles per liter) and then plug it into the average rate formula.

Convert moles of O2 to molarity

The molarity change of O2 is \( \frac{0.015 \ mol}{0.500 \ L} = 0.030 \ M \).

Determine the average rate of reaction

Now plug the molarity change and the time change into the average rate formula: average rate = \( \frac{0.030 \ M}{15.0 \ s} = 0.002 \ M/s \) for O2 production.

Predict the rate of change of N2O

Since for every mole of O2 produced, two moles of N2O are consumed (based on the balanced equation), the change in concentration of N2O is twice that of O2, but in the opposite direction. Therefore, the rate of N2O consumption is: \( \frac{\Delta [\mathrm{N}_2\mathrm{O}]}{\Delta t} = -2 \times \frac{\Delta [\mathrm{O}_2]}{\Delta t} = -2 \times 0.002 \ M/s = -0.004 \ M/s \). The negative sign indicates the concentration of reactant (N2O) is decreasing.

Key concepts

Reaction Rate Expression

Understanding the rate at which chemical reactions occur is fundamental in the study of kinetics. The reaction rate expression provides a quantitative measure of how fast a reaction proceeds. Based on the balanced chemical equation, reaction rates can be expressed in terms of the rate of change in concentration of either the reactants or the products over time.

For instance, consider the reaction: \[ 2 \text{N}_2\text{O}(g) \rightarrow 2 \text{N}_2(g) + \text{O}_2(g) \]
Here, the rate of disappearance of the reactant, N₂O, can be related to the rate of appearance of the products, N₂ and O₂. It's important to note that the rate of reaction is generally a negative value for reactants (since they are being consumed) and positive for products (as they are being formed). Putting this idea into a mathematical form gives us the rate expression for a general reaction involving a reactant A and a product B:\[ \text{rate} = -\frac{1}{a} \frac{\text{d[A]}}{\text{d}t} = \frac{1}{b} \frac{\text{d[B]}}{\text{d}t} \]Where \text{d[A]} and \text{d[B]} represent the change in concentration of A and B, respectively, over the time interval \text{d}t, and a and b are their respective stoichiometric coefficients in the balanced equation.

When we talk about the rate of reaction, we must always be clear about in terms of which reactant or product it is being expressed because the numerical value will differ depending on the substance chosen and its stoichiometric coefficient in the reaction.

Stoichiometric Coefficients

The stoichiometric coefficients in a balanced chemical equation play a key role in determining the relationship between the rates at which reactants are consumed and products are formed. These coefficients represent the proportional amounts of each substance involved in the reaction.

In the reaction mentioned earlier, the stoichiometric coefficient for N₂O is 2, which means for every two moles of N₂O that react, two moles of N₂ and one mole of O₂ are produced. This direct correlation allows chemists to predict how changes in one substance will affect the others. Mathematically, the stoichiometric coefficients are used to link the different rates of change for reactants and products in the reaction rate expression:

Linking Rates

With the coefficients, we can generalize the relationship as follows:\[ \text{rate}(\text{N}_2\text{O}) = -\frac{1}{2} \text{rate}(\text{O}_2) \]The negative sign indicates the consumption of the reactant. While dealing with stoichiometry, it is crucial to balance the chemical equation correctly as this directly affects the rate calculations. Accurate stoichiometric coefficients ensure the correct interpretation of reaction kinetics.

Average Reaction Rate Calculation

Calculating the average reaction rate over a time interval gives insight into the overall speed of the chemical process during that period. To determine the average rate, we use changes in concentration over a specific time. This is different from the instantaneous rate, which would look at a particular moment in time.

From the exercise, to find the average rate of O₂ production, we use the formula:\[ \text{average rate} = \frac{\text{d}[O_2]}{\text{d}t} \]where \text{d}[O₂] is the change in concentration of O₂ and \text{d}t is the time interval.

Putting it Into Practice

Given that 0.015 mol of O₂ is formed in a 0.500 L reaction vessel over 15 seconds, the average rate is calculated in two steps: firstly, we find the molarity change, which is moles of O₂ per liter of solution (\[ 0.030 \text{ M} \]), and secondly, we divide this by the time interval (\[ 0.002 \text{ M/s} \]).

This value represents the average speed at which O₂ appears in the reaction mixture. Similarly, using the stoichiometric relationships, we can calculate the rate at which N₂O is consumed, ensuring to use the appropriate negative sign to indicate a decrease in concentration. It's imperative to always associate the correct units with the calculated values to convey the rates accurately.