Problem 95
Question
The reaction \(2 \mathrm{NO}_{2} \longrightarrow 2 \mathrm{NO}+\mathrm{O}_{2}\) has the rate constant \(k=0.63 \mathrm{M}^{-1} \mathrm{~s}^{-1}\). (a) Based on the units for \(k\), is the reaction first or second order in \(\mathrm{NO}_{2}\) ? (b) If the initial concentration of \(\mathrm{NO}_{2}\) is \(0.100 \mathrm{M}\), how would you determine how long it would take for the concentration to decrease to \(0.025 \mathrm{M}\) ?
Step-by-Step Solution
Verified Answer
(a) The reaction is second order in NO₂, as the units of the rate constant k are M⁻¹ s⁻¹.
(b) To determine the time taken for the concentration to decrease, use the rate equation for a second-order reaction: \(\frac{1}{[A]_t} - \frac{1}{[A]_0} = kt\). Plug in the given concentrations and solve for t: \(t = \frac{3}{0.63} = 4.76 s\). So, it takes 4.76 seconds for the concentration of NO₂ to decrease from 0.100 M to 0.025 M.
1Step 1: Determine the order of the reaction
Given that the rate constant k = 0.63 M⁻¹ s⁻¹. The units of the rate constant can help us determine the order of the reaction with respect to NO₂. For a reaction of the form:
\(aA \longrightarrow products\)
The rate of the reaction is given by:
\(rate = k[A]^n\)
where n is the order of the reaction and [A] is the concentration of the reactant A.
For a first-order reaction (n=1), the units of the rate constant k would be s⁻¹. For a second-order reaction (n=2), the units of the rate constant k would be M⁻¹ s⁻¹. The units of the given rate constant are M⁻¹ s⁻¹, so the reaction is second order in NO₂.
2Step 2: Write the rate equation for the reaction
The rate equation for a second-order reaction is as follows:
\(\frac{1}{[A]_t} - \frac{1}{[A]_0} = kt\)
where [A]₀ is the initial concentration of the reactant, [A]ₜ is the concentration at time t, and k is the rate constant.
3Step 3: Calculate the time taken for the concentration to decrease
We are given that the initial concentration of NO₂ [A]₀ is 0.100 M, and we need to determine the time required for the concentration to decrease to 0.025 M, which is [A]ₜ.
Using the rate equation from Step 2, we can write:
\(\frac{1}{0.025} - \frac{1}{0.100} = 0.63 \times t\)
To find t, first solve the equation within the brackets:
\(\frac{1}{0.025} - \frac{1}{0.100} = \frac{4 - 1}{1} = 3\)
Next, divide both sides by the rate constant 0.63:
\(t = \frac{3}{0.63}\)
4Step 4: Calculate the time value
Now we can calculate the value of t:
\(t = \frac{3}{0.63} = 4.76 s\)
Thus, it takes 4.76 seconds for the concentration of NO₂ to decrease from 0.100 M to 0.025 M in the given reaction.
Key Concepts
Understanding Second-Order ReactionsThe Role of the Rate ConstantConcentration Change Over Time
Understanding Second-Order Reactions
In chemical kinetics, the term "second-order reaction" refers to a reaction where the rate depends on the concentration of one reactant raised to the second power, or on the concentrations of two different reactants each raised to the first power. This is written as:
In the original exercise, we see that the rate constant k has units of M⁻¹ s⁻¹, which is characteristic of a second-order reaction.
This is because, for second-order reactions, the rate constant must have these units to maintain dimensional consistency in the rate equation: Rate = k[A]² (for one reactant).
Therefore, understanding the unit of the rate constant is pivotal in determining the reaction order.
- Rate = k[A]^2 for a single reactant
- Rate = k[A][B] for two reactants
In the original exercise, we see that the rate constant k has units of M⁻¹ s⁻¹, which is characteristic of a second-order reaction.
This is because, for second-order reactions, the rate constant must have these units to maintain dimensional consistency in the rate equation: Rate = k[A]² (for one reactant).
Therefore, understanding the unit of the rate constant is pivotal in determining the reaction order.
The Role of the Rate Constant
The rate constant, denoted as k, is an essential part of the rate equation for any reaction. It is a proportionality factor that links the reaction rate to the concentrations of the reactants.
The value of k is influenced by several factors, including temperature and the presence of a catalyst.
In the second-order reaction from our example, the rate constant is 0.63 M⁻¹ s⁻¹.
These units tell us that the reaction speed is dependent on the square of the concentration of NO₂. This means, as NO₂ concentration decreases, the reaction rate also decreases at a quadratic rate.
The calculation of reaction time utilizing this rate constant involves substituting values into the integrated rate law formula for second-order reactions. This helps to determine how the concentration of a reactant changes over time, which is crucial in kinetic studies.
The value of k is influenced by several factors, including temperature and the presence of a catalyst.
In the second-order reaction from our example, the rate constant is 0.63 M⁻¹ s⁻¹.
These units tell us that the reaction speed is dependent on the square of the concentration of NO₂. This means, as NO₂ concentration decreases, the reaction rate also decreases at a quadratic rate.
The calculation of reaction time utilizing this rate constant involves substituting values into the integrated rate law formula for second-order reactions. This helps to determine how the concentration of a reactant changes over time, which is crucial in kinetic studies.
Concentration Change Over Time
The change in concentration of a reactant over time is a central part of chemical kinetics.
For second-order reactions, the relationship between the concentration at given times and the rate constant is described by the formula:
The example in our exercise demonstrated this by using initial and final concentrations of NO₂, with a calculated time of 4.76 seconds for the concentration to decrease from 0.100 M to 0.025 M.
By substituting into this formula, we can easily solve for time t, offering a precise understanding of concentration dynamics over the course of the reaction.
Overall, mastering these calculations is fundamental in predicting reaction behavior, which is vital in various scientific and industrial applications.
For second-order reactions, the relationship between the concentration at given times and the rate constant is described by the formula:
- \( \frac{1}{[A]_t} = kt + \frac{1}{[A]_0} \)
The example in our exercise demonstrated this by using initial and final concentrations of NO₂, with a calculated time of 4.76 seconds for the concentration to decrease from 0.100 M to 0.025 M.
By substituting into this formula, we can easily solve for time t, offering a precise understanding of concentration dynamics over the course of the reaction.
Overall, mastering these calculations is fundamental in predicting reaction behavior, which is vital in various scientific and industrial applications.
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