Problem 102
Question
Consider a reaction of the type aA \(\longrightarrow\) products, in which the rate law is found to be rate \(=k[\mathrm{A}]^{3}\) (termolecular reactions are improbable but possible). If the first half-life of the reaction is found to be \(40 .\) s, what is the time for the second half-life? Hint: Using your calculus knowledge, derive the integrated rate law from the differential rate law for a termolecular reaction: $$\text { Rate }=\frac{-d[\mathrm{A}]}{d t}=k[\mathrm{A}]^{3}$$
Step-by-Step Solution
Verified Answer
The time for the second half-life (T₁/₂,₂) of the termolecular reaction is equal to \(T_{1/2,2} = \frac{1}{20} [\mathrm{A}_{0}]^2 - 40\) seconds.
1Step 1: Integrate the differential rate law equation
We are given the differential rate law equation:
\[\frac{-d[\mathrm{A}]}{d t}=k[\mathrm{A}]^{3}\]
Integrate both sides of the equation to obtain the integrated rate law equation:
\[ \int \frac{1}{[\mathrm{A}]^3} \, d[\mathrm{A}] = \int k \, dt\]
2Step 2: Evaluate the integrals
To find the integrated rate law, we need to evaluate the integrals:
\[- \frac{1}{2} [\mathrm{A}]^{-2} = kt + C\]
Where C is the constant of integration.
3Step 3: Solve for the constant of integration
At t=0, [A]= [A]₀, the initial concentration of A. So,
\[ - \frac{1}{2} [\mathrm{A}_{0}]^{-2} = C\]
4Step 4: Substitute the constant of integration back into the equation and find the time for the first half-life
We can now write the integrated rate law equation with C:
\[- \frac{1}{2} [\mathrm{A}]^{-2} = kt - \frac{1}{2} [\mathrm{A}_{0}]^{-2}\]
Since the time for the first half-life is given to be 40 s, at this moment [A] = 1/2 [A]₀. Replace the variables in the equation and solve for k:
\[- \frac{1}{2} (1/2 [\mathrm{A}_{0}])^{-2} = 40k - \frac{1}{2} [\mathrm{A}_{0}]^{-2}\]
5Step 5: Solve for k
After solving the above equation for k, we will get:
\[k = \frac{1}{120} [\mathrm{A}_{0}]^2\]
6Step 6: Find the time for the second half-life
Now, we will find the time for the second half-life. At this time, [A] = 1/4 [A]₀. Substitute the variables in the integrated rate law equation, and solve for the time:
\[- \frac{1}{2} (1/4 [\mathrm{A}_{0}])^{-2} = k(T_{1/2,1}+T_{1/2,2}) - \frac{1}{2} [\mathrm{A}_{0}]^{-2}\]
Where T₁/₂,₁ = 40 s (the first half-life) and T₁/₂,₂ (the second half-life) is the unknown value we want to find. Substitute the expression for k and solve for T₁/₂,₂:
\[T_{1/2,2} = \frac{1}{20} [\mathrm{A}_{0}]^2 - 40\]
The time for the second half-life (T₁/₂,₂) is equal to \(\frac{1}{20} [\mathrm{A}_{0}]^2 - 40\) seconds.
Key Concepts
Understanding Termolecular ReactionsRate Law Calculation SimplifiedWhat You Should Know About Half-Life Determination
Understanding Termolecular Reactions
In chemistry, reactions involving collisions between three molecules are termed as "termolecular reactions." These reactions are less common due to the rarity of three molecules colliding simultaneously with the required energy and proper alignment for the reaction to occur. Despite their infrequency, they are possible and follow specific reaction kinetics.
Termolecular reactions are characterized by a rate law that includes the concentration of three reactants raised to specific power(s). Most commonly, such reactions have a rate law that looks like this: \( \text{rate} = k[A]^3 \). This indicates how the rate of the reaction depends on the concentration `[A]` raised to the third power.
A prime aspect of termolecular reactions is their highly specific rate determining step. For these reactions to proceed efficiently, all three molecules must encounter each other in a single step, which makes their kinetics fascinating yet complex. Understanding the mechanics of these interactions is critical for predicting reaction outcomes and rates.
Termolecular reactions are characterized by a rate law that includes the concentration of three reactants raised to specific power(s). Most commonly, such reactions have a rate law that looks like this: \( \text{rate} = k[A]^3 \). This indicates how the rate of the reaction depends on the concentration `[A]` raised to the third power.
A prime aspect of termolecular reactions is their highly specific rate determining step. For these reactions to proceed efficiently, all three molecules must encounter each other in a single step, which makes their kinetics fascinating yet complex. Understanding the mechanics of these interactions is critical for predicting reaction outcomes and rates.
Rate Law Calculation Simplified
Calculating the rate law of a termolecular reaction helps us understand the reaction's kinetics. The rate law for a reaction is expressed as a mathematical equation relating the reaction rate to the concentrations of reactants. It provides insight into how changing the concentration of reactants will impact the rate.
For a reaction where the rate follows \( \text{rate} = k[A]^3 \), integrating the differential rate law is a crucial step for further analysis. The process involves setting up the equation \( \frac{-d[A]}{dt} = k[A]^3 \), and integrating both sides to derive an integrated form. This gives:
Understanding each step in calculating the rate law enables predictions regarding how long it will take for certain concentrations to deplete by half-life, and beyond. Practicing this fundamental skill opens doors to advanced comprehension of complex reactions.
For a reaction where the rate follows \( \text{rate} = k[A]^3 \), integrating the differential rate law is a crucial step for further analysis. The process involves setting up the equation \( \frac{-d[A]}{dt} = k[A]^3 \), and integrating both sides to derive an integrated form. This gives:
- The integrated equation \( - \frac{1}{2} [A]^{-2} = kt + C \) shows how concentration and time are related.
Understanding each step in calculating the rate law enables predictions regarding how long it will take for certain concentrations to deplete by half-life, and beyond. Practicing this fundamental skill opens doors to advanced comprehension of complex reactions.
What You Should Know About Half-Life Determination
Half-life in chemistry refers to the time it takes for half of the reactant concentration to decrease to half its initial amount. Especially in a termolecular reaction, determining the half-life helps in foreseeing how long a reaction will maintain its pace before slowing down significantly.
For our exercise, given the initial half-life as 40 seconds, the task requires determining the second half-life. The first half-life can be determined using the derived constants and initial conditions. For second half-life calculations, where \([A] = \frac{[A]_0}{4}\), it involves setting up the integrated rate law and solving for the unknown time.
Through the formula, \( T_{1/2,2} = \frac{1}{20} [A]_0^2 - 40 \), we can find the specific time for the second half-life. The term \([A]_0^2\) indicates how concentrated initial reactants significantly influence this duration. Calculations reveal the non-linear dynamics of how a reaction changes over continuous half-life periods, useful for planning and industrial uses.
For our exercise, given the initial half-life as 40 seconds, the task requires determining the second half-life. The first half-life can be determined using the derived constants and initial conditions. For second half-life calculations, where \([A] = \frac{[A]_0}{4}\), it involves setting up the integrated rate law and solving for the unknown time.
Through the formula, \( T_{1/2,2} = \frac{1}{20} [A]_0^2 - 40 \), we can find the specific time for the second half-life. The term \([A]_0^2\) indicates how concentrated initial reactants significantly influence this duration. Calculations reveal the non-linear dynamics of how a reaction changes over continuous half-life periods, useful for planning and industrial uses.
- Remember, each half-life is derived from cumulative durations, accounting for increased fractions of depletion.
- Half-life calculations help predict behavior of reactions under various conditions.
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