Lecture 10.2 Half-life

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Lecture 10.2 Half-life

Half-Life of Chemical Reactions

Understanding Half-Life

The lecture focuses on the concept of half-life in chemical reactions, explaining its significance and how to calculate it using experimental data.

Half-life is defined as the time required for half of a reactant to be converted into products, indicating the progress of a reaction.

Graphical Representation

A diagram illustrates that if the initial pressure (or concentration) is 150, after one half-life, it should drop to 75, which is half of the initial value.

The corresponding time for this change in concentration is identified as the first half-life period, approximately 13,000 seconds.

Multiple Half-Lives

After another half-life period, the concentration further decreases to 37.5 (half of 75), marking this as the second half-life period.

The second half-life duration is noted to be around 26,000 seconds.

Calculating Half-Life for First Order Reactions

For first-order reactions, at time T_2 , concentration A should equal half of initial concentration A_0 .

The integrated rate equation used is rearranged: ln(A/A_0) = -kt . Substituting values leads to an expression involving natural logarithm calculations.

Deriving First Order Half-Life Equation

By substituting A = 0.5A_0 , we derive that -ln(0.5)=kt_2 .

This results in the formula for calculating half-life: t_1/2 = 0.693/k , crucial for radioactive decay processes.

Calculating Half-Life for Second Order Reactions

For second-order reactions, similar principles apply where at time T_2 , concentration must also equal half of initial concentration.

The integrated rate law used here is rearranged accordingly to find expressions related to concentrations over time.

Summary and Zero Order Reactions

In zero-order reactions, similar calculations can be performed by substituting values into their respective equations.

Each order has distinct methods for determining their respective half-lives based on their kinetic properties.

Understanding Half-Life in Chemical Reactions

Zero Order Reaction Calculations

The half-life for a zero-order reaction is derived as T_1/2 = a_0/2K , where a_0 is the initial concentration and K is the rate constant.

Integrated rate laws for various reactions have been discussed, emphasizing their importance in determining reaction rates and half-lives.

An example problem involves calculating the concentration of reactant A after 30 seconds for zero, first, and second-order reactions with an initial concentration of 5 mol.

Half-Life Time Periods

For a zero-order reaction with K = 0.01 , the calculated half-life is 250 seconds; this indicates it takes longer to reach half of the reactant compared to other orders.

The first-order reaction's half-life formula yields approximately 69.3 seconds, while the second-order reaction results in a much shorter time of about 20 seconds.

Concentration After Time Elapsed

After 30 seconds, calculations show that only a fraction of reactants has reacted:

Zero order: Concentration drops to 4.7 mol

First order: Concentration drops to approximately 3.7 mol

Second order: Concentration reduces to about 2 mol.

Detailed Calculation Steps

Zero Order Reaction

The integrated rate law for zero-order reactions is expressed as [A]_T = [A]_0 - KT .

Substituting values gives [A] = 5 - (0.01 * 30) = 4.7 text mol .

First Order Reaction

The equation used is ln[A]_T = -KT + ln[A]_0 .

This leads to calculating A_T: using natural logarithm properties results in approximately A_T = e^1.31 ≈ 3.7 text mol .

Second Order Reaction

For second-order reactions, use the formula 1/[A]_T = KT + 1/[A]_0 .

This calculation shows that after substituting values, the concentration becomes approximately [A] ≈ 2text mol .

Summary Insights on Reactant Behavior

Overall observations indicate that different orders of reactions significantly affect how quickly concentrations change over time; zero-order takes longest while second-order reacts quickest within given parameters.