Name: reaction half-life in kinetics
Uploaded: 2020-09-24T15:06:50.000Z
Duration: 2 min 42 s
Channel: JoVE

K12 Standards

Texas Essential Knowledge and Skills

High School

Chemistry and Physics

IPC.7

IPC.7.F

Half-life of a Reaction

During a chemical reaction, the reactant&#8217;s concentration decreases over time. The time taken to reduce it to half of its initial quantity is called the half-life of the reaction, abbreviated as t1/2.
Consider the refrigerant trichlorofluoromethane, which depletes the ozone layer. Its atmospheric residence time is 45 years and it is banned in many countries. But how long does it take for the concentration to drop to 50%? To estimate a reactant&#8217;s concentration after an elapsed time, the half-life can be used.
The half-life differs for each reaction type. Depending on the reaction order, a general half-life expression can be derived from the integrated rate laws. 
Consider the zero-order integrated rate law. At t1/2, the reactant concentration is half of its initial concentration. On substitution of these parameters, an expression for the half-life is formulated.
The formula shows that the half-life for a zero-order reaction is dependent on and directly proportional to the initial reactant concentration. As the reactant concentration declines, the half-life continues to get shorter.
For a first-order reaction, the integrated rate expression is modified by substituting the time as t1/2 and the reactant quantity as half of its initial concentration. On solving, the expression for t1/2 is determined to be a constant.
Thus, the half-life of a first-order reaction is derived to be independent of the initial reactant concentration. Regardless of the decrease in the reactant concentration, the half-life remains constant. 
For a second-order reaction, the integrated rate expression is converted to t1/2 by substituting the time and reactant concentration parameters. 
The half-lives of second-order reactions are inversely dependent on the initial reactant concentration. As the reactant concentration decreases, the t1/2 value increases; meaning, the half-life continues to get longer.
The half-life also helps to estimate the magnitude of the rate constant. It offers a relative comparison between different reaction rates; the shorter the half-life, the faster the reaction rate and vice-versa. 
For instance, the radioactive isotope sodium-24, with a short half-life of 14.7 hours, exhibits a faster decay rate than cobalt-60, with a longer half-life of 5.3 years

half-life of a reaction

The half-life of a reaction (t1/2) is the time required for one-half of a given amount of reactant to be consumed. In each succeeding half-life, half of the remaining concentration of the reactant is consumed. For example, during the decomposition of hydrogen peroxide, during the first half-life (from 0.00 hours to 6.00 hours), the concentration of H2O2 decreases from 1.000 M to 0.500 M. During the second half-life (from 6.00 hours to 12.00 hours), the concentration decreases from 0.500 M to 0.250 M, while during the third half-life, it falls from 0.250 M to 0.125 M. Hence, during each successive period of 6.00 hours, the concentration of H2O2 decreases by half.
The half-life of a first-order reaction is independent of the concentration of the reactant. However, half-lives of reactions with other orders depend on the concentrations of the reactants.
Half-life of First-order Reactions
An equation relating the half-life of a first-order reaction to its rate constant may be derived from its integrated rate law:
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According to the definition of half-life, at time t1/2, the concentration of the reactant A is one-half of its initial concentration. Therefore; t = t1/2 and [A]t = &#189; [A]0.
Substituting these terms into the rearranged integrated rate law and simplifying yields the equation for half-life:
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This half-life equation describes an expected inverse relationship between the half-life of the reaction and its rate constant, k. Faster reactions exhibit larger rate constants and correspondingly shorter half-lives, while slower reactions exhibit smaller rate constants and longer half-lives.
Half-life of Second-order Reactions
Following the same approach as for first-order reactions, an equation relating the half-life of a second-order reaction to its rate constant and initial concentration may be derived from its integrated rate law: 
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On substituting t = t1/2 and [A]t = &#189;[A]0, the integrated rate law is simplified:
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For a second-order reaction, t1/2 is inversely proportional to the concentration of the reactant, and the half-life increases as the reaction proceeds because the concentration of the reactant decreases. Unlike with first-order reactions, the rate constant of a second-order reaction cannot be calculated directly from the half-life unless the initial concentration is known.
Half-life of Zero-order Reactions
An equation for zero-order half-life may be also be derived from its integrated rate law: 
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Substituting t = t1/2 and [A]t = &#189; [A]0, in the zero-order integrated rate law yields:
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The half-life for a zero-order reaction is inversely proportional to its rate constant. However, the half-life of a zero-order reaction increases as the initial concentration increases.
This text is adapted from <a href="https://openstax.org/books/chemistry-2e/pages/12-4-integrated-rate-laws">Openstax, Chemistry 2e, Section 12.4: Integrated Rate Laws.</a>

The half-life of a reaction (t1/2) is the time required for one-half of a given amount of reactant to be consumed. In each succeeding half-life, half of ...

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KEY TERMS AND CONCEPTS

reaction half-life in kinetics

Reaction half-life is the time needed for half of a reactant to be consumed. After each half-life, half of the remaining reactant is used up. A hydrogen peroxide decomposition example shows this pattern clearly. The concentration drops from 1.000 M to 0.500 M in the first 6.00 hours, then to 0.250 M in the next 6.00 hours, and then to 0.125 M in the third period. Each later 6.00-hour interval removes half of what is left.

For first-order reactions, the half-life does not depend on the reactant concentration. An equation for this half-life comes from the integrated rate law and shows an inverse relationship with the rate constant, k. Faster reactions have larger rate constants and shorter half-lives. Slower reactions have smaller rate constants and longer half-lives.

Second-order reactions follow a different pattern. Their half-life depends on both the rate constant and the initial concentration. As the reaction continues and the reactant concentration falls, the half-life gets longer. Unlike a first-order reaction, the rate constant for a second-order reaction cannot be found directly from the half-life unless the initial concentration is known.

Zero-order reactions have their own half-life relationship as well. Their half-life is inversely proportional to the rate constant. It also increases when the initial concentration is higher. These different formulas show that reaction order changes how half-life behaves during a reaction.

Reaction Half-Life in Kinetics

reaction rate

Measuring Reaction Rates

Measuring reaction rates shows how fast reactants are used up or products are formed over time. The rate of reaction is the change in the amount of a reactant or product per unit time. Because of that, scientists measure how a property changes as the reaction proceeds and connect that change to reactant or product amounts. For reactions that make or use a gas, the rate can be followed by changes in volume or pressure. A rate expression is the mathematical way to write how concentration changes...

measuring reaction rates

Tracking Reaction Kinetics with Light and Pressure

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concentration and rate law

Reaction Order from Concentration

Reaction order from concentration shows how reactant amounts affect reaction rate. Rate laws, also called differential rate laws or rate equations, are mathematical expressions that connect reaction rate with the concentrations of reactants. For a generic reaction a A + b B → products, the rate law can be written using the reactant concentrations [A] and [B] and a rate constant k. In a rate law, k is the rate constant for one specific reaction at one specific temperature. The exponents m and n...

determining order of reaction

Finding Rate Law from Initial Rates

Rate laws connect the speed of a chemical reaction to the concentration of its reactants. The rate constant, k, and the reaction orders are not taken from the chemical equation. They are found experimentally by watching how the rate changes when the reactant concentrations change. One common way to do this is the method of initial rates. In this method, several trials are run with different starting concentrations. The measured rates are then compared to determine the reaction orders and,...

the integrated rate law: the dependence of concentration on time

the-integrated-rate-law-the-dependence-of-concentration-on-time

Reading Reaction Progress with Integrated Laws

Integrated rate laws connect reaction concentration to time. They help show how much reactant or product is present after a certain period. They can also help estimate how long a reaction must run to reach a given extent. A common use is finding how long radioactive material must be stored before its radioactivity drops to a safe level. These equations come from the differential rate law after it is integrated with respect to time. For a first-order reaction, where rate = k[A], the integrated...

Reaction half-life is the time needed for half of a reactant to be consumed. After each half-life, half of the remaining reactant is used up. A hydrogen peroxide decomposition example shows this pattern clearly. The concentration drops from 1.000 M to 0.500 M in the first 6.00 hours, then to 0.250 M in the next 6.00 hours, and then to 0.125 M in the third period. Each later 6.00-hour interval removes half of what is left. For first-order reactions, the half-life does not depend on the reactant...

temperature dependence on reaction rate

Collision Theory and Reaction Speed

Collision theory explains why reaction speed depends on how particles collide. Atoms, molecules, or ions must come close together before they can react. The theory says the reaction rate depends on the collision rate, the orientation of the colliding species, and whether the collision has enough energy for bonds to change. A gas-phase reaction between carbon monoxide and oxygen shows why orientation matters. The molecules can collide in more than one way. If the oxygen side of carbon monoxide...

arrhenius plots

Determining Activation Energy from Temperature

The Arrhenius equation links activation energy and the rate constant, k, for chemical reactions. It is written as k = Ae−Ea/RT. In this equation, R is the ideal gas constant with a value of 8.314 J/mol·K, T is the temperature on the kelvin scale, Ea is the activation energy in J/mole, e is the constant 2.7183, and A is the frequency factor. The frequency factor depends on collision frequency and the orientation of reacting molecules. Experimental kinetic data can be used to find the activation...

reaction mechanisms

Elementary Steps in Reaction Mechanisms

Chemical reactions can happen in steps, and a balanced equation does not show the full path at the molecular level. A reaction mechanism, also called a reaction path, gives the step-by-step process that explains how reactants become products. The decomposition of ozone is one example of a reaction that can be described by a sequence of steps. Each step in a mechanism is an elementary reaction. These elementary reactions happen in order, and when they are added together, they give the balanced...

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Reaction Bottlenecks in Chemical Kinetics

Reaction bottlenecks help explain rate-determining steps in multistep reaction mechanisms. In a mechanism with several elementary steps, one step is often much slower than the others. That slowest step is called the rate-limiting step, or rate-determining step. A reaction cannot move faster than its slowest step, so this step controls the overall reaction rate. The idea is like traffic on a four-lane freeway with one lane closed. Cars can only move as fast as the bottleneck allows. In the same...

catalysis

Catalyst Action in Reaction Rates

Catalysts change the rate of a chemical reaction by giving it a faster path. A catalyst increases reaction rate without being consumed during the process. Its effect is often explained with reaction mechanisms and energy diagrams. For an endothermic reaction, the reactants and products start and end at the same energies in both cases. The uncatalyzed path is a one-step mechanism with one transition state. The catalyzed path follows a different mechanism with two transition states and a lower...

enzymes

Enzyme Action and Inhibition

Enzymes are biological catalysts that speed up many reactions in living organisms. They help control cellular metabolism by lowering the activation energy needed for biochemical reactions. When activation energy drops, reaction rates can increase. Enzyme problems can also lead to serious disease. A genetic abnormality that causes G6PD, or glucose-6-phosphate dehydrogenase, deficiency can disrupt the pathway that supplies NADPH to cells. In red blood cells, this can reduce glutathione and...

key terms and concepts

Learn how reaction half-life changes in first-, second-, and zero-order reactions, with a hydrogen peroxide example and key rate law relationships.

Reaction Half-Life in Kinetics | JoVE K12

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NC K-12 Science Standard Course of Study

9-12

Matter and its Interactions

PS.Chm.5.1

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Chemical Reactions

3.2.9-12.D

3.2.9-12.E

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Ohio’s Learning Standards and Model Curriculum

Interactions Of Matter

Chemical reactions

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Matter and Its Interactions: Energy

9-12.C.3.3.d

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Matter and Its Interactions

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SC.912.P.12

SC.912.P.12.12

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HS. Chemical Reactions

HS-PS1-5.

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Reaction Half-Life in Kinetics

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