Half-Life : Definition, Formula, Derivation (Zero & First-Order)

Chemistry is all about changes. How fast is a substance converted into another by a chemical reaction, and is the reaction feasible or not? The extent to which that reaction will proceed is all the questions that decide the course of a chemical reaction. Apart from feasibility and extent, other factors, like the rate of a reaction and the factors that determine the rate, are also crucial for a chemical reaction. For instance, what controls the rate at which the fuel burns in an auto engine? Which parameters determine how rapidly food gets spoiled?

In this module, you learn to perceive the half-life of a reaction and rate equations for zero and first-order reactions. 

What Is the Half-Life Definition of a Reaction?

The half-life definition of a reaction in simple words can be expressed as

“The time it takes for half of the reaction to be completed, i.e., the time in which the reactant concentration is decreased to half of its original value is called the half-life period of the reaction.”

In chemistry and medicine, the half-life of a reaction is used to predict the concentration of a chemical over time. Half-life calculates how quickly a chemical is reduced in the target once it has been absorbed over time or eliminated. The symbol demonstrates that the half-life is t₁/₂. The term, half-life of a reaction, is extremely chronic in nuclear physics. It determines how swiftly atoms might decay due to the radioactive striking.

What Is the Half-Life Formula?

It is a must to remember that the value of half-life varies from order to order. As the half-life formula changes for different reactions, so goes with its unit. The half-life formula and its unit for different order reactions are given below:

1. Half-life formula and unit for zero order reaction:

• The half-life formula used to calculate zero order reaction is t₁/₂ = [A]₀/2k.
• The unit of half-life equation for zero order reaction is ‘second.’

2. Half-life formula and unit for first order reaction:

• The half-life formula used to calculate the first-order reaction is t₁/₂ = 0.693/k.
• The unit of half-life equation for first order reaction is also ‘second.’

3. Half-life formula and unit for nth order reaction:

• The half-life formula used to calculate the nth order reaction is t₁/₂ ∝ 1/[A]₀ⁿ⁻¹.
• The unit of half-life equation for nth order reaction is also ‘M(n-1)s,’ where ‘n’ is the order of the reaction.

Derivation of Half-Life Equation for Zero Order Reaction

For a zero-order reaction, the half-life period (t₁/₂) is when half of the substance has reacted. It implies that

When [A] = [A]₀/2, t = t₁/₂.

Rate equation for a zero order reaction, k = {[A]₀ – [A]}/t

Substituting the values of [A] and t,

k = {[A]₀ – [A]₀/2}/t₁/₂

⇒ t₁/₂ = [A]₀/2k

Hence, the half-life formula for a zero order reaction is directly equivalent to the initial concentration, i.e., t₁/₂ ∝ [A]₀.

If you plot a graph of t₁/₂ vs [A]₀, it will be a straight line passing through the origin and slope = 1/2k.

The unit of k = mol L⁻¹ time⁻¹.

Hence, the unit of t₁/₂ = time.

Solved Example of Half-Life Period for a Zero-Order Reaction

Example: The initial pressure of the gas during the decomposition of gas on the surface of a solid crystal is 5.00 x 10³ Pa. If the rate constant for this zero-order reaction is 8 Pa s⁻¹, calculate the half-life period for this reaction.

Answer: For a zero order reaction,

t₁/₂ = [A]₀/2k

Here, [A]₀ = 5.00 x 10³ Pa and k = 8 Pa s⁻¹

⇒ t₁/₂ = 5.00 x 10³ Pa/ 8 Pa s⁻¹

⇒ t₁/₂ = 312.5 s

Derivation of Half-Life Equation for First Order Reaction

For a first-order reaction, the half-life period is the time taken for any fraction of the reaction to be completely independent of the initial concentration. It implies that

When [A] = [A]₀/2, t = t₁/₂.

Rate equation for a first-order reaction, log{[A]₀/[A]} = {k/2.303} x t

Substituting the values of [A] and t,

log{2[A]₀/[A]₀} = {k/2.303} x t₁/₂

⇒ log 2 = [k x t₁/₂]/ 2.303

⇒ t₁/₂ = {2.303 x log 2}/ k

⇒ t₁/₂ = 0.693/k

Hence, the half-life formula for a first-order reaction is independent of the initial concentration.

If you plot a graph of t₁/₂ vs [A]₀, it will be a horizontal line parallel to [A]₀.

The unit of k = time⁻¹.

Hence, the unit of t₁/₂ = time.

Solved Example of Half-Life Period for a First-Order Reaction

Example: A first-order reaction is found to have a rate constant k = 7.39 x 10⁻⁵ s⁻¹. Find the half-life of this reaction. (log 2 = 0.3010)

Answer: For a first-order reaction,

k = (2.303/t) x log{[A]₀/[A]}

And t₁/₂ = 0.693/k

⇒ t₁/₂ = 0.693/ 7.39 x 10⁻⁵ s⁻¹

⇒ t₁/₂ = 9.38 x 10⁻³ s

Derivation of Half-Life Equation for an nth Order Reaction

From the above derivations, it is concluded that

For a zero-order reaction, t₁/₂ ∝ [A]₀.

For a first-order reaction, t₁/₂ ∝ [A]₀º.

Similarly, for a second-order reaction, t₁/₂ ∝ [A]₀⁻¹.

In general, for an nth order reaction,

t₁/₂ ∝ [A]₀¹⁻ⁿ

⇒ t₁/₂ ∝ 1/[A]₀ⁿ⁻¹

The unit of half-life equation for nth order reaction is also ‘M(n-1)s,’ where ‘n’ is the order of the reaction.

Points to Remember

1. The half-life of a reaction, t₁/₂, is described as the time a reaction takes to reduce its initial concentration to half its exact amount.
2. It is employed to calculate the rate of a chemical reduction while reaching the target once it has been absorbed over a specific period or eliminate the rate constant for the given reaction.
3. The half-life formula for a reaction depends upon the order of a reaction.
4. For a zero-order reaction, the half-life equation is given as

t₁/₂ = [A]₀/2k

5. For a first zero order reaction, the half-life equation is given as

t₁/₂ = 0.693/k

6. For an nth zero order reaction, the half-life equation is given as

t₁/₂ ∝ [A]₀¹⁻ⁿ

7. The unit of t₁/₂ for a zero-order and a first-order reaction is ‘time.’

Did You Know?

• In medical terms, the half-life of medicine is defined as the time a therapeutic drug requires to be present in your body.
• The half-life of a reaction can be related to the disappearance or change in a chemical substance with time.
• Water does not encounter spontaneous degradation. It happens due to the stable character of water. Therefore, it does not get associated with the term half-life.

Conclusion

The half-life of a reaction (t₁/₂) is the time needed for half of a given quantity of reactant to be consumed. In each thriving half-life, half of the remaining concentration of the reactant gets consumed. The half-life of a first-order reaction is unrestrained by the concentration of the reactant. At the same time, the half-lives of reactions with other orders hang on the concentrations of the reactants.

Frequently Asked Questions

1. What is the half-life definition for a radioactive substance?

1. It can be defined in different ways.
2. The time in which half of the total quantity or a total number of atoms of a radioactive substance gets disintegrated is called the substance’s half-life.
3. The time in which a given substance’s radioactivity falls to half its actual value is termed the half-life of that substance.

2. What is a rapid half-life?

A. Rapid is not usually tagged as half-life. The half-life term refers to the time it takes for a product to degrade 50% of its initial effectiveness. It can range from drugs to radioactive elements. At the same time, the speed comparisons with which it takes to get to that 50% are often referred to as “short Half-life” (which means quick) and “long Half-life” (which means long). 

3. What are the examples of zero-order reactions?

A. Some common reactions that follow zero order are as

• Decomposition of HI on the gold surface.
• Decomposition of ammonia on several substances such as platinum, gold, molybdenum, etc.
• Enzymes catalysed reactions are zero-order reactions concerning the substrate and first-order reactions concerning an enzyme.

4. Who proposed the term half-life for the first time?

A. In 1907, Ernest Rutherford tracked down the concept of dating in radioactive elements, which is related to the half-life of the elements. In the 1950s, this dating term was abbreviated to half-life.

Rutherford solicited the concept of the half-life of a radioactive element and studied the age determination process of rocks by determining the decay period of radium to lead-206.

5. What is the significance of half-life in controlling radiation during radioactive decay?

A. In brief, the radiological half-life is critical in radiation control as long-lived radionuclides, once liberated, are all over the place for longer than momentary species. Radionuclides that stay for a long period are revealed to the environment and will exist for a longer time than impermanent nuclides. It may require essential, longer-term observation of the environment to ensure that a person is not significantly unmasked. The value of the half-life of the substance aids in determining the period of existence of that substance in the environment.