Half life: Derivation, Zero, First-Order Reaction

The half-life of a reaction, t1/2, is defined as the time it takes for a reactant concentration to drop by half when compared to its original concentration. In chemistry, the application half-life is used to anticipate the concentration of a chemical over time. Half-life is used to determine how rapidly a drug decreases in the target once it has been absorbed over a period (sec, minute, day) or the elimination rate constant ke (minute-1, hour-1, day-1). It's important to remember that the half-life varies from reaction to reaction.

Key Takeaways: Radioactivity, Zero order reaction, First-order reactions, Second order Reaction, Half-life, Half-life reaction

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Half-Life Formula

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It's important to remember that a reaction's half-life formula changes depending on the order of the reactions. The formula for the half-life of different reactions is given below.

• The half-life of a zero-order reaction, the formula is given as t_1/2 = R0/2k
• The half-life of a first-order reaction is given as t_1/2 = 0.693/k.
• The half-life of a second-order reaction is given by the formula 1/kR₀.

Where,

The half-life of a reaction is referred to as t_1/2 (unit - seconds)

The initial reactant concentration is referred to as R₀ (in mol.L^-1 or M) is [R₀], and

The reaction's rate constant is given as k (unit - M(1-n)s^-1, where 'n' is the reaction's order)

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Derivation of Half-Life Formula for Zero Order Reaction

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The rate constant for a zero-order reaction is measured in mol.L^-1.s^-1.

The following is the formula for a zero-order rate constant: 

k=[R]₀−[R ]/ t

By substituting t = t_1/2, [R] = [R]_0/2 is obtained (at the half-life of a reaction, reactant concentration is half of the initial concentration): 

k=[R]₀−[R]_0/2 / t_1/2

The half-life of a zero-order reaction can be calculated by rearranging the equation: 

t_1/2 = [R]₀ / 2k

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Derivation of Half-Life Formula for First Order Reaction

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The following is the formula for a first-order rate constant expressed mathematically: 

k = 2.303/t log [R]₀ / [R]

At t = t_1/2, [R] = [R]_0/2, according to the reaction half-life definition. The following equation is obtained by substituting these values in the expression for the first-order rate constant: 

k=2.303/ t_1/2 log[R]₀ / [R]_0/2

Rearranging the expression:

t_1/2 = 2.303 log(2) / k = 0.693/k

Thus, the half-life of a first-order reaction is obtained as 0.693/k.  

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Derivation of Half-Life Formula for Second-Order Reactions

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An equation relating the half-life of a second-order reaction to its rate constant and starting concentration may be obtained from its integrated rate law using the same method as for first-order reactions:

\(\frac{1}{[A]_t} = kt + \frac{1}{[A]_0}\)    Or    \(\frac{1}{[A]_t} - \frac{1}{[A]_0} = kt \)

The integrated rate law is simplified when \(t = {t_\frac{1}{2}}\) and \({[A]_t} = {12[A]_0}\) are substituted:

\({kt_{\frac{1}{2}}} = {\frac{1}{\frac{1}{2}[A]_0}} - {\frac{1}{[A]_0}} = {\frac{2}{[A]_0}} - {\frac{1}{[A]_0}}\)

\({kt_{\frac{1}{2}}} = {\frac{1}{[A]_0}}\)

\({t_{\frac{1}{2}}} = {\frac{1}{k[A]_0}}\)

The half-life of a second-order reaction is inversely proportional to the reactant concentration, and the half-life rises as the reaction progresses since the reactant concentration falls. A second-order reaction's rate constant cannot be computed simply from the half-life, unlike first-order reactions, unless the initial concentration is known.

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Effect of Half Life

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• The radioisotope component has a physical Half-life, which means that the amount of radioactivity drops by half with each Half-life. Tc99m, a common radioisotope with a half-life of 6 hours, is an example of a common radioisotope. If it holds 100 MBq of this isotope right now, it will contain 50 MBq in 6 hours and 25 MBq in the next 6 hours.
• The patient can also excrete a little amount of the radiopharmaceutical, usually through the urine and kidneys, thus there is a certain amount that the body gets rid of, and so the radiation can be considered outside the body. If this occurs, that portion of the patient's radiation dose is no longer affected. The biological half-life is the term for this.

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Things to Remember

• The half-life chemistry or reaction half-life, t1/2, is defined as the time it takes for a reactant concentration to decrease by half when compared to its initial concentration.
• Half-life is used to determine how quickly a chemical decreases in the target once it has been absorbed over a period of time (sec, minute, day) or the elimination rate constant 'k' (minute-1, hour-1, day-1).
• A reaction's half-life formula changes depending on the order of the reactions.
• The half-life of a zero-order reaction, the formula is given as t1/2 = R0/2k, The half-life of a first-order reaction is given as t1/2 = 0.693/k, The half-life of a second-order reaction is given by the formula 1/kR0.
• The rate constant for a zero-order reaction is measured in mol.L^-1.s^-1.

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