Half-Life of a Reaction

What is Reaction Half-Life?

The half-life chemistry or a half-life of a reaction, t1/2, is defined as the specific amount of time required for a reactant concentration to decrease by half when compared to its initial concentration. The half-life application is used in chemistry and in medicine to predict the concentration of a substance over time.


The concepts of half-life play a vital role in the administration of drugs into the target, especially in the elimination phase, where half-life is used to discover how quickly a drug decrease in the target once it has been absorbed in a period of time (sec, minute, day) or the elimination rate constant ke (minute-1, hour-1, day-1). It is essential to make note that the half-life is varied between different types of reactions.


Half-Life Formula

It is essential to note that the half-life formula of a reaction varies with the reaction's order. The half-life formula for various reactions is given below.

  • The mathematical expression can be employed to determine the half-life for a zero-order reaction is,  t1/2 = [R]0/2k

  • For the first-order reaction, the half-life is defined as  t1/2 = 0.693/k

  • And, for the second-order reaction, the formula for the half-life of the reaction is given by, 1/k[R]0

Where,

  • t1/2 is the half-life of certain reaction (unit - seconds)

  • [R0] is the initial reactant concentration (unit - mol.L-1 or M), and

  • k is the rate constant of the reaction (unit - M(1-n)s-1, where ‘n’ is the order of  reaction)


Derivation of Half-Life Formula for Zero-Order Reaction

The half-life of a zero-order reaction is explained below on how it is derived, including the expression.

For the half-life of zero-order reaction, the units of the rate constant are mol.L-1.s-1. And, an expression for a half-life of zero-order reaction's rate constant is given by,


\[k = \frac{[R]_{0} - [R]}{t}\]


Substituting the value t = t1/2, at which the point [R] = [R]0/2 (at the half-life of a reaction, the concentration of the reactant is half of the initial concentration)


\[k = \frac{[R]_{0} - [R]_{0}/2}{t}\]


After rearranging the above half-life equation chemistry, the half-life of zero-order reaction expression is found to be,


\[k = \frac{[R]_{0}}{2k}\]


Derivation of First-Order Reaction Half-life Formula

The half-life of first-order reaction is given below on how it is derived, including the expression.

For a half-life of the first-order reaction, the constant rate can be mathematically expressed as follows. The first-order reaction half-life equation is given by,


\[k =  \frac{2.303}{t}\:  log\:   \frac{[R]_{0}}{[R]}\]

 

From the definition of the half-life of a first-order reaction, at t = t1/2, and [R] = [R]0/2. Substituting the values in the expression for the rate constant of half-life first-order reaction, the below-given equation is obtained.


\[k =  \frac{2.303}{t_{1/2}}\:  log\:   \frac{[R]_{0}}{[R]_{1/2}}\]


Rearranging the expression to find t1/2 value.


\[t_{1/2} =  \frac{2.303}{t}\:  log (2)\:   \frac{0.693}{k}\]


Therefore, the first-order half-life reaction is given by 0.693/k.


Probabilistic Nature of half-life Chemistry

The simulation of several identical atoms undergoing radioactive decay, beginning with either four atoms per box (towards the left) or 400 (towards the right). The number at the top is that of how many half-lives have elapsed. Note the consequences of the law of large numbers - with more atoms; the overall decay is more regular and predictable.

Usually, Half-life chemistry describes the decay of discrete entities, such as radioactive atoms. In that case, it doesn't work to use the half-life definition chemistry, stating that "half-life is the required time for exactly half of the entities to decay". For example, if there is only one radioactive atom, and its half-life is just one second, there will not be "half of the atom" left after that one second.


Instead, the half-life definition is defined in terms of probability as "Half-life is the required time exactly for half of the entities to decay on average". To define in other words, the radioactive atom probability decaying within its half-life is about 50%


Several simple exercises can demonstrate probabilistic decay, such as running a statistical computer program or involving flipping coins.


Examples of Half-life Chemical Reactions

Half-life chemistry is demonstrated using dice in a classroom experiment.

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There is a half-life that describes any exponential-decay process.


For example, in the radioactive decay case, the half-life is the length of time, after there is a 50% chance that an atom would have undergone nuclear decay. It differs based on the isotope and atom type and is usually determined as experimentally.


\[t_{1/2} =  \frac{In(2)}{k}\] = τ ln (2)


Here, ln (2) is a natural logarithm of 2 (approximately 0.693)


From the equation given above, the current flowing through an RC or RL circuit decays with a half-life of ln(2) L/R or ln(2) RC, respectively. For this example, rather than "half-life", the term half time tends to be used, but they mean similar.


The half-life of a species in a chemical reaction is the time taken for the concentration of the same substance to fall towards half of its initial value. In the first-order reaction, the half-life of the reactant is ln(2)/k, where k is the rate constant of the reaction.

FAQ (Frequently Asked Questions)

1. Explain the half-life definition chemistry?

The half-life definition chemistry is the time it takes for half an initial amount to disintegrate. The time that is required for half of a reactant to be converted into products. The time it required for half of a given sample to undergo radioactive decay. 


The half-life definition is given by the time that it takes for one-half of the atoms of a nuclide or of an unstable element to decay the radioactively into another nuclide or element.


For a given half-life reaction, the t1/2 of a reactant is the time required for its concentration to reach a value, the arithmetic mean of its initial and final (or equilibrium) value. For an entirely consumed reactant, it is the time taken for the reactant concentration to fall to one half of its initial value. For the first-order reaction, the half-life of the reactant may be known as the half-life of the reaction. In nuclear chemistry, the half-life definition is - the simple radioactive decay process, as the time needed for the activity to decrease to half its value by the same process.

2. What is the half-life of first-order reaction?

The half-life of first-order reaction is defined, in which time the concentration of the reactants becomes half in a chemical reaction. Then it is called a half-life.

Suppose a first-order reaction A is a product.


Integral equation of reaction rate in the given reaction is t=2.303/k.log[A]*/[A]

Where [A] and [A]* is the concentration of reactants in the starring of reaction (t=0) and concentration of reactants after t time, respectively.


As per the definition of half-life, when t=t/2 then [A]=half of [A]*

Substituting the number in the integral equation 


t/2 = 2.3030/k log [A]*/[A]*/2 = 2.303/k log 2 or t/2 = 0.693/k


Where K is the rate constant. This is the first-order half-life reaction.