Derivation of Half-life only for Zero and First-Order Reactions

In numerous chemical processes as well as nuclear reactions, half-life is a specific occurrence that occurs on a daily basis. Imagine that you are at a movie theatre and you are watching a movie. This will give you a good understanding of what half-life is all about. This guy is consuming popcorn from a tub on the floor. After around 15 minutes, about half of the popcorn has been consumed. The remainder of the popcorn is consumed till the conclusion of the film. Most significantly, this demonstrates that popcorn consumption did not occur at a constant rate and that the half-life of popcorn is 15 minutes. Here’s where you can learn about the half-life formula.

Half-Life

The half-life of a sample refers to the amount of time it takes for half of a sample to react with the other half. Also included is the amount of time that a certain quantity requires to lower its initial value to half of its current value.

This is a term that is extremely frequent in nuclear physics, and it represents how quickly atoms might decay as a result of the radioactive bombardment. Furthermore, it could indicate how long an atom will be able to withstand radioactive decay. In addition, the half-life can be used to characterise any sort of decay, whether exponential or non-exponential, with greater accuracy. A notable example is the term “half-life” in the medical sciences, which refers to the amount of time a medicine remains in the human body after it has been administered.

Half-Life Formula

It is crucial to note that the formula for the half-life of a reaction changes depending on the sequence in which the reactions occur.

• The half-life of a zero-order reaction can be calculated using the following mathematical expression: t1/2 = [R]0/2k.
• The half-life of a first-order reaction is provided by the formula: t1/2 = 0.693/k.
• If the reaction is a second-order reaction, the half-life of the reaction is given by the formula 1/k[R0].

Where,

• The reaction’s half-life is denoted by the symbol t1/2 (unit: seconds)
• The starting reactant concentration [R0] is represented by (unit: mol.L-1 or M)
• k is the rate constant of the reaction (measured in units of M(1-n)s-1, where ‘n’ is the order of the reaction).

Half-Life of a First-Order Reaction

Remember that the integral rate law for a first-order reaction is provided by the equation:

[A]=[A]0 e-kt

This can also be written in a different form, which is equivalent:

ln[A]=ln[A]0–kt

If we are interested in determining the half-life of this reaction, we must solve for the time at which the concentration, [A], is equal to half of what it was initially; in other words, we must solve for the time at which the concentration, [A], is equal to half of what it was initially.

 If we substitute this for [A] in our integrated rate law, we get the following:

ln[A]02=ln[A]0−kt

By rearranging this equation and applying the properties of logarithms, we may discover that, for a first order reaction, the following results are obtained:

t12=ln(2)k

What’s interesting about this equation is that it tells us that the half-life of a first-order reaction is independent of the amount of material present at the outset. In terms of time, it takes the same amount of time for the reaction to progress from the entirety of the starting material to half of the starting material as it does to progress from one-fourth of the starting material to half of the starting material. The remaining material is reduced by half in each case in a period of time equal to the constant half-life. Take note that these results are only true for reactions that are first in the chain of events.

Examine the case of a unimolecular first-order reaction with an apparent rate constant of 5.00 s-1. 5.00 s-1 is the half-life of the reaction, and we can calculate it by simply substituting it for k:

t12=ln(2)k

t12=ln(2)5s-1=0.14s

Half-Life for a Zero-Order Reaction

In a zero-order reaction, the integrated rate law is given by the equation:

[A]=[A]0−kt

substituting  in [A]02 for [A], we have:

[A]02=[A]0−kt

We can obtain the following expression for the half-life by rearranging in terms of t:

t12=[A]02k

As a result, for a zero-order reaction, the half-life and the initial concentration are precisely proportional to one another. As the starting concentration of the reaction grows, the half-life of the reaction becomes increasingly longer.

Conclusion

During a chemical process, the half-life of a species is defined as the amount of time it takes for the concentration of the same material to decline to half of its starting value. In a first-order reaction, the half-life of the reactant is equal to ln(2)/k, where k is the rate constant of the reaction and ln(2) is the half-life of the product.

It is critical to understand half-lives because they allow you to calculate when a sample of radioactive material is safe to handle. When a sample’s radioactivity has decreased below detection limits, it is considered safe, according to the rule. And that occurs when 10 half-lives have elapsed.