Half-Life of a Reaction

Chemistry has a lot of conflicting opinions in the student community. Some people think and believe that chemistry is the easiest subject that there ever was, while some people are absolutely scared of even thinking of the topics and chapters from chemistry. So what exactly is the deal here? Is Chemistry really that tough? Or is it just the popular opinion among the students to name every other subject as something that is difficult?

We may not have the answer to any one of those questions but what we do have an answer for is that when it comes to things like studying chemistry, you must not be afraid of it to score really good marks in it. The moment you begin to fully understand and comprehend what the main idea behind many topics are is when you begin to fall in love with the subject as well. 

The half-Life reaction is a very interesting name. Believe us when we say this, but we know you already like the name because of how cool it sounds. And another thing that you should know, is that the concept itself is also very interesting. Number one, it is a very simple and easy topic, the moment you start reading this article’s main content, you would realize this on your own. 

Next, the topic is also very interesting because of what it is about. The main idea and the concept behind the topic is something that many students enjoy and love. 

So we might not be able to answer questions like why is chemistry misunderstood sometimes but one thing that we can assure you is that you will have a great time reading this article. So make sure to take your notepads out and start taking all the essential notes that you need for mastering this topic

The half-life chemistry or a half-life of a reaction, t_1/2, 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 that 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 a 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 a 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 of 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.

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.

Effective Half-life

• The radioisotope component contains a physical Half-life where the radioactivity amount decreases by half with every Half-life. A common radioisotope can be given as Tc99m which contains a Half-life of 6 hours. If it contains 100 MBq of this isotope now, in 6 hours, there would be 50 MBq remaining and 25 MBq 6 hours after this.

• The patient can also excrete a few of the radiopharmaceuticals, often through the urine and kidneys, so there is an amount that the body gets rid of, and hence that radiation can be considered as outside of the body. If this happens, that portion no longer contributes to the radiation dose of the patient. This is called biological Half-life.

The effective Half-life can be described as a combination of these two and relates to the particular dose the patient will receive taking the radioisotope physical properties into account and how long it stays in the body.

Did You Know?

• The half-life of a drug describes how long the therapeutic drug dose will be present in the body. 

• We can relate Half-life either to the change or disappearance in a chemical substance over time. Water is a stable compound and it does not undergo spontaneous degradation. Thus, water does not qualify for the phrase ‘Half-life’.