What is Reaction Half-Life?
The half-life of a chemical reaction can be defined as the time taken for the concentration of a given reactant to reach 50% of its initial concentration (i.e. the time taken for the reactant concentration to reach half of its initial value). It is denoted by the symbol ‘t1/2’ and is usually expressed in seconds.
Table of Contents
- Half-Life Formula
- Derivation of Half-Life Formula for Zero Order Reactions
- Derivation of Half-Life Formula for First Order Reactions
Half-Life Formula
It is important to note that the formula for the half-life of a reaction varies with the order of the reaction.
- For a zero-order reaction, the mathematical expression that can be employed to determine the half-life is: t1/2 = [R]0/2k
- For a first-order reaction, the half-life is given by: t1/2 = 0.693/k
- For a second-order reaction, the formula for the half-life of the reaction is: 1/k[R]0
Where,
- t1/2 is the half-life of the reaction (unit: seconds)
- [R0] is the initial reactant concentration (unit: mol.L-1 or M)
- k is the rate constant of the reaction (unit: M(1-n)s-1 where ‘n’ is the reaction order)
Derivation of Half-Life Formula for Zero-Order Reactions
For a zero-order reaction, the units of the rate constant are mol.L-1.s-1. The expression for a zero-order rate constant is:
\(\begin{array}{l}k = \frac{[R]_{0}-[R]}{t}\end{array} \)
Substituting t = t1/2, at which point [R] = [R]0/2 (at the half-life of a reaction, reactant concentration is half of the initial concentration).
\(\begin{array}{l}k = \frac{[R]_{0}-[R]_{0}/2}{t_{1/2}}\end{array} \)
Rearranging the equation, the expression for the half-life of a zero-order reaction is found to be:
\(\begin{array}{l}t_{1/2} = \frac{[R]_{0}}{2k}\end{array} \)
Derivation of Half-Life Formula for First-Order Reactions
For a first-order reaction, the rate constant can be mathematically expressed as follows:
\(\begin{array}{l}k = \frac{2.303}{t}log\frac{[R]_0}{[R]}\end{array} \)
From the definition of reaction half-life, at t = t1/2, [R] = [R]0/2. Substituting these values in the expression for the first-order rate constant, the following equation is obtained:
\(\begin{array}{l}k = \frac{2.303}{t_{1/2}}log\frac{[R_0]}{[R]_0/2}\end{array} \)
Rearranging the expression to find the value of t1/2:
\(\begin{array}{l}t_{1/2} = \frac{2.303}{k}log(2) = \frac{0.693}{k}\end{array} \)
Thus, the half-life of a first-order reaction is given by 0.693/k.
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Zero Order Reactions
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FAQs
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].
What is the deriving formula for half-life? ›
Derivation of Half-Life Equation for Zero Order Reaction
When [A] = [A]₀/2, t = t₁/₂. Hence, the half-life formula for a zero order reaction is directly equivalent to the initial concentration, i.e., t₁/₂ ∝ [A]₀.
What is the half-life formula for first-order? ›
The half-life of a first-order reaction is a constant that is related to the rate constant for the reaction: t1/2 = 0.693/k.
What is the formula for calculating half-life? ›
If you are given a problem where you are told how many half-lives have elapsed as well as how much time has passed, you can solve for the length of a half-life by using the equation T=t/n, where T is the length of a half-life, t is how much time has passed, and n is the number of half-lives that have passed.
What is the mean half-life derivation? ›
The time taken for half of the original population of radioactive atoms to decay is called the half-life. This relationship between half-life, the time period, t1/2, and the decay constant λ is given by t12=0.693λ t 1 2 = 0.693 λ . Break down tough concepts through simple visuals.
How do you derive the half-life of a zero order reaction? ›
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].
What is deriving the formula? ›
To derive a formula means to deduce, obtain, or prove the formula from a set of already-known or already-established principles or observations. What derivation looks like can be a little different in pure, rigorous mathematics, vs. in science, but the idea is similar.
What is the formula for half-life in calculus? ›
( ln 2 ) / k . Systems that exhibit exponential decay follow a model of the form y=y0e−kt. y = y 0 e − k t . Systems that exhibit exponential decay have a constant half-life, which is given by (ln2)/k.
How do scientists calculate half-life? ›
The half-life is then determined from the fundamental definition of activity as the product of the radionuclide decay constant, λ, and the number of radioactive atoms present, N. One solves for λ and gets the half-life from the relationship λ = ln2/T1/2.
What is the general expression for the half-life of the nth order reaction? ›
The general expression for the half-life period of an nth order reaction is: t1/2=2n−1−1k(n−1)(A0)n−1.
The equation for half-life is T 1 / 2 = l n ( 2 ) λ , where T 1 / 2 is the half-life, and λ is the decay constant, which is a value specific to each chemical. Half-life follows exponential decay because half-life involves multiplying the remaining quantity by the same number repeatedly.
What are the 2 definitions of half-life? ›
Half-life is the time it takes for half of the unstable nuclei in a sample to decay or for the activity of the sample to halve or for the count rate to halve. Count-rate is the number of decays recorded each second by a detector, such as the Geiger-Muller tube.
How to derive first order reaction? ›
For first-order reactions, the equation ln[A] = -kt + ln[A]0 is similar to that of a straight line (y = mx + c) with slope -k. This line can be graphically plotted as follows. Thus, the graph for ln[A] v/s t for a first-order reaction is a straight line with slope -k.
What is the formula for deriving a function? ›
Definition: Derivative Function
f′(x)=limh→0f(x+h)−f(x)h. A function f(x) is said to be differentiable at a if f′(a) exists. More generally, a function is said to be differentiable on S if it is differentiable at every point in an open set S, and a differentiable function is one in which f′(x) exists on its domain.
What is the formula for half-life uncertainty? ›
2.4.
The relative uncertainty of the slope equals the relative uncertainty on the decay constant (or the half-life), i.e. σ b / b = σ λ / λ .
How do you derive the half-life equation from the integrated rate law? ›
Re: Half Life
The half-life of a reaction is the time it takes for an initial concentration to get to half its value. Thus, we can adjust the 1st order integrated rate law to get ln(1/2)*[A]0 - ln[A]0 = -kt(1/2). This can be rewritten as: This can be simplified to ln(0.5) = -kt(1/2), or -0.693 = -kt(1/2).
