The Half Life Formula: Understanding Exponential Decay

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When dealing with chemical reactions or radioactive processes, one concept that frequently arises is half-life (t1/2). Essentially, it refers to the time required for exactly half of an initial quantity to undergo a specific transformation. This could be disintegration in nuclear physics, decay in chemistry, or any other type of process where matter transforms into something else.

In this article, we will delve deeper into what half-life means and how it can be mathematically described using the concept of exponential decay. Specifically, we'll focus on deriving an equation that allows us to calculate the half-life time (t1/2) given certain parameters about our system.

What is Exponential Decay?

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Before diving headfirst into the mathematics behind half-life, let's quickly review what exponential decay actually means. Imagine you have a container filled with some sort of substance A which over time transforms into another substance B at an rate that is constant and independent from how much remains untransformed.

Mathematically speaking, this situation can be represented by the following differential equation:

dA/dt = -k \* A

where:

- dA/dt represents the change in amount of substance A per unit time (in units like moles/second)

- k is called the decay constant and it has a value that depends on specific conditions such as temperature or pressure

- t is time measured from some reference point, usually when there was still 100% remaining

Now let's solve this differential equation to find an expression for A(t), which represents how much of substance A remains at any given time t:

A(t) = A0 \* exp(-kt)

where:

- A0 denotes the initial amount present initially (i.e., when t=0)

- k is again the decay constant we introduced earlier

- e is Euler's number approximately equal to 2.71828...

The Half Life Formula: Derivation and Explanation

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Now that we have a solid understanding of exponential decay, let's turn our attention back towards deriving an expression for half-life time (t1/2). Recall from above:

A(t) = A0 \* exp(-kt)

We want to find the value of t at which exactly half of the initial amount remains. Mathematically speaking this means setting up and solving the following equation:

A(t1/2)/A(0)= 1/2

Substituting in our expressions for A(t) gives us:

exp(-k \* t_{1/2}) / exp(0) = 1/2

Simplifying further using properties of exponents (specifically, exp(x)-exp(-x)=(e^{x}-1)/(e^{x}+1)), we arrive at the following expression for half-life time:

t_{1/2}= ln(2)/k

where:

- l is natural logarithm

- k is again the decay constant

In words: The half life of a process depends inversely on its rate constant.

A Simple Example

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To make this concept more concrete, let's consider an example. Suppose we have a radioactive substance with an initial amount of 326 grams which decays according to first-order kinetics (i.e., following the exponential decay law) at a half-life time of about 15 years and 8 months.

Given these parameters:

- Initial Amount: A0 = 326 g

- Half-Life Time: t1/2 ≈ 11.69315 years

We can now use our formula to calculate both k (the rate constant) and the actual decay constant λ:

k = ln(2)/t_{1/2} \approx 3.56 x10^{-4} yr^-1

λ= k\*ln(2)\approx8.67x10^(-5)s^-1

In conclusion, we've seen how to derive an expression for half-life time (t1/2) in terms of the initial amount A0 and the final amount remaining after a certain period t using nothing but basic principles from calculus and exponential functions.

The main takeaway is that understanding these concepts can help us better predict what happens when dealing with systems involving radioactive decay or other types of transformations where matter disappears over time at constant rates independent of how much remains untransformed.

Half life formula