LN2.CONST

Application:

Calculating the time to halve a quantity in exponential decay

In physics, chemistry, and other fields, many processes are modeled using exponential decay. The formula for exponential decay is:

N(t)=N0​∗e−kt

where:

• N(t) is the amount of the substance at time t.
• N0​ is the initial amount of the substance.
• k is the decay constant.
• e is Euler's number (the base of the natural logarithm).

A common question is: "What is the half-life (t1/2​), the time it takes for the substance to decay to half its initial amount?"

To find the half-life, we set N(t)=N0​/2:

N0​/2=$N_0\cdot e^{-k{t}_{1/2}}$​

1/2=$e^{-k{t}_{1/2}}$

Taking the natural logarithm of both sides:

ln(1/2)=ln($e^{-k{t}_{1/2}}$​)

−ln(2)=−kt1/2​

ln(2)=kt1/2​

t1/2​=ln(2)/k

This is a fundamental formula for calculating half-life. In a real-world application, such as a program simulating radioactive decay, you would use a predefined constant for ln(2) to avoid recalculating it every time, which is both more efficient and more precise.

Scenario with a table:

Imagine a nuclear power plant simulation. We have several radioactive isotopes with different decay constants. We need to calculate their half-lives to determine when they are safe to handle. We'll use the constant for ln(2), which is approximately 0.693147.