what is the formula to calculate half life?

Question

Tritium has a half-life of 12.5 years. If you had a sample of 8.00 grams of tritium today, how many grams of tritium would you have in 50.0 years?

Answer 1

Hi Leah,

To describe radioactive decay, we can use the concept of natural decay from mathematics. The derivation comes from Calculus, so I will skip that unless you would like me to show you (let me know)

The equation is:

N = N₀ e^-kt

where N is the amount of material at any point in time, N₀ is the starting amount of the material (at t=0), e is Euler's number, k is the decay constant (specific to the material of interest), and t is time

You are given that the half-life is 12.5 years, this means that if we start with X amount of material and wait 12.5 years, we will only have (1/2)X material left.

Let's model this half-life using our equation above:

N = N₀ e^-kt

N/N₀ = e^{-k(12.5 years)} = 1/2

note: N/N₀ = 1/2 because this is what half-life means: the final N is half of N₀, so dividing them gives 1/2

So this becomes

1/2 = e^{-k(12.5 years)}

We can solve this for k by taking the natural logarithm of both sides, this gives:

ln(1/2) = ln (e^{-k(12.5 years)} )

ln(1/2) = -k(12.5 years)

k = - ln(1/2) / 12.5 years = 0.055 1/years (1/years is the unit for this decay constant)

We just derived our decay constant for this material, so now our decay equation can be written:

N = N₀ e^-0.055t

To solve our problem, we plug in 8.00 grams for N₀, and 50.0 years for t in the above equation and solve for N as our answer