The half-life of a radioactive isotope is the amount of time it takes for a quantity
of radioactive material to decay to one-half of its original amount.
i) The half-life of Carbon 14 (C-14) is 5230 years. Determine the decay-rate
parameter � for C − 14.
To determine the decay-rate parameter (λ) for Carbon-14 (C-14), we can use the formula for exponential decay:
N(t) = N₀ * e^(-λ * t)
Where:
N(t) = the quantity of radioactive material remaining at time t
N₀ = the initial quantity of radioactive material
λ = decay-rate parameter
t = time
In this case, we are given the half-life (T₁/₂) of C-14, which is 5230 years. The half-life is the time it takes for the material to decay to one-half of its original amount.
Using the formula for half-life:
T₁/₂ = (ln(2))/λ
We can rearrange the equation to solve for λ:
λ = (ln(2))/T₁/₂
Substituting the given value of T₁/₂ = 5230 years into the equation:
λ = (ln(2))/5230
Calculating the value using a calculator or math software, we find that:
λ ≈ 0.0001322 years^(-1)
So the decay-rate parameter for C-14 is approximately 0.0001322 years^(-1). This value represents the rate at which C-14 decays over time.