Answer:
The function that models the mass of the carbon-14 sample remaining t years since the initial measurement is
M(t) = 741 e⁻⁰•⁰⁰⁰¹²¹ᵗ
with M(t) in grams and t in years.
Step-by-step explanation:
Radioactive reactions always follow a first order reaction dynamic
Let the initial concentration of Carbon-14 be M₀ and the concentration at any time be M
(dM/dt) = -kM (Minus sign because it's a rate of reduction)
(dM/dt) = -kM
(dM/M) = -kdt
∫ (dM/M) = -k ∫ dt
Solving the two sides as definite integrals by integrating the left hand side from M₀ to M and the Right hand side from 0 to t.
We obtain
In (M/M₀) = -kt
(M/M₀) = e⁻ᵏᵗ
M(t) = M₀ e⁻ᵏᵗ
Although, we can obtain k from the information on half life.
For a first order reaction, the rate constant (k) and the half life (T) are related thus
T = (In2)/k
The half life is the time taken for the radioactive substance to decay to hAlf of its original amount, and according to the question, T = 5730 years
k = (In 2)/5730 = 0.000120968 /year. = 0.000121 /year
M(t) = M₀ e⁻ᵏᵗ
k = 0.000121 /year, M₀ = 741 grams
The equation then becomes
M(t) = 741 e⁻⁰•⁰⁰⁰¹²¹ᵗ
with M(t) in grams and t in years.
Hope this Helps!!!
