82.8k views
0 votes
Carbon-141414 is an element which loses exactly half of its mass every 573057305730 years. The mass of a sample of carbon-141414 can be modeled by a function, MMM, which depends on its age, ttt (in years). We measure that the initial mass of a sample of carbon-141414 is 741741741 grams. Write a function that models the mass of the carbon-141414 sample remaining ttt years since the initial measurement.

2 Answers

1 vote

Answer:

M(t)=741*(1/2)^t/5730

User Trond Kristiansen
by
8.7k points
3 votes

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.

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!!!

User Andre Pastore
by
7.8k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories