Question

It is known that the mass, m(t), of a radio-active substance decreases as it decays, and that the equation governing this is m′(t)=−0.015m(t) where t is in years. What is the half-life? (Hint: Find an exponential function which solves the equation. You will then have a formula for the mass, so you can find the half-life.)

Answer 1

Answer:

m(t) = m₀ e⁻⁰•⁰¹⁵ᵗ

Half-Life = 46.21 years.

Step-by-step explanation:

Radioactive reactions always follow a first order reaction dynamic

Let the initial mass of radioactive substance be m₀ and the mass at any time be m

(dm/dt) = -Km (Minus sign because it's a rate of reduction)

The question provides K = 0.015 from the given differential equation

(dm/dt) = -0.015m

(dm/m) = -0.015dt

 ∫ (dm/m) = -0.015 ∫ 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) = -0.015t

(m/m₀) = (e^(-0.015t))

m = m₀ e^(-0.015t)) = m₀ e⁻⁰•⁰¹⁵ᵗ

m(t) = m₀ e⁻⁰•⁰¹⁵ᵗ

At half life, m(t) = (m₀/2), t = T(1/2)

(m₀/2) = = m₀ e⁻⁰•⁰¹⁵ᵗ

e⁻⁰•⁰¹⁵ᵗ = (1/2)

In e⁻⁰•⁰¹⁵ᵗ = In (1/2)

-0.015t = - In 2

t = (In 2)/0.015

t = (0.693/0.015)

t = 46.21 years

Half life = T(1/2) = t = 46.21 years.

Hope this Helps!!!