Question

The half-life of an isotope is 100 years. Use this information to determine the differential equation that describes the mass as a function of time. In other words m' = km where k is a constant and m(t) is the mass after t years.

Answer 1

Step-by-step explanation:

Half life is 100years

Given that

Rate of decay = In2/half life

Then,

k=In2/100

k=0.00693/year.

The same of decay of the mass is 0.00693/year.

The differential equation that describe this is

dm/dt=-km

Using variable separation

1/m dm =-kdt

Integrate both side

∫1/m dm = ∫-kdt

Inm = -kt+c

Take exponential of both side

m=exp(-kt+c)

m=exp(-kt)exp(c).

exp(c) is a constant, let say A. Then,

m=Aexp(-kt)

When t=0 the mass is m(0)

Then A=m(0)

m=m(0)exp(-kt)

For the material to decay to 10% of it original

i.e m/m(0)=10%=0.1

m/m(0)=exp(-kt)

0.1=exp(-kt)

Take In of both side

In(0.1)=-kt

t=-In(0.1)/k

Since k=0.00693/year

t=-In(0.1)/0.00693

t=332.26years