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.

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