A certain radioactive material is known to decay at a rate p

A certain radioactive material is known to decay at a rate proportional to the amount present. Initially there is 50 milligrams of the material present and after 2 hours it is observed that the material has lost 10% of its original amount. (a) Find an expression for the amount of material remaining at any time t . (b) The amount of the material after 4 hours. (c) The time at which the material has decayed to one half of its initial amount.

Solution

dN/dt = -kN

dN / N = -k dt
integrate

ln(N) = -kt + C
initial conditions:
t = 0, N = 50mg
t = 2 hrs, N = (100 - 10)% of 50mg = 45mg

for t = 0
ln(N) = -kt + C
ln(50) = C

for t = 2
ln(N) = -kt + C
ln(45) = -2k + ln(50)
-2k = ln(45) - ln(50)
-2k = ln(45/50)
k = -0.5 ln(0.9)

after 4 hours
ln(N) = -kt + C
ln(N) = -[ -0.5 ln(0.9) ]t + ln(50)
ln(N) = 2ln(0.9) + ln(50)
ln(N) = ln(0.81) + ln(50)
ln(N) = ln(40.5)

N = 40.5mg