Natural Growth and Decay dP = kP → P = P0 e ! kt dt
Example 1• The annual growth rate of a town is 3% of its population. In 2000 the population is 13000. • Show that the population can be calculated by the general growth and decay model • Hence ﬁnd the population in 2010 (to nearest 100)
Example 2 - Half LifeThe half life a a radio-active substance is 200years. Initially there are 200g. How muchmass remains after 300 years.
Law of Limited Growth and DecayIn reality the population growth may berestricted due to death, diseases orlimitations in food, water etc DP If = k(P − A) → P = A + Be ! kt dt
Wilhelmy’s LawThe velocity of a reaction is proportionalto difference between the initialconcentration of the reagent (a) and theamount transformed at any time (x) dx = k(a − x) 0 ≤ x ≤ a ! dt
Wilhelmy’s Law ExampleIn a particular reaction theconcentration of a reagent is10 (g) and 4 (g) have alreadybeen transformed. Find theamount left after 5 minutes
General ExampleBrine containing 2 g of salt per litre ﬂowsinto a tank initially ﬁlled with 50 litres ofwater containing 10 g of salt. If the brineenters the tank at 5 litres /minute, theconcentration is kept uniform by stirring, andthe mixture ﬂows out at the same rate, ﬁndthe amount of salt in the tank after 10minutes.