3. For a series of experiments a tank is split into two compartments by a semi-permeable membrane. The first component has a volume of. V] litres and is filled with a solution of a chemical with a concentration of CI moles per re. The second con1Partilleiit has volume of V2 litres and is filled with a solution of the same chemical, but with a concentration of C2 moles Per litre. The chemical diffuses through the membrane Lt 1.1, rate proportional to the difference in concentration between the two solutions and 111 a direction such as to equalise the two concentrations. The differential equations modelling the diffusion process are:
dC1 k
dt
dC2 k — C2) and dt V2
where I is the time in hours and k is a constant of diffusion measured in litres per hour.
.. \, In a particular experiment VI = 20, V2 = 5, k = 0.2 and, imtially, = 3 and C2 .= 0.
(a) Using matrix methods, determine the particular solution to these equations.
(b) Determine both concentrations after 4 hours, their final value and the approximate time the concentrations reach that final value.
