An initial amount ? of a tracer (such as a dye or a radi

An initial amount ? of a tracer (such as a dye or a radioactive isotope) is injected into Compartment 1 of the two-compartment system shown in Figure 6.1.10. At time t > 0, let x1(t) and x2(t) denote the amount of tracer in Compartment 1 and Compartment 2, respectively. Thus under the conditions stated, x1(0) = ? and x2(0) = 0. The amounts are related to the corresponding concentrations ?1(t) and ?2(t) by the equations x1 = ?1V1 and x2 = ?2V2, (i) where V1 and V2 are the constant respective volumes of the compartments. The differential equations that describe the exchange of tracer between the compartments are dx1 dt = ?k21?1 + k12?2 (ii) dx2 dt = k21?1 ? k12?2, or, using the relations in (i), dx1 dt = ?L21x1 + L12x2 (iii) dx2 dt = L21x1 ? L12x2, where L21 = k21?V1 is the fractional turnover rate of Compartment 1 with respect to 2 and L12 = k12?V2 is the fractional turnover rate of Compartment 2 with respect to 1. (a) Use Eqs. (iii) to show that d dt[x1(t) + x2(t)] = 0 and therefore x1(t) + x2(t) = ? for all t ? 0, that is, the tracer is conserved. (b) Use the eigenvalue method to find the solution of the system (iii) subject to the initial conditions x1(0) = ? and x2(0) = 0. (c) What are the limiting values x?1 = limt?? x1(t) and x?2 = limt?? x2(t)? Explain how the rate of approach to the equilibrium point (x?1, x?2) depends on L12 and L21. (d) Give a qualitative sketch of the phase portrait for the system (iii). (e) Plot the graphs of x?1?? and x?2?? as a function of L21?L12 ? 0 on the same set of coordinates and explain the meaning of the graphs.