An initial amount ? of a tracer (such as a dye or a radi
Question and Solution
An initial amount ? of a tracer (such as a dye or a radioactive isotope) is injected into Compartment 1 of the two-compa
83 % (127 Review)
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.
Your answer will be ready within 2-4 hrs. Meanwhile, check out other millions of Q&As and Solutions Manual we have in our catalog.
Get immediate access to 24/7 Homework Help, step-by-step solutions, instant homework answer to over 40 million Textbook solution and Q/A
Pay $7.00/month for Better Grades
Crazy for Study is a platform for the provision of academic help. It functions with the help of a team of ingenious subject matter experts and academic writers who provide textbook solutions to all your course-specific textbook problems, provide help with your assignments and solve all your academic queries in the minimum possible time.
Disclaimer: Crazy For Study provides academic assistance to students so that they can complete their college assignments and projects on time. We strictly do not deliver the reference papers. This is just to make you understand and used for the analysis and reference purposes only.