Differential Equation with 4 states

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SUMMARY

The discussion centers on solving a differential equation system for a reaction scheme with four states: A, B, C, and D, with specific forward and reverse rate constants (K(1), K(-1), K(2), K(-2), K(3), K(-3)). The equations governing the system include conservation of mass and rate equations for each state. The user seeks expressions for A, B, C, and D as functions of time (t) and the rate constants but is struggling with rearranging and solving the equations. The need for potential substitution methods or integration factors is highlighted as a possible approach to derive the required expressions.

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  • Understanding of differential equations and their applications in chemical kinetics.
  • Familiarity with rate constants and equilibrium concepts in reaction schemes.
  • Knowledge of integration techniques, particularly integration factors.
  • Basic grasp of conservation laws in chemical systems.
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  • Study the method of integration factors for solving linear differential equations.
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This discussion is beneficial for chemists, physicists, and students studying chemical kinetics, particularly those dealing with complex reaction mechanisms and differential equations.

Coomb Raider
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Dear Physics Forums,

I have a reaction scheme with 4 states where A<->B, B<->C and B<->D:
C
/
A - B
\
D

This proceeds to equilibrium from a start point of A = 1 and B, C, D = 0. Forward and reverse rate constants for A<->B are K(1) and K(-1), for BC: K(2), K(-2) and BD: K(3), K(-3). I can therefore write out quite a few equations:

A(t)+B(t)+C(t)+D(t) = 1
dA/dt = B(t) * K(-1) - A(t) * K(1)
dC/dt = B(t) * K(2) - C(t) * K(-2)
dD/dt = B(t) * K(3) - D(t) * K(-3)
dB/dt = A(t) * K(1) + C(t) * K(-2) + D(t) * K(-3) - B(t) *( K(3)+K(2)+(K(-1) )

at t=0: A=1, B=0, C=0, D=0, dA/dt = -A(t)K(1)
at t=inf: A*K(1) = B*K(-1), C*(K-2) = B*K(2), D*(K-3) = B*K(3)

What I need are 4 expressions defining A, B, C and D in terms of t and the rate constants. I've hit a brick wall pretty quickly just trying to rearrange and solve using e.g. integration factors, do I have enough expressions to pull this off? Perhaps a substitution factor approach is needed, but it's been 15 years since I was doing calculus on a regular basis.

Any level of help appreciated from gentle pointers to just doing it for me, I'm not proud!
 
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OK, the diagram didn't work. The slashes were supposed to run from state B upwards to C and downwards to D.

Apologies
 

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