of Solutions rate equations
The analytic method:
dN = ! kN Write the differential equation dt dN = ! kdt Rearrange N N Integrate ln = !kt N0
Simplify
N = N0 e! kt
The integration step depends, among other things, on having only one variable on each side of the equation.
where the analytic method fails
Consider the following reaction mechanism for combustion of hydrogen in a chlorine atmosphere:
k1 Cl2 !!"Cl + Cl k2 Cl + H 2 !!" HCl + H
k3 Cl2 + H !! HCl + Cl "
k4 Cl + H + M !!" HCl + M
k5 Cl + Cl + M !!"Cl2 + M k6 H + H + M !!" H 2 + M
where the analytic method fails
Now examine the related rate equations:
Cl2 !!"Cl + Cl
k1 k2 Cl + H 2 !!" HCl + H
d[Cl2 ] = !k1 [Cl2 ] dt ! k3 [Cl2 ][ H ]
2
Cl2 + H !! HCl + Cl "
k3
+ k5 [Cl2 ] [ M ]
k4 Cl + H + M !!" HCl + M
k5 Cl + Cl + M !!"Cl2 + M k6 H + H + M !!" H 2 + M
d[ HCl] = k2 [Cl ][H 2 ] dt +k 3 [Cl2 ][H ]
+k 4 [ H ][Cl][ M ]
where the analytic method fails
...the related rate equations...
k1 Cl2 !!"Cl + Cl
Cl + H 2 !!" HCl + H
k2
k3 Cl2 + H !! HCl + Cl "
d[Cl] = 2k1 [Cl2 ] dt !k 2 [ H2 ][Cl ] +k 3 [Cl2 ][H ]
2
Cl + H + M !!" HCl + M
k4
Cl + Cl + M !!"Cl2 + M
k5 k6 H + H + M !!" H 2 + M
!k 4 [Cl ][ H ][ M ] !2k 5 [Cl] [ M ]
where the analytic method fails
...the related rate equations...
k1 Cl2 !!"Cl + Cl k2 Cl + H 2 !!" HCl + H
d[ H ] = k 2 [H 2 ][Cl] dt ! k3 [Cl2 ][ H ]
2
! k4 [Cl ][H ][M ] !2k 6 [ H ] [ M]
k3 Cl2 + H !! HCl + Cl "
k4 Cl + H + M !!" HCl + M
k5 Cl + Cl + M !!"Cl2 + M k6 H + H + M !!" H 2 + M
d[ H 2 ] = ! k2 [ H 2 ][Cl] dt +k 6 [H ]2 [M ]
where the analytic method fails
k1 Cl2 !!"Cl + Cl k2 Cl + H 2 !!" HCl + H
k3 Cl2 + H !! HCl + Cl "
k4 Cl + H + M !!" HCl + M
k5 Cl + Cl + M !!"Cl2 + M k6 H + H + M !!" H 2 + M
And, if that isn't bad enough, the rate constants of all of these reactions will change during the course of the reaction unless the reaction is isothermal.
where the analytic method fails
k1 Cl2 !!"Cl + Cl k2 Cl + H 2 !!" HCl + H
k3 Cl2 + H !! HCl + Cl "
k4 Cl + H + M !!" HCl + M
k5 Cl + Cl + M !!"Cl2 + M k6 H + H + M !!" H 2 + M
And, if that isn't bad enough, the rate constants of all of these reactions will change during the course of the reaction unless the reaction is isothermal. This is, of course, what makes chemistry interesting.
Numerical integration
Euler's method: Write the differential equation
dN = ! kN dt !N = "kN !t
Replace with a nite difference equation Rearrange
!N = "kN!t
Solve N (t + !t) = N + !N Repeat until done
N ( t + 2!t ) = N ( t + !t ) + !N !
N ( t + 3!t ) = N ( t + 2!t ) + !N
Numerical integration
Euler's method: Advantages: dN/dt can be any function whatsoever. We don't need its integral, only its value.
Numerical integration
Euler's method: Advantages: dN/dt can be any function whatsoever. We don't need its integral, only its value. dN/dt can depend on any number of variables.
Numerical integration
Euler's method: Advantages: dN/dt can be any function whatsoever. We don't need its integral, only its value. dN/dt can depend on any number of variables. Disadvantages: We don't gain the benet of all that known is about the analytic solutions, if they exist.
Numerical integration
Euler's method: Advantages: dN/dt can be any function whatsoever. We don't need its integral, only its value. dN/dt can depend on any number of variables. Disadvantages: We don't gain the benet of all that is known about the analytic solutions, if they exist. The method is BORING!
Igor implementation
Set initial conditions Dene rate constants Set integration parameters
Igor implementation
Set initial conditions Most chemical kinetic problems are initial condition differential equations in which the starting concentrations are known.
Igor implementation
Set initial conditions Dene rate constants Some estimate of the rate constants will be required to make this method work. These will often be determined from experiments on simpler systems.
Igor implementation
Set initial conditions Dene rate constants Set integration parameters Decide what t will be. Decide how often you want to examine the results of your integration
Igor implementation
Set initial conditions Dene rate constants Set integration parameters Integrate Use the Euler method for simplicity of implementation; other methods may be faster.
Igor implementation
Set initial conditions Dene rate constants Set integration parameters Integrate Monitor the results Either numerical or graphical methods may be used, depending on the objective.
Igor implementation
Set initial conditions Dene rate constants Set integration parameters Integrate Monitor the results ...Now decide what you have learned about numerical integration
Igor implementation
Set initial conditions Dene rate constants Set integration parameters Integrate Monitor the results ...Now decide what you have learned about numerical integration about chemistry
Igor implementation
Set initial conditions //create a wave to hold the answer Wave nO3 //and a wave for the change in [O3] Wave dnO3 initial concentrations: variable nO3Start=1e12 nO3[0]=...