reaction order

(A -> B)

A(t)
linear graph
t 1/2

Oth

A = - k t + Ao

A vs. time

slope = - k

[A]o / (2k)
1st

ln A = - k t + ln Ao

ln A vs. time

slope = - k

ln 2 / k
2nd (1 / A) = k t + (1 / Ao)

(1 / A) vs. time

slope = k

1 / (k [A]o)

 


Derive equation for 1st and 2nd order reaction with a single reactant [having more than a single reactant, raises the complexity, so it won’t be shown]

For a 1st order reaction: A -> B,

[1]

where describes the slope / derivative of the curve in a graph of [A] vs. time and is referred to as a differential equation. Given the expression for , our goal is to determine A(t), which is a function that describes the graph plotting [A] versus time, t.  To "solve" the above differential equation, “take the integral” of [1]:

1_ODE_soln_1

1_ODE_soln_2

ln A = - k t + C

at t = 0, [to evaluate the value of C]

ln Ao = C

substitue

ln A = - kt + ln Ao

ln A - ln Ao = - k t

1_ODE_soln_3

or A = A o e- k t  [see below for an alternative equation]

 

For the 2nd order reaction: A -> B,

where describes the slope / derivative of the curve in a graph of [A] vs. time. As above, solving this differential equation:

2_ODE_soln_1

2_ODE_soln_2

2_ODE_soln_3

at t = 0,

2_ODE_soln_4

substitue,  

2_ODE_soln_5

2_ODE_soln_6

 

A similar method would be used to treat a 0th order reaction: A -> B [not shown]

Notice that A(t) for a 1st& 2nd order reaction is on the "ap equation sheet".

non-calculus derivation of rate law.

derive rate law from Reaction mechanism  -  steady state approximation method applied to a biology example.  [file in pdf format]

 

a slightly more realistic treatment for the reaction A <--> B, which unfortunately has a somewhat complicated analysis; mathematica simulation.

 

Alternative equation describing the solution to the above 1st order differential equation (supplied by Su Tran, Ms. Saucedo, & Mr. Ring;  2006)

 

 

Using “my equation”

 

 

where

 

At the half-life,

 

 

substituting [2] into [1],