Lecture 3 - Integrated rate laws: how concentrations vary with time Integrated rate laws: how concentrations vary with time over the course of a reaction Flashcards
What info does integrated rate laws and half-lives give you
• Integrated rate laws
– give us concentrations as a function of time
– allow us to determine the order of reaction and rate coefficient from a graph of conc vs time
• Half-lives
– another way of characterising reaction rates
What is the area under the graph of rate = -d[A]/dt give you?
Gives you how much reactant has reacted after time t
Integrated rate laws: zeroth-order
How would you plot on a graph?
Rate = -d[A]/dt = k
d[A] = - k dt
[A] t
∫ d[A] = -k ∫ dt
[A]₀ 0
[A] - [A]₀ = - kt
[A] = [A]₀ - kt
Y = c mx
Integrated rate laws: first-order
How would you plot on a graph?
Rate = -d[A]/dt = k[A]
-( 1/[A] )d[A] = k dt
[A] t
∫ 1/[A] d[A] = -k ∫ dt
[A]₀ 0
Ln([A]/[A]₀) = - kt —> ln[A] = ln[A]₀ - kt (look at fraction in ln to know if gradient is +ve or -ve) units for k = s^-1
[A] = [A]₀e^-kt
Integrated rate laws: second-order
How would you plot on a graph?
For A -> P
Rate = -d[A]/dt = k[A]²
[A] t
∫ 1/[A]² d[A] = -k ∫ dt
[A]₀ 0
-1/[A] + 1/[A]₀ = - kt (multiply eq. by -1 to give +ve k on graph)
Y mx
1/[A] = 1/[A]₀ + kt
Y c mx
Units k = mol-1 dm3 s-1
Integrated rate laws: second-order
How would you plot on a graph?
For A + B -> P
When [A]₀ = [B]₀ also follows -d[A]/dt = k[A]²
If not -d[A]/dt = k[A][B]
Ln(([B]/[B]₀) / ([A]/[A]₀)) = ([B]₀-[A]₀)kt
Ln([B]/[A]) = ln([B]₀/[A]₀ + ([B]₀-[A]₀)kt
Ie a plot of Ln([B]/[A]) against time gives a straight line
Other bits of info on IRL
How to work out the order of a reaction from the graph plots
This is normally a much better method than the method of initial rates, especially for fast reactions.
There is always some ‘dead-time’ (starting at t=0 to a small value of t), so that initial rates are inherently inaccurate.
(Only use for slow reactions)
Plot [A] vs t, ln[A] vs t and 1/[A] vs t and whichever gives you the straightest line is the correct order ( just do to 2 sf)
Integrated rate laws -
when you monitor concentration of a product or
monitoring the absorbance of a reactant or product
For a reaction of A -> P or A + B -> P
A) where [P] is measured
Use eqs. - [P] = [A]₀ - [A] or [A] = [P]∞ - [P]
(And, as long as [A]₀ < [B]₀, the final conc of product is
[P]∞ = [A]₀ (if reaction goes to completion - either K is large or products are removed from reactants))
B) where abs is measured
Determine concentrations using abs is prop to conc (beer lambert law) A = Ecl
For gas phase reactions, gas pressures can be used instead of gas concentrations as pressure is prop conc
Always plot f[A] vs t EVEN if you are given [P] vs t as data!
Alternate formulation for [A]₀, [B]₀ and [A]₀ - [A]
[A]₀ = a
[B]₀ = b
[A]₀ - [A] = x
Ie first order —> ln(a-x) = lna - kt
Second order —> 1/a-x = 1/a + kt
What is a half life?
What would the half life be for first order
The half-life, t1/2 of a reactant is the time taken for it’s
concentration to drop to half its original value
ln[A] = ln[A]₀ - kt
Ln(0.5[A]₀) = ln[A]₀ - kt1/2
Kt1/2 = ln([A]₀/0.5[A]₀)
t1/2 = ln(2)/k - it a constant
How many half-lives does it take for a first order
reaction to be 99% complete?
% No. Of half lives
100 - 0
50 - 1
25 - 2
12.5 - 3
6.25 - 4
3.125 - 5
1.56 - 6
0.78 - 7
0.39 - 8
Therefore about y
Half lives - second order
1/[A] = 1/[A]₀ + kt
1/0.5[A]₀ = 1/[A]₀ + kt1/2
2 [A]₀ = 1/[A]₀ + kt1/2
t1/2 = 1/([A]₀k)
• The half-life depends on the initial concentration
• It will increase as reaction proceeds, i.e. after each
half-life, the next half-life will be doubled. (Ie first half life at 1s, then next at 3s then next at 7s)
• Not as useful as for 1st order
