Kinetics

Reaction rates

reaction rate $v$ desribes the speed of a reaction
unit: $mol\cdot{dm^{-3}}\cdot{s^{-1}}$

Calculation

Method 1

to calculate the rate of a reaction, the changes of concentration over time are used

$$v=\cfrac{\Delta{c}}{\Delta{t}}$$

where:
- $\Delta{c}$ is the change in concentration
- $\Delta{t}$ is the change in time
for a general reaction, the equation is more complex, since it includes all the possible reactants and products $$v=-\sum{\cfrac{\Delta{c_R}}{\nu_R\cdot\Delta{t}}}=\sum{\cfrac{\Delta{c_P}}{\nu_P\cdot\Delta{t}}}$$
where:
- $\Delta{c_R}$ is the change in concentration of a reactant
- $\Delta{c_P}$ is the change in concentration of a product
- $\Delta{t}$ is the change in time
- $\nu_R$ is the stoichiometric coefficient of a reactant
- $\nu_P$ is the stoichiometric coefficient of a product

Change in concentration over time

as the concentration of the product increases, the concentration of the reactant decreases as the time goes on
this relationship is not linear

Change of concentration in time of a reactant and a product

Method 2

the rate of a reaction is directly proportional to the concentration of the reactant
to calulate the rate of a reaction, the concentration in a given moment in time is used

$$v=k[A]^a$$

where:
- $k$ is the rate constant
- $[A]$ is the concentraion of the reactant $A$
- $a$ is the order of the reaction

Order of a reaction

order of a reaction is an experimentally derived value, which describes how much does the concentration of a reactant influence the rate of the reaction
Zeroth order reaction
- the concentration of the reactant does not affect the rate of the reaction
- $a=0$, hence:

$$v=k$$

First order reaction
- the rate of the reaction is directly proporitional to the concentration of the reactant
- as there is less reactant, the reaction gets slower and vice versa
- $a=1$, hence:

$$v=k[A]$$

Second order reaction
- the rate of the reaction is directly proportional to the square value of the concentration of the reactant
- as there is less reactant, the reaction gets slower and vice versa
- it is similar to the first order reaction
- $a=2$, hence:

$$v=k[A]^2$$

Changes in rate constant units

the $[k]$ changes with the different orders of the reaction:
- $a=0=>[k]=mol\cdot{dm^{-3}}\cdot{s^{-1}}$
- $a=1=>[k]=s^{-1}$
- $a=2=>[k]=dm^3\cdot{mol^{-1}}\cdot{s^{-1}}$

Rate law

rate law combines both methods of calculating the rate of a reaction

$$\cfrac{-\Delta[A]}{\Delta{t}}=k$$

Zeroth order reaction

$$[A]=-[A]_0-kt$$

First order reaction

$$[A]=[A]_0\cdot{e^{-kt}}$$

Second order reacion

$$[A]=[A]_0+\cfrac{1}{kt}$$

where $[A]_0$ is the concentration of reactant $A$ when $t=0$

Half-life of a reaction $t_{1/2}$

it is used to describe what time does it take till half of the reactant has undergone a reaction
Zeroth order reaction

$$t_{1/2}=\cfrac{[A]_0}{2k}$$

First order reaction

$$t_{1/2}=\cfrac{ln2}{k}$$

Second order reacion

$$t_{1/2}=\cfrac{1}{k[A]_0}$$

Collision model

Elementary reactions

elementry reactions are reactions which do not have an intermediate step
intermediate is a species which is neither a reactant nor a product
- it is usually quite unstable, although, in some cases, it can be isolated
these reactions can be represented by simple notation, yet their mechanism is very complex
their stoichiometric coefficients correspond with their order
- this is not the case for non-elementary reactions

Movement of particles resulting in a reaction

paticles react with one another by collidng - collision model
in monomolecular reactions (where one particle undergoes a chemical change), nothing collides
in bimolecular reactions (where two different particles interact), a collision happens
- whether a collision results in a reaction is determined by the space orientation of the particles
- the reaction also needs to overcome an energy barrier - activation energy
in reactions with more than two different particles, a succession of elementary reaction happens before the final product is formed
collision which do result in a chemical reaction are called effecive (otherwise, they are ineffective)

Reaction enrgy profile

activation energy $E_a$ is the energy needed to start a chemical reaction

Reaction coordinate

graphing the total energy levels of a reaction results in a reaction energy profile (reaction coordinate)

Elementary reaction coordinate

Elementary reaction energy profile

on it, three stages are distinguishable:
- R - energy of the reactants
- TS - energy of the transition state
- P - energy of the products
the difference between the energy of the products and the transition state is the activation energy

Multi-step reaction coordinate

Multi-step reaction energy profile

Transition state

transition state is a state, where the reactans for an activated complex
it is an intermediate which exists for a very short ammount of time and cannot be measured very well

Calculation of the rate constant using the activation energy

the rate constant can be calculated by the Arrhenius equation

$$k=Ae^{\frac{-E_a}{RT}}$$

where:
- $A$ is the pre-exponential factor
  - it is uniqe for every reaction
  - it needs to be determined experimentally
- $R$ is the universal gas constant
- $T$ is the absolute temperature
a newer equation was developed, which does not use any experimentl values - Eyrling equation
- it is more useful in the theoretical field

Implications of the Arrhenius equation

The higher the activation energy, the lower the rate constant and the lower the rate of the reaciton.
The higher the temperature, the faster the reaction.

Reaction mechanisms

reaction mechanism shows how exactly a reaction works with all of its intermediate steps
rate-determining step is the slowest reaction in the chain of all the intermediate elementary reactions
- to make the reaction go faster, this step needs to be adjsuted

Steady-state approximation

steady-state approximation assumes there is a moment in a more complex reaction where an intermediate is in an equilibrium where it gets created as fast as it gets consumed
it helps up determine the rate law for a complex reaction

Example

reaction: $2\ N_2O_5\longrightarrow{4\ NO_2+O_2}$
steps:
1. $2\ N_2O_5\leftrightharpoons{NO_2+NO_3}$
2. $NO_2+NO_3\longrightarrow{NO+NO_2+O_2}$
3. $NO_3+NO\longrightarrow{2\ NO_2}$
rates:
1. $v_1=k_1[N_2O_5]$, $v_{-1}=k_{-1}[NO_2][NO_3]$
2. $v_2=k_2[NO_2][NO_3]$
3. $v_3=k_3[NO_3][NO]$
calculating $[NO]$:
- $v_2=v_3$
- $k_2[NO_2][NO_3]=k_3[NO_3][NO]$
- hence: $[NO]=\cfrac{k_2[NO_2]}{k_3}$
calculating $[NO_3]$:
- $v_1=v_{-1}+v_2+v_3$
- $k_1[N_2O_5]=k_{-1}[NO_2][NO_3]+k_2[NO_2][NO_3]+k_3[NO_3][NO]$
- hence: $[NO_3]=\cfrac{k_1[N_2O_5]}{k_{-1}[NO_2]+k_2[NO_2]+k_3[NO]}$
- after adjustment ($[NO]$ substitution): $[NO_3]=\cfrac{k_1[N_2O_5]}{k_{-1}[NO_2]+2k_2[NO_2]}$
claculating the overall rate:
- $v=\cfrac{\Delta[O_2]}{\Delta{t}}=k_2[NO_2][NO_3]$
- after adjustment ($[NO_3]$ substitution): $v=\cfrac{\Delta[O_2]}{\Delta{t}}=\cfrac{k_1k_2[N_2O_5]}{k_{-1}+2k_2}=k[N_2O_5]$
- where $k=\cfrac{k_1k_2}{k_{-1}+2k_2}$

Catalysis

catalyst is a substance which bonds with the molecules of the reactant (either via covalent bonds or intermolecular forces) to form an intermediate which requires less activation energy than the activation energy of the reaction without it
catalyst forms an intermediate which then breaks down to the product and the catalyst or is kicked out by another substance
catalyst itself does not undergo any reaction and is regenerated in the end
catalysis needs to be taken into account when calculating the overall rate of the reaction
the reaction profile changes

Catalized reaction energy profile
catalysis is important in most natural processes and is enabled primarily by enzymes
- it is also used in countelss synthesis processes