# 5.2.1 Rate Equations

### Rate Equations

• The rate of reaction refers to the change in the amount or concentration of a reaction or product per unit time and can be found by:
• Measuring the decrease in the concentration of a reactant OR
• Measuring the increase in the concentration of a product over time
• The units of rate of reaction are mol dm-3 s-1 #### Rate equation

• The thermal decomposition of calcium carbonate (CaCO3) will be used as an example to study the rate of reaction

CaCO3 (s) → CaO (s) + CO2 (g)

• The rate of reaction at different concentrations of CaCO3 is measured and tabulated • A directly proportional relationship between the rate of the reaction and concentration of CaCO3 is observed when a graph is plotted Rate of thermal decomposition of CaCO3 over the concentration of CaCO3 • The rate of reaction for the thermal decomposition of CaCO3 can also be written as:

Rate of reaction = k x [CaCO3]

• The proportionality constant k is the gradient of the graph and is also called the rate constant
• The rate equation is the overall expression for a particular reaction without the ‘x’ sign

Rate of reaction = k [CaCO3]

• Rate equations can only be determined experimentally and cannot be found from the stoichiometric equation

Rate of reaction = k [A]m [B]n

[A] and [B] = concentrations of reactants

m and n = orders of the reaction

• For example, the rate equation for the formation of nitrogen gas (N2) from nitrogen oxide (NO) and hydrogen (H2) is rate = k [NO]2 [H2]

2NO (g) + 2H2 (g) → N2 (g) + 2H2O (g)

rate = k [NO]2 [H2]

• As mentioned before, the rate equation of the reaction above cannot be deduced from the stoichiometric equation but can only experimentally be determined by:
• Changing the concentration of NO and determining how it affects the rate while keeping [H2] constant
• This shows that the rate is proportional to the square of [NO]

Rate = k1 [NO]2

• Then, changing the [H2] and determining how it affects the rate while keeping [NO] constant
• This shows that the rate is proportional to [H2]

Rate = k2 [H2]

• Combining the two equations gives the overall rate equation (where k = k1 + k2)

Rate = k [NO]2 [H2]

#### Order of reaction

• The order of reaction shows how the concentration of a reactant affects the rate of reaction
• It is the power to which the concentration of that reactant is raised in the rate equation
• The order of reaction can be 0, 1,2 or 3
• When the order of reaction of a reactant is 0, its concentration is ignored
• The overall order of reaction is the sum of the powers of the reactants in a rate equation
• For example, in the following rate equation, the reaction is:

Rate = k [NO2]2[H2]

• Second-order with respect to NO
• First-order with respect to H2
• Third-order overall (2 + 1)

#### Half-life

• The half-life (t1/2) is the time taken for the concentration of a limiting reactant to become half of its initial value

#### Rate-determining step & intermediates

• The rate-determining step is the slowest step in a reaction
• If a reactant appears in the rate-determining step, then the concentration of that reactant will also appear in the rate equation
• For example, the rate equation for the reaction below is rate = k [CH3Br] [OH]

CH3Br + OH → CH3OH + Br

• This suggests that both CH3Br and OH take part in the slow rate-determining step
• This reaction is, therefore, a bimolecular reaction
• Unimolecular: one species involved in the rate-determining step
• Bimolecular: two species involved in the rate-determining step
• The intermediate is derived from substances that react together to form it in the rate-determining step
• For example, for the reaction above the intermediate would consist of CH3Br and OH The intermediate is formed from the species that are involved in the rate-determining step (and thus appear in the rate equation)

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