Kinetics

Kinetics is the study of the rate at which chemical reactions occur. Reactions occur because molecules collide with one another, but not all collisions between molecules produce a reaction. This is because molecules must have at least the activation energy and collide with the right orientation. To understand kinetics, we must first understand what we mean by 'activation energy'.

k₁

A + B ⇌ C + D

k₂

In a typical chemical reaction, reactants are converted into products, and the forward rate of reaction is the number of moles of C (or D) produced per litre per second. k₁ is called the forward reaction's rate constant and k₂ is the backward reaction's rate constant. At constant temperature, the rate of reaction is proportional to the rate constant and the concentration of reactants.

The rate of reaction is the gradient of the tangent of the concentration versus time graph.

The rate of reaction slows over time, because the reactants run out and because products often give negative feed-back in metabolic pathways. We do our rate measurement at the very moment we mix the reactants, and call this the initial rate of reaction, for which we will use the symbol V.

The rate of reaction slows over time.

There are several factors affecting the rate constant, one of which is the activation energy. These factors are summarised in the Arrhenius equation, which allows us to calculate the rate constant:

k = Z p e ^{− ∆G‡} ⁄ R T

Z - frequency of collisions (Z) between molecules (s⁻¹). [It is a common misconception that increasing the temperature increases Z. Although true, the effect is very small, and insignificant over the range of physiological temperatures].
p - fraction of molecules colliding with the correct orientation. Z p is often combined and written as 'A'.
∆G^‡ (also known as E_a) - activation energy of the reaction (J mol⁻¹).
T - Temperature (K).

The activation energy is the energy required to form the transition state from the reactants:

H₂ + ½O₂ (reactants) → 2H + O (transition state) → H₂O (products)

This is often expressed as a reaction co-ordinate diagram:

Reaction coordinates show the progress of a reaction from high G reactants via very high G intermediates to low G products.

In general, a reaction coordinate diagram shows the progress of a reaction: the x-axis shows the reactants, intermediates and products of the reaction, and the y-axis shows the energy of these chemical species.

Coordinate diagrams show the progress of a reaction on the x-axis, and the energy of the state on the y-axis.

The proportion of molecules in a system with energy greater than or equal to the activation energy is determined by the Maxwell-Boltzmann distribution. The Boltzmann distribution shows how many molecules can react at a given temperature. The area under the graph shows the number of molecules with a given energy.

The Maxwell-Boltzmann distribution shows that the most probable molecular energy increases with temperature.

At elevated temperatures, note that the area under the curve with velocity (i.e. kinetic energy) sufficient to react (say, 40) increases: the area bounded by the curve and the line x = 40 increases. This means that at higher temperatures, more molecules are able to react and the reaction rate will increase.

The Boltzmann distribution shows the proportion of molecules with a particular energy or speed at a given temperature.

If you've not done A-level biology, you may not have come across Q₁₀, but if you have, you'll almost certainly have been told that:

The rate of a chemical reaction doubles for every ten degree rise in temperature.

This is what we proper scientists term a 'lie'. It's is approximately true that for an average chemical reaction, the rate (k) of reaction roughly doubles or even triples for every ten degree rise in temperature (T), but all those italics are important. The equation determining how fast two reactions run when compared to one another is:

Q10 equation.

Q₁₀ is the k₁/k₂ ratio when T₁ and T₂ are 10°C apart. For a reaction that doubles in rate between 0°C and 10°C, the Q₁₀ 'law' will only be true if the activation energy (E_a) of the reaction is 44.5 kJ mol⁻¹. Note that Q₁₀ is also dependent on temperature: it actually gets smaller as the temperature increases. The Q₁₀ 'law' is a lie: laws are there for the guidance of the wise and the blind obedience of fools. See here for a more mathematical deconstruction of this fallacy.

The rate constants are themselves related to the equilibrium constant. For the reaction:

A ⇌ B

With forward rate constant k₁ and backwards rate constant k₋₁:

d[A] ⁄ dt = − k₁[A]

d[B] ⁄ dt = − k₋₁[B]

At equilibrium, these rates will be equal, i.e.:

d[A] ⁄ dt = d[B]/dt

− k₁[A_eq] = − k₋₁[B_eq]

The equilibrium constant is defined as the mass action ratio at equilibrium, hence, it is the ratio of the rate constant for the forward reaction to theat of the reverse reaction:

K_eq = k₁ ⁄ k₋₁ = [B]_eq ⁄ [A]_eq

Order of reaction

The rate of reaction (ROR) depends both on the rate constant (and hence the temperature and activation energy), and also on the concentrations of reactants. The way the rate varies with the concentration of reactants is called the order of reaction. Reactions may be described as zeroeth, first, second, etc., order.

If ROR does not depend on the concentration of A, it is said to be zeroeth order w.r.t. A.
If ROR increases linearly with the concentration of A, it is said to be first order w.r.t. A.
If ROR increases with the square of the concentration of A, it is said to be second order w.r.t. A.
If ROR depends linearly on both A and B, it is first order w.r.t. A and B, and second order overall.

Zeroeth order reaction

A → C

ROR = k

A zeroeth order reaction does not speed up with increasing reactant concentration.

In a zeroeth order reaction, the rate of reaction is independent of the reactant concentrations. Although this sounds weird, it is very common in enzyme catalysed reactions: when there's only a little enzyme present, adding more substrate makes no difference, because the enzyme is saturated.

First order reaction

A → C

ROR = k [A]

A first order reaction shows a linear relationship between reactant concentration and the rate of reaction.

First order reactions are very important, because they are typical for reactions of the form A → products, where there is only one reactant involved e.g. radioactive decay. The concentration vs. time curve is exponential, with a half life t_½ equal to ln 2 ⁄ k. There is an exponential fall in reactant concentration over time.

Half life is the time taken for half of the reactants to be used up.

Second order reaction

2A → C + D

ROR = k [A]²

A second order reaction shows parabolic kinetics.

Although a simple second order reaction like this isn't so common,

A + B → C + D

ROR = k [A] [B]

is common. The reaction overall is termed second order, because it depends on the concentration of two reactants. It is first order with respect to either reactant individually.

Pseudo-first order reaction

A + B → C + D

ROR = k [A] [B]

If [A] is very much higher than [B], the reaction rate will appear to be almost entirely governed by [B]. The reaction rate will appear to look like:

ROR = k′ [B]

This is termed pseudo-first order w.r.t. B and is important in understanding enzyme kinetics.

Putting it all together

For a simple one-stage reaction like:

NH₂CH₂COO⁻ + H⁺ → NH₂CH₂COOH

The rate of reaction will be:

ROR = k [NH₂CH₂COO⁻] [H⁺]

This is first order w.r.t. both glycine and the proton, and second order overall. See that the ROR depends on both the rate constant k and on the concentrations of reactants.

Catalysis

Catalysts are chemicals that increase the rate of a chemical reaction, without being themselves permanently altered by the reaction. They work by lowering the activation energy, and/or binding reactants in favourable orientations. Catalysts do not get used up by the reaction (although 'wear-and-tear' may eventually 'poison' them), nor do they change the equilibrium constant K_eq.

k₁

A + B ⇌ C + D

k₂

At equilibrium:

k₁ [A] [B] = k₂ [C] [D].

K_eq= [C] [D] ⁄ [A] [B] = k₁ ⁄ k₂.

Catalysts must increase both the forward and the backward rates equally.

The way that catalysts actually work is by providing an alternative route to the products, which has a lower activation energy than the uncatalysed reaction, as in this reaction coordinate:

Note that the transition state is of lower energy in the catalysed reaction.

A lower, catalysed activation energy means that more molecules have the energy to react, as you can see in the Boltzmann distribution.

With a lower activation energy, the area under the Boltzmann distribution increases.

Test yourself

Draw a reaction coordinate diagram to compare the dissolution of carbon dioxide into a bucket of water, and the same reaction catalysed by the enzyme carbonic anhydrase.
CO₂ + H₂O → H⁺ + HCO₃⁻
Write down the rate law for this equation, and state what order it has. Why does the catalysed reaction run faster?

Answers

Science

Reference

Steve's Place

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