### Level

Chemistry background for second- or third-year physics degree students who have not previously studied chemistry to at least GCE A-level or equivalent.

### Content

• Concentration: definitions and units
• Rate equations
• Zeroth, first and second order reactions and their rate constants
• Collision theory
• Boltzmann factor
• Arrhenius equation
• Eyring theory
• Dynamic equilibrium
• Principle of detailed balance
• Equilibrium constant
• Acid-base equilibrium
• ### Prior knowledge

Chemistry

• The mole
• Symbols and notation for chemical reactions
• Acids and bases
• Molecular kinetic theory

• Maxwell-Boltzmann distribution
• Mean speed
• Boltzmann factor
• Diffusion
• Thermodynamics

• Gibbs energy
• ### Summary

• Concentration is defined in terms of the number of particles, or moles, of a substance per unit volume. The rate of a chemical reaction refers to the rate of change of concentration of products and reactants.
• In general, reaction rate is a power-law function of the concentration(s) of reactant(s). The constant of proportionality is known as the rate constant, and the power denotes the order of the reaction.
• The order of an elementary reaction can always be deduced from its stoichiometry.
• A steady state will result if the production rate of a substance is equal to its rate of removal in a further reaction.
• A bimolecular (elementary) reaction is described by simple collision theory in which reaction rate is proportional to collision frequency and to the fraction of ‘successful’ collisions. In such reactions, collision frequency is proportional to the concentrations of reactants and is also a function of other parameters such as the sizes of the reacting particles.
• The fraction of successful collisions is given by a Boltzmann factor involving the activation energy of an intermediate transition state. The Arrhenius equation relates the rate constant to the Boltzmann factor, and a pre-exponential factor incorporates all the other relevant parameters.
• Eyring theory provides a more precise thermodynamic interpretation of activation energy in terms of Gibbs energy.
• For any chemical reaction, the reverse reaction is also possible. An undisturbed closed system will eventually reach a dynamic equilibrium in which forward and reverse reaction rates are equal.
• At equilibrium, the concentrations of reactants and products are related via an equilibrium constant. One common type of chemical equilibrium is acid-base equilibrium, where the equilibrium constant is known as the acidity constant; pH is related to concentration and to acidity constant.
• The equilibrium constant is a function of temperature, and is conveniently expressed in terms of temperature and the Gibbs energy for the reaction at that temperature.
• ### Main narrative

This lecture introduces reaction kinetics (the study of reaction rates) and chemical equilibrium.

### 1 Reaction rates

1.1 How much?

In this section we are concerned with how the amount of a substance changes over time in a chemical reaction. An important quantity here is the concentration of a substance, namely the amount per unit volume, which can be expressed in various ways (Slide 2).

One way to express concentration is simply to state the number of particles (e.g. molecules, atoms, ions, nuclei) per unit volume. This number density is usually given the symbol n, often with a subscript to indicate the relevant particle e.g. nH if we are dealing with hydrogen atoms. Number density has units of volume-1, usually either m-3 or (particularly if density is low) cm-3. (Chemists often use the term ‘molecule’ here, rather than ‘particle’, for all entities including molecules, atoms, ions and radicals.)

When reactions take place in solution, molar concentration is often used, which refers to the number of moles per cubic decimetre, mol dm-3 (1 dm3 = 10-3 m3= 1 litre). 1 mole means NA of the relevant particles, where NA is Avogadro number i.e. 6.02 x 1023 mol-1.

(A solution containing 1 mole of a substance in 1 litre of solution is sometimes referred to as a molar solution, denoted 1M; similarly a 2M (2 molar) solution has 2 moles in 1 litre of solution, and so on. However this terminology is becoming outdated and is now rarely used by chemists, though older texts frequently use it.)

Chemical concentration is often denoted using square brackets, e.g. [H+] refers to the concentration of hydrogen ions. When concentration is denoted in this way, it is usually expressed in units of mol dm-3 or equivalent.

More generally, we will use c to represent concentration, with a subscript to indicate the substance of interest. Equations describing reaction rates, and concentrations as a function of time, can often be more neatly expressed using c than square brackets, but there are no firm rules and you will find a variety of notations in use. The units of c can be chosen according to context.

1.2 How fast?

The rate of a chemical reaction is defined as the rate at which a given concentration changes with time and is given the symbol v or J (we will use J here). Suppose there is a reaction

A + B → P

Equation 1

where one particle (or mole) of substance A reacts with one particle (or mole) of B to produce one particle (or mole) of a product P. The reaction rate can be defined in terms of the increasing concentration of P:

Equation 2

However, as cP increases, the concentrations of A and B decrease at the same rate, so we can also write the reaction rate in terms of cA or cB:

Equation 3a

Notice that J is always defined in such a way as to be positive. As A and B are being removed by the reaction, the rates of change of their concentrations, dcA/dt and dcB/dt, are negative.

More generally (Slide 3), a reaction might have the form

aA + bB → pP + qQ

Equation 4

i.e. a particles (or moles) of A react with b of B to produce p particles (or moles) of P and q of Q. The reaction rate is then defined as

Equation 3b

In general the reaction rate will depend on the concentrations of the reactants A and B as described by the reaction’s rate equation:

J = k cAx cBy

Equation 5

Here, x is the order of the reaction with respect to A, and y the order with respect to B. The overall order is x + y.

The parameter k is the rate constant or rate coefficient; the former is the more common terminology, but some insist that the latter is more correct as in general k is not constant but depends on temperature. If several reactions are being discussed, then a different symbol might be used for each rate constant (e.g. q, µ or λ); alternatively, constants can be distinguished using subscripts e.g. k4 would be the rate constant for Reaction 4 above.

In an elementary reaction (a single-step reaction involving one or two reactants), the order can always be correctly deduced from the stoichiometry of the reaction (the relative proportions of reactants). If Reaction 4 is elementary, then

J = k cAa cBb

Equation 6

The order is a with respect to A, b with respect to B, and the overall order is a + b.

The molecularity of an elementary reaction is defined as the number of molecules (or other particles) coming together in the reaction (they form an intermediate transition state - see Box 2). A bimolecular reaction refers to two particles coming together (a = b = 1 in Reaction 5), and a unimolecular reaction involves the decay or disintegration of a single particle.

However, in a complex reaction, which proceeds by a sequence of elementary chemical reactions, the rate equation cannot be written in terms of the chemical equation and the orders x and y in Equation 4. In such cases x and y must be determined experimentally e.g. x can be found by using a large excess of B (so that cB is kept almost constant as the reaction proceeds) and seeing how the rate varies with cA. In complex reactions, non-integer orders are possible.

It is not meaningful to define the overall molecularity of a complex reaction, though intermediate elementary reactions may be so characterised.

Some complex reactions have rate equations that happen to correspond to Equation 6, but more detailed kinetic studies have revealed that they proceed via two or more steps. So, while it is possible to deduce the form of the rate equation from stoichiometry if a reaction is known to be elementary, it is not possible to say whether a reaction is elementary just by looking at the form of its rate equation.

1.3 Orders of reactions

Mathematically, the simplest reaction is a zeroth order reaction. Suppose a substance, A, is entering a system from an external source at a constant rate (e.g. diffusing into a cell). Such a reaction has x = 0 and the rate equation for A is simply:

J = k

Equation 7

Here, k has dimensions of [concentration] [time-1]. On a molecular scale, suitable units for k would be cm-3 s-1 (or molecules cm-3 s-1). On a larger scale, the concentration and reaction rate would be expressed in terms of moles and k would have units mol dm-3 s-1 (Slide 4).

The (trivial) solution of the differential equation at (7) describes how the concentration of A changes with time:

cA(t) = csub>A(0) + kt

Equation 8

A first order reaction (x = 1) has a rate proportional to a single concentration. Examples of first order reactions include unimolecular chemical decomposition and radioactive decay (Slide 5). A general first order rate equation is:

J = kc

Equation 9

In such cases the dimension of k are [time-1] and units are usually

s-1. The solution to such equations is well known, and describes how concentration varies as a function of time:

c(t) = c(0)e-kt

Equation 10

The timescale for a first-order reaction can conveniently be characterised by a half-life:

t1/2 = ln(2)/k

Equation 11

Wiring down the rate equations for higher order reactions is also straightforward. For example, putting x = y = 1 into Equation 6 yields the rate equation for an elementary reaction of the form A + B → C (Reaction 1). Such a second order reaction is described by collision theory (see Box 2) and has a rate constant k with dimensions [concentration-1] [time-1], molecular units cm3 -1, molar units mol-1 dm3 s-1.

1.4 Reaction schemes

Sometimes the products of one reaction go on to take part in further reactions e.g.

A + B → C

Equation 12

C + D → E

Equation 13

In such reaction schemes (sequences of reactions), concentrations are described by a set of simultaneous differential equations. Sometimes these equations can be solved analytically, but often a numerical solution is required.

If the production and removal rates of a given substance are equal then its concentration remains in a steady state. A simple special case occurs when a substance is produced at a constant rate in a zeroth order reaction (with rate constant k0) and is removed in a first-order process (whose rate constant is k1) (Slide 6).

Equation 14

The concentration cB is described by the equation

Equation 15

and a steady state is reached when

cB = k0/k1

Similar reasoning can be applied to a sequence of first order reactions:

Equation 16

If k2 >> k1, B is very short-lived and its concentration is always small. When its rate of formation and removal become equal:

Equation 16

k1cA - k2cB = 0

Equation 17a

i.e.

cB = k1cA / k2

Equation 17b

An applet on the Thomas Group website from the Oxford University Chemistry Department illustrates the effect of varying the rate constants in such a sequence of two first-order reactions:

http://rkt.chem.ox.ac.uk/tutorials/kinetics/kinetics.html

Examples of reaction schemes in biological physics, and solutions to the appropriate differential equations, are discussed in the lecture The Physics of Biological Regulation.

### 2 Theories of reaction kinetics

This section discusses factors that determine the rate of a bimolecular reaction.

2.1 Simple collision theory

For a reaction to occur, two reactant particles must collide with one another, but not every collision results in a reaction. Analysis of the collision process reveals other factors, in addition to concentration, that influence a rate constant.

The simplest collision theory describes encounters between neutral particles in a fluid (gas or liquid). The frequency of collisions between two types of particle, A and B, can be derived by considering the situation shown in Slide 7. It is usual to consider the frequency of A-B collisions per unit volume, denoted by ZAB.

Suppose a spherical particle of type A, radius rA, moves at speed v through a collection of type-B particles, radius rB. At a time Δt, particle A will collide with all the B particles whose centres lie within a cylinder radius rA + rB, length vΔt. More generally rA + rB can be replaced by a single parameter, the collision diameter dAB which depends on the sizes of the two particles, giving a collision cross-section πdAB2.

The cylinder’s volume is πdAB2 vΔt so, if cB is the concentration of B, the number of collisions with B that a single particle of A experiences in this time is πdAB2 v Δt cB. The collision frequency for a single A particle is thus πdAB2 v cB. If there are cA particles of A per unit volume, then the overall A-B collision frequency per unit volume is

ZAB = πdAB2cAcB

Equation 18

where ⊽ is the mean relative A-B speed. For a collection of particles A and B in thermal equilibrium, ⊽ can be derived from molecular kinetic theory:

⊽ = (8kBT/πµ)1/2

Equation 19

where kB is the Boltzmann constant, T the absolute temperature and µ the reduced mass of A and B. (Note that ⊽ is the mean speed, not the root-mean-square speed.) Hence the collision frequency per unit volume is:

ZAB = πdAB2 (8kBT/πµ)1/2 cAcB

Equation 20

Notice that Equation 20 predicts that the collision frequency and hence the reaction rate should be directly proportional to both cA and cB; that is, the reaction between A and B is predicted to be second order (see Section 1 above).

2.2 Fraction of successful collisions

Not all collisions will result in a reaction. In many reactions, the key parameter that determines whether a collision will be ‘successful’ is the temperature and its relation to the energy changes that occur as a reaction takes place. We can illustrate the principle using the simple bimolecular reaction shown in Slide 8.

The reaction proceeds via an intermediate activated complex, or transition state, which can be represented as A- -B- -C, where all three atoms are close together and interacting. The potential energy here is greater than when the atom and molecule are widely separated, and the 3-D plot on Slide 8 shows how this energy might be plotted as a function of the particles’ separations. The lowest-energy path along this energy surface is known as the reaction coordinate. (Other paths are also known as reaction coordinates, but they are not favoured energetically so we don’t need to consider them.)

Energy changes in a reaction are often depicted using a reaction coordinate diagram – a plot of energy along the reaction coordinate. The reaction coordinate diagram shows an energy barrier, height Δε, where Δε is the activation energy for the a pair of particles forming a product. While we have so far considered reactions at the molecular level, activation energy is commonly measured and expressed using molar quantities, in which case it is generally represented as ΔE where

ΔE = NA Δε

Equation 21

and NA is Avogadro number, 6.02 x 1023 mol-1.

The reaction shown in shown in Slide 8 has a symmetrical reaction coordinate diagram, but in general the energy of the products differs from that of the reactants as in Slide 9. The presence of a catalyst can increase the probability of a reaction (and hence increase the reaction rate) by providing an intermediate stage with a reduced activation energy, as shown on the right of Slide 9.

Encounters between A and B will only result in a reaction if the initial kinetic energy of the particles (when widely separated) is greater than Δε. In thermal equilibrium, the probability, p, that the colliding particles have enough energy can be calculated from the Maxwell-Boltzmann distribution:

p = exp(-Δε/kBT)

Equation 22

where kB is the Boltzmann constant and T the temperature.

2.3 The Arrhenius equation

The collision frequency, Z, and the probability, p, of a collision resulting in a reaction can be brought together in a general expression for reaction rate, J:

J = Z exp(-Δε/kBT)

Equation 23

If collision frequency is as described by the simple theory above, then using (20) we can write:

J = πdAB2 (8kBT/πµ)1/2 cAcB exp(-Δε/kBT)

Equation 24

This can usefully be compared with the rate equation for a bimolecular reaction (Equation 4 with x = y = 1):

J = cAcB

Equation 25

Hence we can identify an expression for the second-order rate constant, k:

k = πdAB2 (8kBT/πµ)1/2 exp(-Δε/kBT)

Equation 26

Equation 26 (Slide 10) is one particular case of the Arrhenius equation for a rate constant:

k = A exp(-Δε/kBT)

Equation 27a

In this equation, named after Swedish chemist Santé Arrhenius (1859-1927), A is called by various names such as the A-factor, the frequency factor or the pre-exponential factor and depends on the type of reaction. Equation 27a is sometimes written in terms of the molar activation energy ΔE (Equation 21) and the molar gas constant R (R = NA kB) i.e.:

k = A exp(-ΔE/RT)

Equation 27b

In general A is only weakly temperature dependent, unlike the strongly temperature-dependent Boltzmann factor. Slide 10 shows an example of an Arrhenius plot (a graph of ln(k) against 1/T) which is close to a straight line, as would be the case for A independent of temperature. Such a plot can be used to determine values of activation energy and the ‘constant’ A.

Further developments in theories of reaction kinetics have sought to give a more precise thermodynamic interpretation of ‘activation energy’ as well as deriving appropriate frequency factors for various types of reaction. In practice, rate constants are measured empirically rather than being calculated from theory. But theoretical approaches are still very useful for predicting the likely magnitude of a rate constant, and comparison between theory and measurement can provide insight into the ways in which reactions actually proceed.

One example is Eyring-Polanyi theory, often simply known as Eyring theory (Slide 11) developed almost simultaneously by Henry Eyring, Michael Polanyi and M G Evans. This theory, which is applicable to bimolecular reactions, electronic transitions and light absorption, produces the equation:

k = kBT/h exp(-ΔG*/RT)

Equation 28

Here h is the Planck constant, ΔG* is the molar Gibbs energy of activation for the transition state, and R the molar gas constant.

Since

ΔG = Δ HT ΔS

Equation 29

we can write

k = kBT/h exp(-ΔS*/R) exp(-ΔH*/RT)

Equation 30

Comparing Equation 30 with the Arrhenius equation, we can identify the Eyring theory A-factor:

A = kBT/h exp(-ΔS*/R)

Equation 31

Although this A is temperature dependent, the dependence is relatively weak compared with the exponential Boltzmann factor.
If T ≈ 300 K, A ≈ 6 x 1012 s-1.

Finally, we should note that temperature is not always the sole factor that determines whether a collision is ‘successful’. If the reaction involves complex molecules (as is often the case in biological systems), a reaction might only occur if molecules approach one another with the correct orientation, which can considerably reduce the reaction rate below that predicted by simple collision theory. For such reactions, a modified A-factor needs to include the probability of molecules being correctly orientated.

3 Equilibrium

This section looks at situations where there is chemical equilibrium between a forward and a reverse reaction.

3.1 Forward and reverse

For any chemical reaction, for example:

aA + bB → pP + qQ

Equation 32a

the reverse reaction is also possible. (Here a, b, p and q represent the numbers of particles, or numbers of moles, of A, B, P and Q respectively.)

In a closed system, left to itself, concentrations will eventually settle down in such a way that there is dynamic equilibrium between the forward and reverse reactions; that is, their rates are equal, and the concentrations remain constant. (The equality between the forward and reverse reaction rates is known as the principle of detailed balance.) The system is represented thus:

aA + bB ⇌ pP + qQ

Equation 32b

If any of the reacting substances are added to, or removed from, the system, the resulting change of concentrations means that the reaction rates change, upsetting the equilibrium. But, as described by Le Chatelier’s principle, the system always responds in such a way as to restore equilibrium. For example, adding more A or B increases the forward reaction rate, which results in C and D being produced at a greater rate until equilibrium is restored when the forward and reverse reaction rates again become equal.

For an animation that shows a system reaching the same dynamic equilibrium from a variety of initial conditions, see Dr Nutt’s pages at the Chemistry Department of Davidson College, North Carolina:

At equilibrium, the relative proportions of concentrations in a given pair of forward-reverse reactions always have the same value so for the system at (32) we can define an equilibrium constant, Kc (Slide 12):

Kc ={[P]p[Q]q [/A]a[B]b}equ

Equation 33

which can be extended and generalised to reactions involving any number of products and reactants. Kc is expressed in terms of equilibrium concentrations; the curly brackets and the subscript ‘equ’ are often omitted and taken as read. For gaseous reactions (less relevant for reactions within cells) the equivalent expression uses partial pressures (number densities) and the equilibrium constant is denoted Kp.

The units of Kc depend on the stoichiometry of the reaction,
e.g. if a = b = p = q = 1, Kc is a dimensionless number.

A common type of reaction involves dissociation and recombination:

AB ⇌ A + B

Equation 34

Here, the dissociation constant Kd has units of concentration e.g. mol dm-3.

Kd = [A][B]/[AB]

Equation 35

An acid-base equilibrium (Slide 13) shows the general form of such a reaction involving a conjugate acid-base pair.

HA + H2O ⇌ H3O+ + A-

Equation 36

Here HA is an acid (proton donor) and A- is a base (proton acceptor). pH is defined as

pH = -log10 [H3O+]

Equation 37

and the acidity constant is

Ka = [H3O+][A-]/[HA]

Equation 38

It is often useful to relate pH to acidity constant via pKa, where

pKa = -log10 Ka

Equation 39

hence

pH – pKa = log10([A-]/[HA])

Equation 40

A special case is the water acid-base equilibrium (Slide 14) with water acting as both a weak acid (proton donor) and a weak base.

H2O + H2O ⇌ H3O+ + OH-

Equation 41

In pure water

[H3O+] = [OH-] = 10-7 mol dm-3

Equation 42

hence (from Equation 37) pH = 7.

Since 1 dm-3 of water has mass 100 g, and 1 mol has mass 18 g, the molar concentration of H2O must be 55.5 mol dm-3. Hence from Equation 38 the acidity constant of pure water is

Ka = [H3O+][OH-]/[H2O] = 1.8 x 10-16 mol dm-3

3.2 Equilibrium, rate and temperature

We can relate the equilibrium constant to the relevant rate constants (Section 2), and show how an equilibrium constant depends on temperature.

For elementary bimolecular reactions (i.e. where the reactions shown at (32) are described by collision theory) we can write:

Jforward= kforward [A]a[B]b

Equation 43

and

Jreverse= kreverse [P]p[Q]q

Equation 44

(Sometimes J12 and k12 are used for the forward reaction rate and rate constant, J21 and k21 for the reverse.)

By applying the principle of detailed balance (i.e. equating Jforward and Jreverse) and comparing with Equation 2 we have:

Kc = [P]p[Q]q/[A]a[B]b = kforward/kreverse

Equation 45

Applets such as this one developed at the University of California, Irvine, show how the forward and reverse reaction rates affect equilibrium:

The ratio of forward and reverse rate constants depends on temperature and can be expressed using an appropriate theory of reaction kinetics, e.g. the Eyring equation:

k = kBT/h exp(-ΔG*/RT)

(Equation 28)

which, with Equation 45, leads to

Kc = exp(-ΔG/RT)

Equation 46

where ΔG is the difference in molar Gibbs energy between the two sides of the reaction (Slide 15), R is the molar gas constant and T the temperature.