Category: Advanced Chemistry

The rate law for the 2nd order reaction of the type A + 2B → P is:

Using the same logic described in the previous article, we can rewrite the rate law as:

where a = [A₀] and b = [B₀].

Rearranging the above equation and integrating throughout, we have

Substituting the partial fraction expression in eq13 and after some algebra, we have,

which is equivalent to

 

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The transition state theory provides a theoretical way to calculate the pre-exponential factor A and the activation energy E_a, and therefore the rate constant k of an elementary chemical reaction. This is in contrast with the Arrhenius equation, where A and E_a are obtained empirically. The theory is based on statistical thermodynamics and can be illustrated for a gas-phase bimolecular reaction using the following assumptions:

1) The reacting system, expressed as , proceeds along an energy path that includes a point of highest potential energy called the saddle point, where a distinct species known as the activated complex  is formed. The conversion of the activated complex to the product P can be seen as an asymmetric vibrational stretch:

2) A rapid pre-equilibrium is established between the activated complex and the reactants: .

3) The rate of the reaction is attributed to the rate-determining step, which is the rate of conversion of the activated complex to the product: , where is proportional to the vibrational frequency of the activated complex.

The expression for the gas-phase rapid equilibrium from assumption 2 is:

which in terms of molar concentration is:

Substituting eq1 into the rate equation , , yields:

In general, the equilibrium constant, when written in terms of standard molar partition functions, is (see this article for derivation), where is the respective stoichiometric coefficient of the J-th species. For our bimolecular reaction,

where .

The activated complex in our example is a linear molecule with N = 3 atoms and therefore has 3N – 5 = 4 modes of vibration (2 bending, 1 symmetric stretch and 1 asymmetric stretch). The standard molar partition function for the activated complex is the product of the partition functions of different modes of motion: , which we can rewrite in the form: , where .

The standard molar partition function for the asymmetric vibration is , where is the vibrational frequency that leads to the conversion of the activated complex to the product. Assuming , . Therefore,

Substituting eq3 into eq2 gives:

where

Substituting eq4 into the rate constant equation yields:

From assumption 3, is proportional to the vibrational frequency of the activated complex,

where is the proportionality constant called the transmission coefficient, which accounts for the notion that not every oscillation of the activated complex leads to its conversion to the product.

Substituting eq6 into eq5 results in:

The standard Gibbs energy is . By analogy, we define the standard Gibbs activation energy as  , which we shall substitute into eq7 to give:

 

 

Eq7 and eq8 are different forms of the Eyring equation. Substituting eq8 in the definition of activation energy and differentiating, noting that and are standard states, which are temperature-independent, gives:

Substituting eq9 into eq8 yields:

Comparing eq10 and the Arrhenius equation, we have:

Repeating all the above steps for a gas-phase unimolecular reaction, we arrive at an activation energy of instead of eq9. Therefore, the transition state theory predicts that the activation energy for a gas-phase elementary reaction is

where for a unimolecular reaction and for a bimolecular reaction.

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The rate law for the reaction A + B → P is:

Unlike the 2nd order reaction of the type A → P with a rate law of , eq11 cannot be integrated unless we express [B] in terms of [A], or [A] and [B] in terms of a third variable x. Since A and B react in a ratio of 1:1, the remaining concentrations of A and B at any time of the reaction are:

where [A₀] and [B₀] are the initial concentrations of A and B respectively.

Differentiating both sides of [A] = [A₀] – x with respect to time yields

Therefore, we can rewrite eq11 as

where a = [A₀] and b = [B₀]

Integrating eq12 using the partial fraction expression of gives

After some algebra, we have,

which is equivalent to

 

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The equilibrium constant of a chemical reaction is the reaction quotient of the reaction at dynamic equilibrium.

The derivation of the equilibrium constant involves the following steps:

1. 1. Derive the expression for the Gibbs energy of a multi-component reaction.
   2. Derive the chemical potential of a multi-component system.
   3. Combine the two expressions.

Step 1

From eq14 of the article on the Nernst equation, the reaction Gibbs energy for a reaction is

and at standard state

Step 2

From eq24 of the article on the Nernst equation, the chemical potential of a multi-component system is

Step 3

Combining eq1 and eq3

Substitute eq2 in eq4

Since lnx^a +lnx^b +… = ln(x^ax^b …), eq5 becomes

Let , where Q is the reaction quotient.

At equilibrium, a reversible reaction is spontaneous in neither direction. Hence the change in Gibbs energy with respect to the change in the extent of the reaction, i.e. the reaction Gibbs energy , is zero. Eq7 becomes

where K is the reaction quotient at equilibrium, i.e.

Since the value K of does not change for a particular reaction at constant temperature, we call it the equilibrium constant.

But how does , which is dimensionless, relate to and ? and are approximations of the thermodynamic equilibrium constant . For an ideal gas,

where .

Similarly, for solutes in dilute solutions,

Eq21a and eq21b show that and are only unitless when . Equating the two expressions gives:

 

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In 1887, Walther Nernst, a German chemist, developed the Nernst equation, which describes the relationship between an electrochemical reaction’s potential and its standard electrode potential under standard and non-standard conditions.

The derivation of the Nernst equation involves the following steps:

1. 1. Derive the formula for the reversible open circuit potential , E^o, of an electrochemical cell at constant temperature and pressure under standard conditions.
   2. Derive the expression for E^o in terms of chemical potentials of the reaction species at constant temperature and pressure.
   3. Derive a ‘correction factor’ for E^o in step2. This final expression, the Nernst equation, describes how the resultant E^o varies with activities of chemical species.

 

Step 1

From the definition of enthalpy, H = U + pV, we have

Substitute the definition of the internal energy of a system undergoing reversible change, dU = dq_rev + dw_rev in eq1

Substitute the definition of the change in Gibbs energy of a system, dG = dH – d(TS) in eq2

At constant temperature and pressure, dT = dp = 0

Substitute the definition of the change in entropy of a system, dS = dq_rev/T in eq3

Substitute the definition of total reversible work, w_rev = w_ex + w_add, where w_ex is reversible expansion work and w_add is reversible additional work other than reversible expansion work, in eq4

Substitute the definition of the change in reversible expansion work, dw_ex = –pdV in eq5

Substitute the definition of the reversible electrical work between two points in an electric circuit, dw_add = –nFEdξ in eq6

where n is the number of moles of electrons, F is the Faraday constant, E is the open circuit potential of the electrochemical cell and Δ_rG = dG/dξ is the Gibbs energy of the electrochemical cell reaction.

When the open circuit potential of an electrochemical cell is measured at standard conditions (1 bar, 298K, 1M), we write eq7 as:

 

Step 2

The chemical potential of the j-th species, in a multi-component system is

where n‘ is all n other than nj.

The total differential of the Gibbs energy of the system at constant temperature and pressure is

Substituting eq9 in eq10

Eq11 can be expressed in another way by considering the reversible reaction

with the change in Gibbs energy at constant temperature and pressure of the system given by eq11. Next, let’s define

where vj is the stoichiometric number of the j-th species in the reversible reaction and dξ is the amount of substance that is being changed in the reaction.

ξ is called the extent of a reaction and has units of amount in moles. Note that the stoichiometric numbers for reactants are negative by convention and those for products are positive. Substituting eq12 in eq11

Substitute the definition of the Gibbs energy of an electrochemical cell reaction, , into eq13

Substitute eq7 from Step1 in eq14

When the open circuit potential of an electrochemical cell is measured at standard conditions (1 bar, 298K, 1M), we write eq15 as:

 

Step 3

Firstly, let’s consider a system with a single pure substance. Substitute the definition of the change in Gibbs energy of a system, dG = dH – d(TS), in the change of enthalpy of the system dH = dU + d(pV)

The reversible change in internal energy of a pure substance (a system with a single pure substance only does expansion work) can be expressed as

Substitute the definitions of entropy, dS = dq_rev/T, and expansion work, –pdV, in eq18

Substitute eq19 in eq17

At constant temperature, dT = 0 and eq20 becomes dG = Vdp. Integrating this new expression on both sides from  to ,

Let p_i = 1 bar = p^o (i.e. standard conditions) and p_f = p

Secondly, we extend the above working to a multi-component system of ideal gases. Since G is an extensive property, we add the Gibbs energies of all components to give the total Gibbs energy of the system:

Let’s assume the reference pressures of the component gases are the same since they are all ideal gases, i.e. p_a^o = p_b^o = p^o 

For simplicity, let G_Total and G^o_Total be G and G^o respectively.

Taking the partial derivative of the above with respect to n_j, at constant temperature, pressure and the amount of the other components, n‘, we have

For the last partial derivative of the above equation, only the j-th term survives.

Substitute eq9 from Step 2 in eq21

Similarly, for a solute in a dilute solution satisfying Henry’s law, we can write

where c_j and c^o are the j-th solute’s concentration and molar concentration respectively.

We can fine-tune eq22 and eq23 by replacing p_j/p^o and c_j/c^o with γ_j(p_j/p^o) and γ_j(c_j/c^o) where γ_j is a factor called the activity coefficient that can account for both ideal and non-ideal fluids (γ = 1 and γ < 1 for ideal and non-ideal fluids respectively). Next, we combine the modified equations into a single equation by letting a_j = γ_j(p_j/p^o) and a_j = γ_j(c_j/c^o), where a_j is the activity of species j.

Substituting eq24 in eq15 from Step 2,

Substituting eq16 from Step 2 in eq25

where Q is the reaction quotient with Q = . 

Hence, is the ‘correction factor’ for E^o, as the activities of reaction species vary. Eq26 is the Nernst equation.

For an electrochemical equation of the form , the Nernst equation can be written as:

Note that since E^o is by definition a reduction potential, the reference equation is always and hence eq27.

 

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A typical experiment setup for pH measurement consists of a glass electrode (indicator electrode), a reference electrode (a regular AgCl electrode), the analyte and a potentiometer. The potentiometer ensures that current flow is negligible in the electrochemical circuit so that IR drop and overpotential are minimised, resulting in more accurate measurements. Such a method in electrochemistry where current is negligible is called potentiometry.

Comparing the setup with a circuit diagram (see above diagram), the overall potential of the electrochemical cell is:

The liquid junction potential arises due to the difference in mobility of K⁺ and Cl^– in the controlled flow of the saturated KCl solution out of the porous frit of the external reference AgCl electrode (to complete the circuit). E_analyte = IR = 0 since current flow is negligible in potentiometric methods.

 

 

Lastly, before a pH meter is put to use, it is calibrated using solutions of known pH. The reason for this is as follows:

Substituting eq30 from the previous article in eq31,

where L’ = L – E_{ext ref elec} + E_{liq jn}.

Since E_{liq jn} cannot be calculated or measured directly, the value of L’ must be determined by calibrating the pH meter with solutions of known pH.

 

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IR drop in electrochemistry is the voltage drop due to the resistance of the analyte (and the internal KCl solution if a calomel or AgCl electrode is used) in a close circuit.

In the previous article regarding pH measurement, which involves potentiometry, no net current flows in the circuit. When a current flows, e.g. in coulometry and electrogravimetry, we have to consider two effects:

• • IR drop
  • Overpotential

We can represent IR drop as an internal resistance of a battery in the following circuit diagram:

The general equation for a galvanic cell is:

where E_{cell, open} = E_R – E_L.

Eq32 when applied to a Zn electrochemical cell gives the following values:

E_{cell, open} and E_galvanic for the above example are recorded as negative values by the voltmeters because of the way the terminals of the voltmeters are connected to the respective circuits. From eq32, is always less than .

For an electrolytic cell,

If no current is flowing in the electrolytic cell, eq33 reduces to E_applied = E_{cell, open}. For electrolysis to proceed, . For example, the variation of E_applied for the electrolytic reduction of Zn²⁺ is illustrated by the following diagram:

Once again, E_applied for the above example is recorded as a negative value by the voltmeter because of the way the voltmeter is connected to the circuit.

 

 

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A pH glass electrode is an ion-selective electrode that is sensitive to H⁺, and therefore, used to measure pH of solutions. It is composed of a glass bulb that is H⁺-selective and an internal AgCl electrode.

The glass bulb is specially manufactured with a composition of 70+% SiO₂ and 20+% Na₂O. It has a silicate network within the glass matrix, with some Si atoms singly bonded to O atoms and other Si atoms coordinated with Na⁺ (see diagram above). On the outer and inner surfaces of the bulb, some of the Si–O^– arms are also coordinated with Na⁺, while the rest of the Si–O^– are uncoordinated.

Before the pH glass electrode can be used, it must be hydrated (soaked in water of pH = 7) to form two layers of gel. This is to allow H⁺ to diffuse into the gel layers to convert the Si–O^–Na⁺ groups on both surfaces to Si–O^–H⁺. Since the inner surface of the bulb is already in contact with a KCl buffer solution at pH 7, both the inner and outer surfaces develop the same ion-exchange equilibrium when the electrode is dipped in water:

This ion-exchange occurs because H⁺ binds more strongly to SiO^– than Na⁺. Therefore, the position of the equilibrium is almost entirely to the right. On the inner surface, the displaced sodium ions enter the buffer solution, and on the outer surface, displaced sodium ions enter water. The hydrated surfaces now consist of a combination of SiO^–H⁺ and SiO^– arms, and is ready for use. The electrode is immersed in an analyte of unknown pH, establishing the following equilibrium on the outer surface:

and the following equilibrium on the inner surface equilibrium:

If the pH of the analyte is less than 7, the outer surface will have fewer uncoordinated Si–O^– arms and hence, less negative charges as compared to the inner wall, whose equilibrium remains unchanged. The difference in charges on the surfaces produces a potential difference across the glass bulb. Furthermore, it confers enough activation energy for sodium ions within the glass matrix to dissociate from the silicate network. These mobile sodium ions then migrate interstitially across the bulb and along the potential gradient, to complete the circuit.

Even though the above equilibria are ion-exchanges and not redox reactions, the potential across the bulb E_b is found empirically to be Nernst-like, where:

 

 

Comparing with a circuit diagram (see above diagram), the overall potential of the glass electrode is:

where E_{int elec} is the potential of the internal AgCl electrode and E_{int soln} is the potential drop across the saturated KCl solution.

Since no current flows in potentiometric methods,

Furthermore, manufacturing defect or damage of electrode over time results in the electrode recording a non-zero potential when the analyte is pH = 7. We call this an asymmetry potential, which must be included in E_glass:

Substituting eq28 in eq29, where a_inner, E_asym and E_{int elec} are constants,  E_{glass eletrode} at rtp is:

where L = E_{int elec} + E_asym – 0.0592 log a_inner.

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Polarisation is the deviation of the electrode potential from its equilibrium value, E_eqm , when a certain level of current flows. It is composed of two components: activation (or kinetic) polarisation and concentration polarisation.

Activation polarisation is the change in electrode potential from its equilibrium value when a certain level of current flows, resulting from the overcoming of the activation energy of the rate-determining step in an electrochemical reaction. Concentration polarisation is the change in electrode potential from its equilibrium value when a certain level of current flows, caused by inefficiencies in electrode reactions due to sluggish mass transfer between the electrode surface and the solution.

The extent of the polarisation of an electrode or an electrochemical cell is measured by a quantity called overpotential, η, where

It is difficult to directly measure the individual components of overpotential when an electrochemical cell is operating. However, by making certain assumptions, it is possible to derive formulas attributing to η_act and η_con, and design experiments based on these assumptions to measure the two quantities.

 

 

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Activation polarisation is the change in electrode potential from its equilibrium value when a current flows, as a result of overcoming the activation energy of the rate-determining step in an electrochemical reaction.

To understand the concept of activation polarisation, let’s analyse an experiment with the following electrode reaction:

where r_red is defined as the flow of the number of moles of Ox per unit area of the electrode per unit time, i.e. the rate of reduction of Ox, and r_ox is the flow of the number of moles of Red per unit area of the electrode per unit time, i.e. the rate of oxidation of Red. We represent r_red and r_ox with the following rate equations:

where k_c and k_a are the cathodic rate constant and anodic rate constant respectively (each with units of length per time). From the combined Faraday’s laws of electrolysis, we have

Divide eq38 throughout by the area of electrode,

where j = I/A, is the current density. With reference to eq35, z = 1, and so we can rewrite eq36 and eq37 as:

where j_c and j_a are the cathodic current density and the anodic current density respectively.

To apply the above equations, consider a piece of metal M immersed in an M⁺ solution. Over time, an electric double layer is formed, with the development of an equilibrium electrode potential between the metal and its ionic solution, such that r_red = r_ox or j_c = j_a (i.e. no net current flowing). If we connect the M⁺/M half-cell to an SHE, the circuit is closed.

Assuming that M lies below hydrogen in the electrochemical series, a net current flows towards the SHE, i.e. electrons flow towards M, the cathode. This results in a less positive electrode reduction potential and the equilibrium in eq35 shifting to the right, with r_red > r_ox. Since the electrode potential deviates from its equilibrium value when a current flows, we say that the electrode is polarised.

In terms of current density, j_c is no longer equal to j_a, and because it is a reduction reaction, j_c > j_a, i.e. a net cathodic current flows. The net current density j is given by:

 

 

Substituting the Eyring equation where is the activation Gibbs energy in eq42,

with

When the electrode is at equilibrium, j_a = j_c and we can rewrite eq44 and eq45 as:

where j₀ denotes the equal current densities at equilibrium and is called the exchange current density; and and are the anodic activation Gibbs energy at equilibrium and the cathodic activation Gibbs energy at equilibrium respectively.

Next, we shall investigate the energy profile of the reaction of eq35 at the cathode at equilibrium (see Fig I below). For simplicity, we assume that the transition state has equal resemblance to M⁺ and M and therefore the activation Gibbs energy required for the reduction reaction is the same as that of the oxidation reaction.

As mentioned, when the electrode is connected to an SHE, its potential becomes less positive, which in turn causes a decrease of the activation Gibbs energy for the reduction reaction (electrode has stronger attraction for M⁺) and an increase in the activation Gibbs energy for the oxidation reaction (more difficult for M⁺ to form). The resultant energy profile is shown in Fig II. Note that the peaks in Fig I and Fig II are aligned at the same Gibbs energy level for convenience.

 

 

By letting be the magnitude of the total change in Gibbs energy between the state when a current flows and the state when the electrode is at equilibrium, we have:

where is a fraction of and is called the transfer coefficient. If you recall, the formula is derived based on the definition of the reversible electrical work between two points in a circuit with a potential difference E. The change in Gibbs energy in our case is due to the change in electrode potential when a current flows, E – E_eqm and so, we have:

Substituting eq34 from the previous article in eq50, the magnitude of the change in Gibbs energy for the passage of one mole of electrons (n = 1) is:

Substituting eq51 in eq48 and eq49,

Substituting eq53 and eq52 in eq44 and eq45 respectively,

Substituting eq46 and eq47 in eq54 and eq55 respectively,

where f = F/RT

The net current density is obtained by substituting eq56 and eq57 in eq42:

Eq58 is known as the Butler-Volmer equation. If the experiment that we have described so far consists of an electrolyte that is well stirred (so that mass transfer effects are minimised, i.e. no concentration polarisation), the polarisation of the electrode arising from the flow of current is attributed entirely to the change in activation energies of the reduction and oxidation reactions at the electrode, i.e. activation polarisation. Since η is a measure of activation polarisation, η in eq58 is η_act,cat, the activation overpotential at the cathode. Note that η_act,cat < 0 , because η = E – E_eqm where E decreases at the cathode upon polarisation.

If M is more electropositive than hydrogen, it becomes the anode when connected to the SHE. The flow of electrons away from M results in a less negative electrode reduction potential. Using the same logic, we arrive at eq58 with η being η_act,an, the activation overpotential at the anode, where η_act,an > 0.

In general, for the electrochemical cell (see this article for notation),

where E_{cell, open} is the open-circuit potential of the cell at equilibrium, which is usually the standard electrode potential of the cell, E⁰. Since η_R < 0 and η_L > 0, we can rewrite eq59 as:

where . Eq60 shows that the relationship of E_galvanic < E_cell,open is always true when a current flows in an electrochemical cell.

For an electrolytic cell, E_cell,open is usually negative. If we also consider IR drop, the magnitude of the externally applied potential must be greater than the sum of the magnitudes of the cell open-circuit potential, potential due to IR drop and the overpotential, for electrolysis to proceed:

 

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