Category: Advanced Chemistry

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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Concentration polarisation is the change in electrode potential from its equilibrium value when a certain level of current flows, due to inefficiencies in electrode reactions caused by sluggish mass transfer between the electrode surface and the solution.

When a piece of metal is immersed in its ionic solution, an equilibrium electrode potential is established with no net current flowing. If the metal is connected to another half-cell, e.g. an SHE, and the metal is below hydrogen in the electrochemical series, a net cathodic current flows.

Furthermore, if the metal electrode is consuming species faster than reactants reaching the electrode surface, a concentration gradient arises between the bulk solution and the electrode surface. Since diffusion is a slow process, the depletion of reactants at the electrode surface leads to a lower electrode potential (similarly, when the electrode is producing species faster than the products leaving the electrode surface, the electrode again becomes polarised). The difference between the modified potential and the equilibrium potential is called the concentration overpotential η_con.

To derive a mathematical relationship between current (or current density) and concentration overpotential, we have to assume that activation overpotential of the cell is negligible. This is achieved by using of non-polarisable electrodes.

At equilibrium, the potential of the cathode is given by the Nernst equation:

When a current flows and polarisation of the electrode occurs, [M^z+] becomes [M^z+’] where [M^z+’] < [M^z+]. The corresponding open-circuit electrode potential is

The concentration overpotential of the polarised electrode is

According to Fick’s first law of diffusion, the flux of ions, r, towards the cathode is

where D is the coefficient of diffusion with SI units m²s^-1 and x is the distance between the bulk solution and the electrode surface.

 

 

Substituting eq65 in eq39 of the previous article, the cathodic current density is:

Note that current in a cell not only flows from one electrode to the another electrode along a wire, but also continues to flow back to the first electrode through the electrolyte, and eq66 represents this flow. Rearranging eq66,

Substituting eq67 in eq64 and rearranging,

where f = F/RT.

Concentration overpotential, like activation overpotential, leads to a lower potential than the equilibrium potential of a galvanic cell when a current flows.

where . In the case of an electrolytic cell, a higher applied potential is needed to maintain the desired current. If we include IR drop, the formula is:

 

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A three-electrode cell is used for electrochemical measurements involving the passage of current. It is a good set-up to evaluate electrochemical quantities like overpotential and IR drop. A typical three-electrode system consists of a working electrode, a reference electrode and a counter (or auxillary) electrode (see diagram below).

Generally, the potential of a reference electrode must remain constant in order to measure the potential of a working electrode. However, electrodes become polarised when a current flows. The solution is to introduce a third electrode called a counter electrode to the cell for dynamic measurements. A high impedance voltmeter displays the potential of the working electrode and is connected between the working electrode and the reference electrode so that very negligible current flows between them. Experiments involving the polarisation of the working electrode at various current levels can therefore be conducted with respect to the counter electrode. Furthermore, the reference electrode is fitted with a movable column of electrolyte called the Luggin capillary, which allows the measurement of IR drop by varying the distance d.

It is important to note that the counter electrode serves only to complete the circuit for current flow. Ideally, it should not to interfere with reaction at the working electrode and is designed to be non-polarisable. One way to minimise the formation of overpotential at the counter electrode is to increase its surface area, A. This in turn reduces its current density (a fixed amount of current from the potentiostat is spread over a larger area), which is described by the Butler-Volmer equation:

Since j₀ is an intensive property,  is small if A is large. This implies that

Since f ≠ 0, η ≈ 0.

 

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The three-electrode cell is used to measure the activation overpotential of a cell at various levels of current. Consider the following experiment setup:

The opening end of the Luggin capillary is brought very close to the working electrode (d≈0) and the potential of the working electrode E_eqm is measured with the potentiometer and counter electrode removed from the circuit. The two are then reconnected to the circuit and the externally applied potential is turned on to a magnitude greater than 0.57 V (e.g. 0.8V), resulting in Fe²⁺ being oxidised to Fe³⁺ at the anode and H⁺ reduced to H₂ at the cathode. The potential of the working electrode E_pol is again recorded. Assuming that the electrolyte is well-stirred, the difference between E_pol and E_eqm is the activation overpotential of the anode with respect to the current flowing at 0.8 V. By increasing the applied potential, we can tabulate the increasing levels of current, I, flowing in the circuit (see table below) and consequently record the corresponding overpotentials η_act,anode using eq61.

If the overpotential at the anode is greater than +100 mV, the second exponential in the Butler-Volmer equation tends to zero, giving:

This means that the net current density at the anode mainly comprises of anodic current density. Taking natural logarithm on both side of eq69,

Eq70 is known as the Tafel equation, named after the Swiss chemist, Julius Tafel. The plot of ln(I/A) against η is called a Tafel plot and it allows us to find the value of the transfer coefficient  from the gradient of the line and the exchange current density j₀ from the intercept. Such an experiment that monitors current at various potentials is called voltammetry.

The three-electrode cell can also be modified slightly to measure the total overpotential of a cell, . Consider the following experiment with metallic copper for both the anode and cathode, and AgCl electrode as the reference electrode.

With the potentiometer set to 0 V,

• • Measure potential of the left electrode E_{L, eqm} by connecting the reference electrode to junction X.
  • Measure potential of the right electrode E_{R, eqm} by connecting the reference electrode to junction Y.

We’d expect E_{L, eqm} = E_{R, eqm}. Next, turn the potentiometer to 1.0 V and

• • Measure potential of the left electrode (anode) E_{an, pol} by connecting the reference electrode to junction X.
  • Measure potential of the right electrode (cathode) E_{cat, pol} by connecting the reference electrode to junction Y.

We’d expect E_{an, pol} > E_{L, eqm} and E_{cat, pol} < E_{R, eqm}.

We can also measure IR drop (in mV/cm) for the above setup by setting the external applied voltage at 1.0 V and measuring the anode potential at different Luggin capillary distances, d.

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A Pourbaix diagram is a reduction potential versus pH graph. It was invented by Marcel Pourbaix, a Belgian chemist, in the 1930s. A typical Pourbaix diagram illustrates the equilibria of electrochemical and non-electrochemical reactions. It consists of plots of equations derived either from the Nernst equation (for electrochemical reaction equilibria) or the equilibrium constant relation (for non-electrochemical reaction equilibria), with all reactions related to a particular metal or non-metal. For example, the equilibria of acidified water are:

E₁ :

E₂ :

with the following Nernst equations at rtp, , and :

The Pourbaix diagram for acidified water is the E/E_SHE-pH graph of eq71 and eq72 (where E/E_SHE is the electrode potential of a reaction relative to the Standard Hydrogen Electrode):

A Pourbaix diagram is therefore a map of the relative stability of species in a reaction with respect to E/E_SHE and pH. Acidified water is unstable above the upper line with regard to the evolution of oxygen gas. Below the lower line, acidified water is again unstable, as it is converted to hydrogen gas. In other words, acidified water is relatively stable in the region between the two lines. Understanding the electrochemical stability of water is important in many industrial processes, e.g. the refining of copper, which has the following Pourbaix diagram:

The equilibria involved are:

1)

2)

3)

4)

5)

6)

7)

and their corresponding equations are:

a)

b)

c)

d)

e)

f)

g)

Equations (a), (c), (d), (e) and (g) are derived using the Nernst equation, while equations (b) and (f) originated from the equilibrium constant relation. We assume all activities of aqueous species are equivalent to their concentrations and that the diagram is plotted with all aqueous copper species concentrations of 10^-6 M.

With reference to the copper Pourbaix diagram, the horizontal line corresponds to eq(a), which is independent of pH (i.e. E doesn’t change by varying pH). The two vertical lines correspond to eq(b) and eq(f), both of which are independent of E (i.e. pH doesn’t change by varying E). The slanted lines correspond to eq(c), eq(d), eq(e) and eq(g). Furthermore, the Pourbaix diagram for the electrolysis of acidified water is superimposed on the diagram for copper, as the electrolyte consists of water.

The Pourbaix diagram allows us to select the appropriate E and pH for copper refining, which is represented by equilibrium(1). The preferred pH is between 2 and 4 (for [Cu²⁺] = 10^-6 M), to avoid the formation of oxides. Furthermore, the more negative the cathodic potential is (by application of an external potential:  see diagram below), the greater equilibrium(1) shifts to the right. However, if the potential is too negative, the deposition of copper has to compete with the evolution of hydrogen gas at the cathode, which reduces the yield. Hence, the thermodynamically preferred E value is roughly between 0.00 V and 0.16 V for 2 < pH < 4 and [Cu²⁺] = 10^-6 M. In practice, the preferred E value is also influenced by kinetics, as a result of overpotential, IR drop, etc.

Finally, the copper Pourbaix diagram is also useful for corrosion prevention. According to the diagram below, a layer of oxide is formed to prevent further corrosion of Cu (passivation) when the solution has a pH of between 8 to 12. In acidic solutions, the corrosion of Cu is prevented by bringing the potential of the metal to less than 0.1 V.

 

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