Relating ΔG to Q, K and EMF

29th Dec 2020

The main purpose of this article is to show you the relationship between Gibbs free energy change ΔG, the reaction quotient Q, the equilibrium constant K and electromotive force E. With the exception of the reaction quotient Q, I will assume that you are familiar with the other values listed.

We start with a fundamental thermodynamic equation, without justification, and then proceed from there. You will find that authors will sometimes justify their derivations differently. In this article, there will also be some simplifications applied. Overall, I believe this material will tie up a few loose ends and provide more closure.

The reaction quotient Q and equilibrium constant K

The reaction quotient Q is similar to the equilibrium constant K except that it describes the composition of the system which is not at dynamic equilibrium. We can use a single axis to illustrate this (Figure 17.1).

Figure 17.1 Measuring the composition of the system which is not at equilibrium using the reaction quotient Q

Approaching from either the reactant side or product side, you can see that at any point before we reach dynamic equilibrium we can define the system's composition with Q. Potentially, there are a near-infinite number of reaction quotients that could be expressed. This is in contrast to K where at a given temperature, there is only one defined value and expression of K. The reaction quotient is a measure which depends on the particular reaction conditions and composition at a given time, whereas the equilibrium constant K is characteristic of the overall reaction at a given temperature.

Like K, the reaction quotient is more correctly defined in terms of activities instead of concentration or partial pressure alone. Activity is a dimensionless quantity and it follows then that, strictly speaking, reaction quotients and equilibrium constants are also dimensionless. For our purposes, the concentration of a pure solid or liquid is assigned the dimensionless value of 1.

As always, you must continue to follow whatever you are taught in your course and not try to get ahead by stating that Q and K are actually dimensionless because I or other authors say so! The concept of activity is usually taught during the first year of a chemistry degree programme and is not something we need to explore here. In many but not all cases, activity is numerically equivalent to concentration and partial pressure. This is why concentration and partial pressure can be used instead of activities at the pre-university level. If you are asked to calculate a value for Q then you can probably just apply what you know about K, using concentration or partial pressure as before.

Gibbs free energy change and Q

If you recall from my article which explains ΔG I did not discuss the differences between non-standard and standard conditions. The main reason why was because I wanted to focus on the relationship between ΔG, ΔS and ΔH and the concepts of spontaneity as applied to chemical systems and leave out other considerations until later. Overall, if the enthalpy change ΔH and entropy change ΔS are quoted under standard conditions, then the value for ΔG also applies to standard conditions. The equation to calculate free energy change for most pre-university courses more often than not is of the form ΔG^o = ΔH^o - TΔS^o.

With the reaction quotient defined, we can now state the relationship between ΔG and Q (Figure 17.2). There are a number of assumptions which apply to the equation below. As with the equation ΔG = ΔH - TΔS, it is assumed that the temperature and pressure of the system is constant.

Figure 17.2 The equation which relates standard and non-standard forms of ΔG (when calculating Q, the concentration of a pure solid is assigned a unit value, 1)

The constant R is the molar gas constant and T is the temperature (K). The quantity ΔG^o is the standard Gibbs free energy change. It is the change in Gibbs free energy under standard conditions when the reactant is completely converted into product, with the reactant and product in their standard state. ΔG is the change in free energy under non-standard conditions. The standard state of a substance is somewhat arbitrarily chosen but its assignment is rarely an issue at this level. I have not seen any pre-university programme clearly define what constitutes the standard state of a given element and there is probably no need to. It seems that students only need to recall a few allotropes (different forms of the same element) e.g. graphite vs. diamond vs. C₆₀ etc., oxygen vs. ozone, and recall which is the standard state.

There are probably more experiments carried out under non-standard conditions than there are under standard conditions. The usefulness of the equation in Figure 17.2 is the interconversion between standard and non-standard conditions.

The graphical representation and differences of the "ΔG's" is summarised in Figure 17.3. In short, ΔG^o represents the free energy change between pure reactants and pure products. As such, it is a reflection of the nature of the reaction. The non-standard ΔG is the slope at a given point along the curve, and is dependent on the chemical composition at a given point along the reaction pathway. ΔG^o values are unique to a given reaction whereas ΔG values can vary depending on the composition.

Figure 17.3 The graphical difference between ΔG and ΔG^o

The derivations which confirm the connection between ΔG and ΔG^o, and more accurate descriptions of the x-coordinate can be found in university-level textbooks. Let us highlight a few points regarding these two quantities.

• ΔG^o can be positive, zero or negative, depending on the nature of the reaction.
• Starting from pure reactants, ΔG is always negative (recall that the ΔS_total increases when the reaction starts from pure reactants). On passing the point of dynamic equilibrium, ΔG = 0 (take care here, ΔG⁰ is not necessarily zero). After this, ΔG is always positive.
• We can see at the point of dynamic equilibrium, ΔG = 0 and Q = K. Hence ΔG^o = -RT ln K. We will return to this shortly.
• Throughout the curve, as different Q values become more positive, ln Q becomes more positive. Consequently, ΔG becomes more positive from left to right.
• When ΔG^o = 0, then ΔG = RT ln Q. At the point of dynamic equilibrium, Q = K, and hence ΔG = RT ln K = 0. Hence, K = 1 (note, T > 0 K) which would represent equal proportions of reactant and product and be the half-way point on the x-axis between the reactants and products.

Standard free energy change ΔG^o and K

As shown above, the relationship between ΔG^o (not ΔG) and K is given by ΔG^o = -RT ln K. You can see that both the ΔG^o and K are essentially measures of how spontaneous a reaction is. The more spontaneous a reaction, the more product is present in the equilibrium mixture, which is also reflected in a higher value for K.

Gibbs free energy change and E_cell

As alluded to in my article about Gibbs Free Energy and E_cell, a spontaneous reaction drives an electric current. The absolute potential energy of the test-electron at the anode is positive (Coulombic repulsion results in a positive potential energy) and steadily decreases to some value greater than or equal to zero. The work done (an ordered form of energy) originates from the change in potential energy between the anode and cathode. In other words, the potential energy change (E_cell) is converted into work. This energy is 'free energy' because it is a form of non-expansion work, or, work that is not involved in the expansion or contraction of a gas.

We can then reason that the magnitude of E_cell is proportional to the free energy. Currently, the units do not match so we proceed to use known constants to address this. The derivation is given in Figure 17.4.

Figure 17.4 Relating ΔG with EMF

The value E_cell is measured in Volts, which is equivalent to Joules per Coulomb. Faraday's constant represents the charge of one mole of electrons. The negative sign follows a convention needed to account for the different roles of the system and surroundings with regard to work done. The details of this convention are overall unnecessary for our discussion and are omitted here. The multiplication of -F converts Volts into Joules per mole of electrons. We then multiply the expression by n, the stoichiometric number of electrons transferred (not the actual number of moles of electrons transferred), because the energy required to push (or pull) electrons depends on the number of electrons transferred. The value of n is specific to the process that the quantity E_cell represents (when ready, see Figure 17.5 for an example).

The equation ΔG = -nFE_cell shows us that the gradient of the curve (ΔG, Figure 17.3) is related to non-standard E_cell, with the point of equilibrium being ΔG = 0 and consequently, E_cell = 0. The forward reaction continues while ΔG < 0 and E_cell > 0. The value of E_cell, like ΔG and Q, depends on the chemical composition.

In my derivation I have chosen to use ΔG and pair it with E_cell but you could also pair ΔG^o with E_cell^o and perform the same derivation. With that, the equation ΔG^o = -nFE_cell^o indicates that a spontaneous reaction occurs with E_cell^o > 0. Let us consider the possible results under more common, non-standard conditions.

1. What happens if you connected two identical half-cells under non-standard conditions? Try it. Through experimentation you would find E_cell = 0 meaning ΔG = 0, a point of equilibrium. Once again there is no preferred direction of the electron flow. With the non-standard concentrations of the electrolytes being equal, then K = 1 (inspect the Nernst equation if in doubt). As a result, ln K = 0 and so E_cell^o would have to be zero.
2. If you connected two different half-cells under non-standard conditions, then you should be able to observe a non-zero E_cell. Under the conditions, if E_cell = 0, then you are at dynamic equilibrium. To find the E_cell^o, you would need to know the chemical composition and then use the Nernst equation, with Q = K, to find E_cell^o = (RT/nF) ln K. Incidentally, calculating the spontaneous E_cell^o from tabulated E^o values enables one to predict K of the spontaneous direction.
3. If you connected two half-cells that differ only in the concentration of the electrolyte, then you should be able to record a non-zero potential difference. If you want to know more, look up concentration cells.

To summarise, we have ΔG = -nFE_cell and ΔG^o = -nFE_cell^o. The measures E_cell^o and ΔG^o are fixed quantities, specific to the given reaction under standard conditions. As the cell discharges, the conditions become non-standard with E_cell > 0 decreasing to zero and ΔG < 0 increasing to zero. The electrochemical reaction does not continue naturally beyond the point of dynamic equilibrium since ΔG > 0 and E_cell < 0, which signify a non-spontaneous direction.

The values ΔG and ΔG^o can be treated like ΔH in numerical, Hess' law type calculations. When you reverse the direction of the 'forward' reaction, then you will need to change the sign of ΔG. Such properties then reveal other useful results. The last part of this article explores these results.

Using ΔG^o to predict E^o

One can predict the value of a reduction electrode potential using two other reduction electrode potentials. Below is an example related to the reduction of copper ions, in the +1 and +2 oxidations states. One could calculate using electrochemical cell E_cell^o values but I have applied a slightly more direct method using electrode E^o values. In addition, you can also use non-standard ΔG to predict non-standard electrode potentials E in the same way.

Figure 17.5 Predicting reduction potentials using other reduction potentials. Notice how the value of n is only in regard to the process which E represents.

Simply adding the copper(II) and copper(I) reduction half-equations and their respective reduction potentials would not yield the correct electrode potential of copper(II) to copper(0).

Using ΔG to show E_cell = E_cathode - E_anode

We can borrow the ideas behind the above cycle to present a more general form and show that E_cell = E_cathode - E_anode. In this example, we are recognising that the Gibbs free energy change of a forward reaction is the negative of that of the backward reaction (Ox = oxidised form, Red = reduced form). This examples also shows us that we need not multiply E values when determining E_cell.

Figure 17.6 Showing E_cell = E_cathode - E_anode

It is also possible to show the formula which assumes standard conditions, E_cell^o = E_cathode^o - E_anode^o, by applying the same method with standardised ΔG^o.

Deriving the Nernst equation

We can also use the equations ΔG = -nFE_cell and ΔG^o = -nFE_cell^o to derive the Nernst equation from the more fundamental equation given in Figure 17.2. Note that the value of n on line 2 of Figure 17.7 is the same for both terms because it refers to the number of electrons transferred in the redox reaction. If the voltaic cell includes a standard hydrogen electrode, then we can also pass a non-standard and standard electrode potential instead.

Figure 17.7 Deriving the Nernst equation (when calculating Q, the concentration of a pure solid is assigned a unit value, 1). The above expression for Q also applies to K when Q = K.

This also gives us an opportunity to show that we can use electrochemical cell potentials in the Nernst equation, as opposed to electrode potentials, discussed previously. We assign the concentration (as defined by the activity) of solids as unity and thus reveal the expression for the reaction quotient and equilibrium constant of a redox reaction equation.

To finish here, we have looked at the relationships between ΔG and other thermodynamics quantities such as Q, K and EMF. While it is very unlikely that you would be expected to reproduce these connections at this stage, having some explanation to the underlying principles will help with your own understanding when applying the equations.