# Electrochemistry 2: Electromotive force

27th Dec 2020

In part 1 of this series, we looked at the construction of a voltaic cell and the basic idea of potential difference. In this part, we will look at the some of the interpretation of potential difference as applied to all pre-university level chemistry courses. As mentioned in part 1 of this series, I will not be covering all key areas that are normally taught at this level so instead you can view this material as a supplement to your own study notes.

In the previous part we looked at the notion of potential difference between two electrodes. This was based on the idea that different half-equations yielded different degrees of electron density on the electrode. Species which are easily oxidised lead to a greater build-up of electrons on the electrode and are more likely to perform the function of the negatively charged anode.

A more complete redox reaction can occur if both electrolytes are connected using a salt bridge and the electrodes are connected together by a conducting wire. Instead of using a wire, one can connect both electrodes using a high-resistance voltmeter. This will immediately give us an indication of the potential difference, usually in units of Volts. The Volt is another measure of energy and more precisely it is the potential energy change per unit charge. We will learn why a high-resistance voltmeter is used and what the Volt means to chemists in due course.

It turns out that the potential difference of a voltaic cell depends on quite a few factors and so it becomes necessary to have in place a standard procedure which enables us to document and then compare values. This is where the topic as a whole starts to get more intricate and potentially more confusing. We will go through these points steadily in turn.

### Measuring the potential difference

The recorded potential difference depends on how the voltmeter is connected to the electrodes. If the black/negative pole and the red/positive pole of the voltmeter matches the polarity or charge of the electrodes, then the potential difference will be recorded as a positive number. This is handy if you do not know beforehand which electrode is the negative or which is the positive. Connecting the voltmeter to the voltaic cell until a positive reading is displayed is a quick way of determining the polarity of the cell (Figure 16.1a).

Knowing the sign on the voltmeter is important since it indicates the direction of the flow of electrons. Let us first go over the other factors which affect the recorded potential difference and return to this point later.

As electrons migrate from the anode to the cathode one will find that, over time, the potential difference changes, tending to zero. This occurs because the electron density on both electrodes starts to equalise or the reagents are nearly consumed. This is equivalent to saying that a cell or battery is going 'flat' or 'dead'. The question is, when should we record the potential difference that is characteristic of a given cell? Chemists address this by documenting the initial potential difference, using freshly prepared electrodes and electrolytes (anything prepared within the past few days is usually fine for most purposes). This approach is easy to repeat and partly ensures that the potential difference is at its greatest magnitude. Using fresh reagents is especially important for voltaic cells which inherently produce a very small potential difference and are difficult to record.

The other major factor which can affect the potential difference is the connection of the voltmeter itself. By closing the circuit, the electron densities at each electrode are already beginning to equalise. To temporarily counter this, we restrict the flow of most of the electrons (Figure 16.1b) and thus try to preserve the initial differences of electron density and potential energy for as long as is needed. This is why chemists document potential differences with a high-resistance or high-impedance voltmeter.

### Comparing half cells

At this stage, we can bring in a more convenient label to describe a part of the voltaic cell. Instead of referring specifically to the electrode or electrolyte, chemists refer to both the electrode and its respective electrolyte as a half-cell. This, as you can tell, signifies effectively half of the voltaic cell.

With the background to the voltaic cell in place, we can now begin to investigate different half cells and complete the discussion about the factors which affect the potential difference.

If we swap the copper half-cell for a magnesium half-cell, then based on our knowledge of the reactivity of metals, we would expect the zinc half-cell to function as the cathode. If we left the black/negative pole of the voltmeter connected to the zinc half-cell, then the potential difference would be negative because the role of the zinc half-cell has changed from being a negative anode to being a positive cathode (Figure 16.2).

Using this same voltmeter setup is an important step because it enables us to determine the direction of electron flow of any half-cell compared to the zinc half-cell and allow us to make conclusions. Take each step slowly.

• The magnesium half-cell is the negative pole and therefore functions as the anode when connected to the zinc half-cell. That is, magnesium is more easily oxidised than zinc. Similarly, zinc is more easily oxidised than copper. The mini-series of reactivity, starting with the strongest reductant is: Mg, Zn then Cu.
• Magnesium(II) ions are less easily reduced compared to zinc(II) ions. Zinc(II) ions are less easily reduced than copper(II) ions. A mini-series of reducing ability, starting with the strongest oxidant is: Cu(II), Zn(II) then Mg(II).
• When zinc is connected to the black/negative pole of the voltmeter and the reading is...
• ...positive, then the other half-cell is the positive pole (the cathode) and undergoes reduction
• ...negative, then the other half-cell is the negative pole (the anode) and undergoes oxidation

In the above series of experiments, the zinc half-cell is forming a reference point for the other two half-cells, namely copper and magnesium. We could add more to the reactivity series, as long as we connect the red/positive pole of the voltmeter to the test half-cell and leave the black/negative pole connected to the reference, zinc half-cell. The table below summarises the results.

 Connected to Zn half-cell Voltage Copper +1.10 V Cu less easily oxidised than Zn Zinc 0.00 V Magnesium -1.61 V Mg more easily oxidised than Zn

To finish this brief exploration, let us compare three half-cells which all record a positive potential difference (Figure 16.3). See if you can draw up a reactivity series.

In all cases, the zinc half-cell is still connected to the black/negative pole of the voltmeter. What this particular set of data is showing is the difference in the magnitude of the potential difference. The greater the potential difference, the greater the difference of the electron density between the cathode and the zinc anode. As a result, the extent of reduction at the cathode is greater. Hence, the voltaic cell with the more positive potential difference utilises a stronger oxidant at the cathode. You could deduce similar findings for a series of voltaic cells which generate negative potential energy differences: the more negative the potential difference, the more the anode will get oxidised.

The above conclusions could have been based on any reference half-cell connected to the black/negative pole of the voltmeter and are not limited to zinc half-cells. The findings are important so here is a short summary of a test half-cell connected to the red/positive pole of the voltmeter, with the reference half-cell connected to the black/negative pole:

• A positive potential difference indicates that the test half-cell is the positive pole and therefore represents the cathode (reduction)
• A negative potential difference indicates that the test half-cell is the negative pole and therefore represents the anode (oxidation)
• The more positive the potential difference, the more the test half-cell undergoes reduction
• The more negative the potential difference, the more the test half-cell undergoes oxidation

### The standard hydrogen electrode

The compilation of lists of potential differences from a range of voltaic cells serves multiple uses, which we will state when ready. So far we have demonstrated how the zinc half-cell can function as a reference point and help build a reactivity series. Chemists, however, do not use the zinc half-cell as the reference point and instead have decided to use what is referred to as the standard hydrogen electrode, often abbreviated to SHE, as the reference point.

The half-cell of the SHE is composed of hydrogen gas passed at 101 kPa over a platinum black electrode, immersed in an aqueous 1.00 M solution of a strong, monoprotic acid (a compound which produces one mole of protons from each mole of molecules e.g. hydrochloric acid). The potential difference is recorded at a temperature of 298 K. At this level, the 'standard' in 'standard hydrogen electrode' reflects these conditions. You should be able to find diagrams of the standard hydrogen electrode in almost any pre-university level textbook.

The potential difference is recorded using freshly prepared reagents and a high-resistance voltmeter. These last two conditions are not part of 'standard' conditions and not always stated but are usually followed nonetheless. Furthermore, the black/negative pole of the voltmeter is connected to the (reference) SHE. As a result, the above summaries still apply.

### The electrode potential

Having introduced the SHE, we can now fine-tune the terminology used thus far. When the SHE is connected to a half-cell with all of the other components and aforementioned conditions that make up a voltaic cell, then the resultant potential difference (in Volts) is referred to as the standard electrode potential, symbolised as Eelectrodeo or just Eo (sometimes read as 'E nought').

The Eo values are specific to the other test half-cells and are normally listed in a data booklet. Some data booklets will list Eo values alphabetically, making it easier to find values of particular half-cells. Other lists will be sorted in order of increasing or decreasing potential difference, making oxidising/reducing abilities easier to compare.

You will also see that the half-equations are all written as reduction reactions. This is convention, and like all conventions, is essentially a wise choice grounded by experience. We normally read equations left-to-right and then record a value (enthalpy change ΔH, equilibrium constant K, entropy change ΔS etc.) which directly relates the forward process to this value. In this case, the extent of forward (reduction) reaction is represented by the sign and magnitude of Eo.

As we saw in Figures 16.2 and 16.3, the extent to which reduction takes places depends on the sign and magnitude of the potential difference. With that, the next two related bullet-points are worth memorising:

• The more positive Eo, the more likely the forward, reduction reaction takes place
• The more negative Eo, the less likely the forward, reduction reaction takes place (and hence the reverse, oxidation reaction will more likely take place)

You should then see from data booklet tables that, as expected, molecular fluorine is very easy to reduce while potassium ions are very difficult to reduce.

The Eo of the SHE is 0.00 V. This is because we would have connected two identical SHE half-cells which, as with all voltaic cells composed of two identical half-cells, have zero potential difference. Notice how the distribution of all electrode potentials is quite even when using the SHE as the reference, typically ranging from approximately -3 V to +3 V. If we had chosen to use the zinc half-cell as a reference, then the potential difference trends would have been the same except that the range of the voltages recorded would differ.

The 'E' in Eo is a reference to the first 'e' in electromotive force or EMF. The electromotive force is the same as the maximum potential difference.

### Standard electrochemical cell potential

Having been presented a list of Eo values, one can immediately deduce the EMF of all voltaic cells where one of the half-cells is connected to the SHE. Just read the values directly from the list. This assumes that the black/negative pole is always connected to the SHE.

The main application of the Eo values is the design of voltaic cells with a particular cell EMF, many of which do not involve the SHE. A more general measure of potential difference of a voltaic cell under standard conditions is the standard cell potential, Ecello. Let us emphasise the similarities and differences here: Eo and Ecello represent EMF under given conditions and can exhibit the same trends. The only difference is the choice of half-cells. The value Eo measures the potential difference of a voltaic cell where at least one of the half-cells is the SHE, under standard conditions. The value Ecello is a more general measure of the potential difference of a voltaic cell which may or may not be composed of a SHE. At this level, Ecello can be determined by experiment or calculated, in contrast to Eo values which are determined by experiment.

Measuring Ecello by experiment is easy, simply use a voltmeter as described. The sign of the experimental Ecello will depend on the polarity of the voltmeter and the polarity of the cell. Both factors are needed when deciding the natural direction of the flow of electrons. Calculating Ecello requires some work. The background to many of the formulae involved are explained in this article so I will state them here without justification.

What I will write here is a brief overview of how I approach problems related to Ecello and highlight another convention. Always check with your teacher about syllabus requirements and what examiners are looking for.

Say you wanted to calculate Ecello of a voltaic cell with a magnesium half-cell and a copper half-cell. You could draw the cell diagram with labels for the conditions included. When drawing voltaic cells like those shown in Figure 16.2, it usually does not matter which half-cell is drawn on the left or which is drawn on the right (unless you are required to draw conventional cell diagrams, below). I think most readers would understand that a labelled value for the EMF and, '+' and '-' signs to denote the polarity of the voltaic cell are enough. I very much doubt examiners will ask you to label the poles of the voltmeter.

I would use the formula: Ecello = Ecathodeo - Eanodeo. There are other variants of this equation but I will leave them out here. The values for Ecathodeo and Eanodeo are reduction potentials, that is, they are Eo values of their half-equation when written as a reduction reaction. This means you can copy the numerical values directly from the list. If you have understood the trends behind Eo, then you should be able to assign reduction half-equations to Ecathodeo and Eanodeo such that Ecathodeo > Eanodeo and hence Ecello > 0.

Back to our magnesium-copper example. I use the notation EOx|Redo, where Ox = oxidised form and Red = reduced form; hence, the order of Ox and Red appears in the same order as a reduction half-equation. So, EMg2+|Mgo = -2.37 V and ECu2+|Cuo = +0.34 V. Hence, copper(II) is more easily reduced compared to magnesium(II). Therefore, Ecello = Ecathodeo - Eanodeo = 0.34 - (-2.37) = 2.71 V. When balancing redox equations, sometimes you may need to multiply half-equations to balance the number of electrons transferred. Multiplying half-equations is not a problem but do not multiply Eo values. As explained later, Eo values are independent of the number of electrons transferred in a redox reaction equation.

Optional: spontaneous and non-spontaneous Ecello values. The last point regarding calculated Ecello > 0 is not assessed on all courses. In summary, the inequality Ecello > 0 signifies the redox reaction, based on your assignment of Ecathodeo and Eanodeo, is spontaneous. If you have made the opposite assignment where Ecello < 0, then your choices represent the non-spontaneous process. We will revisit this in another article.

Optional: conventional cell diagrams. A conventional cell diagram (or cell representation) is one where the anode is always drawn on the left-hand side (LHS) and the cathode is drawn on the right-hand side (RHS). There is neither the need to include '+' or '-' symbols here nor is the polarity of the voltmeter required. Oxidation is understood to take place on the left and the electrons migrate towards to the right. This diagram is considered a shorthand form of the above cell diagrams. A '|' symbol represents a change in phase or state and the symbol '||' represents the salt-bridge. Figure 16.4 gives a couple of examples.

Some authors quote the formula Ecello = ERHSo - ELHSo, in which case they are probably expecting you to refer to or draw a diagram with the anode on the LHS. This form is equivalent to that shown previously: Ecello = Ecathodeo - Eanodeo.

### Non-standard potentials

More often than not, chemists build voltaic cells under non-standard conditions. Consequently, the Eo values listed in the data booklet are not directly applicable to problems posed under non-standard conditions. One way of finding non-standard values comes from the application of the Nernst equation (Figure 16.5; different courses sometimes use different forms) named after the German chemist, Walther Hermann Nernst.

The terms are given below. Use values pertaining to your course or book.

• E - non-standard electrode potential
• Eo - standard cell potential
• R - molar gas constant
• T - temperature (K)
• z - stoichiometric number (coefficient) of electrons transferred

There are variants of the Nernst equation which can be used for electrode potentials Eo or electrochemical cell potentials Ecello, though I will focus on the former in this article and the latter in another article. The equation given above can be applied to a half-equation Ox + ne- ⇌ Red in the data booklet. 'Ox' represents the oxidised form and 'Red' represents the reduced form. You may find that the natural logarithm is expressed as '+ ln([Ox]/[Red])' instead of '- ln([Red]/[Ox])'. Some courses combine R, T and F, and then re-define the Nernst equation at temperature T = 298 K.

You will probably be thinking about how to express the concentration of the reduced form of the metal cation e.g. what would [Cu(s)] be? The short answer is the concentration terms given in the Nernst equation are really the numerical equivalent of a quantity known as activity. The background to activity is beyond our scope here. I will simply state that by definition, the activity of pure solids and pure liquids is unity, or '1'. I have not seen questions which ask about the activity of pure gases so I would expect the relevant information to be laid out for you if you were asked. If we take [Cu2+(aq)] = 1.00 M and assign [Cu(s)] = 1 (also, the number of electrons transferred z = 2), then you can see that E = E0.

You may be asked to calculate electrochemical cell potentials (symbolised without the nought as Ecell) using non-standard electrode potentials. The equation for the non-standard electrochemical cell potential Ecell = Ecathode - Eanode is valid, where at least one of the reduction potentials is non-standard.

### Electromotive force in Volts

One of the common questions asked of students is the construction of redox equations and then the calculation of the potential difference, i.e. Ecell. Each electrode is assigned a half-equation with a given electrode potential. Many students ask if by doubling the half-equation (in order to build a balanced redox equation) does it mean that the electrode potential, measured in Volts, should also be doubled? This question probably arises because students have previously calculated other Hess' law type energy changes and then assume that the same method would be applied here. The short answer is no, the electrode potentials remain the same. Electrode potentials (which are potential differences), when measured in Volts, are independent of the number of electrons involved and is summarised next.

The Volt, named after the Italian physicist Alessandro Volta, can be defined as the change in potential difference (in Joules) between two points (e.g. anode and cathode, Figure 16.6) for 1 Coulomb's worth of ions passed (equivalent to about 6.24 × 1018 electrons). Numerically, Volt = Energy change (J) / charge passed (C).

The electrode potential is a value that is characteristic of the given half-cell, a measure of the potential energy change between the half-cell and the SHE under the conditions provided. The value of the electrode potential can, for example, depend on the concentration or temperature of the electrolyte or the identity of the reductant/oxidant involved but does not depend on the number of moles of electrons transferred.

If you have previously worked through Hess' law or free energy cycle calculations then you will be aware that when any equation is doubled then the enthalpy change, entropy change and free energy change would also double. In a separate article I show the relationship between free energy change and electrode potentials (and EMF in general), where we see that the value of the electrode potential, unlike the free energy change, does not depend on the number of 'moles of equation' applied.

### Uses and concluding remarks

One of the uses of electrochemistry is the determination of metal ion concentrations. Such techniques are alternatives to titrations and spectrophotometry (or colorimetry). Suppose you were given an unknown but also very dilute solution of magnesium chloride, and were required to determine the concentration. You could assemble a voltaic cell using the salt solution with another suitable half-cell and then record the non-standard electrochemical cell potential. Working backwards, you could then determine the non-standard electrode potential of the magnesium half-cell and then use the Nernst equation to deduce the concentration of magnesium(II), or in this case, [Ox]. Additionally, electrochemistry can be used to determine the concentration of salts which are only sparingly soluble and then through calculation determine the solubility product, Ksp.

We have focused mostly on the background to the voltaic cell, an electrochemical cell which induces an electrical current as a result of a spontaneous reaction. Along the way, we have looked at some of the underlying ideas behind potential energy and potential energy change. As you may have noticed after reading this and other materials, there are many ways of solving the same problems. As always, check with your teacher about which method is expected when working through problems. In another article, we will take a brief look at the relationship between Gibbs Free Energy and EMF.