Standard electrode potentials E⦵, standard cell potentials E⦵ cell and the Nernst equation

Electrochemical Cells: \(E^\ominus\), \(E^\ominus_{\text{cell}}\) and the Nernst Equation

Welcome to one of the most exciting (and sometimes challenging!) topics in Physical Chemistry: Electrochemistry. This chapter is all about understanding how chemical reactions can generate electrical energy, and vice versa. Think of it as the chemistry behind batteries—how do we measure the "chemical push" that makes electrons move?

Don't worry if this seems tricky at first. We will break down the difficult concepts into simple, manageable steps, starting with definitions and building up to the calculations!

Section 1: Standard Electrode Potential (\(E^\ominus\))

1.1 Defining the Standard Electrode (Reduction) Potential (24.2.1a)

In electrochemistry, we look at reactions that involve electron transfer (redox reactions). We often study these reactions in two separate compartments called half-cells.

The Standard Electrode Potential (\(E^\ominus\)) is a measure of the tendency of a species to gain electrons (i.e., undergo reduction).

• \(E^\ominus\) is always written for the reduction half-reaction.
• It is measured in Volts (\(\text{V}\)).
• The value tells you the potential difference relative to a standard reference electrode.

Analogy: Think of \(E^\ominus\) as the "greediness" of a chemical species for electrons.

• A large positive \(E^\ominus\) (e.g., \(+2.87\text{ V}\) for \(\text{F}_2\)) means the species loves electrons and is easily reduced. It is a powerful oxidising agent.
• A large negative \(E^\ominus\) (e.g., \(-2.37\text{ V}\) for \(\text{Mg}^{2+}\)) means the species does not like electrons and prefers to be oxidised. It is a powerful reducing agent.

1.2 Standard Conditions (Quick Review)

For the potential to be truly "Standard" (\(E^\ominus\)), the half-cell must be set up under specific conditions:

• Temperature: \(298\text{ K}\) (\(25^\circ\text{C}\)).
• Concentration: All aqueous ions must have a concentration of \(1.0\text{ mol dm}^{-3}\).
• Pressure: Any gases involved must be at a pressure of \(101\text{ kPa}\) (\(1\text{ atm}\)).

1.3 The Standard Hydrogen Electrode (SHE) (24.2.2)

We need a fixed zero point to measure all these potentials against. This reference point is the Standard Hydrogen Electrode (SHE).

The SHE is assigned an electrode potential of exactly \(0.00\text{ V}\).

How the SHE is constructed:

1. A piece of inert Platinum (Pt) metal (which conducts electrons but doesn't react) is used as the electrode.
2. The Pt electrode is dipped into a solution containing hydrogen ions (\(\text{H}^+\)) at a concentration of \(1.0\text{ mol dm}^{-3}\).
3. Hydrogen gas (\(\text{H}_2\)) is bubbled over the electrode at a pressure of \(101\text{ kPa}\) and a temperature of \(298\text{ K}\).

The half-reaction occurring at the SHE is:

\(2\text{H}^+(aq) + 2e^- \rightleftharpoons \text{H}_2(g) \quad E^\ominus = 0.00\text{ V}\)

Section 2: Measuring \(E^\ominus\) (Half-Cells) (24.2.3)

To measure the \(E^\ominus\) of any half-cell, we must connect it to the SHE to form a complete circuit (a cell).

2.1 Setup for Metal/Metal Ion Half-Cells (e.g., Copper) (24.2.3a)

This consists of a metal dipping into a solution of its own ions.

• Half-cell setup: A copper strip immersed in \(1.0\text{ mol dm}^{-3}\) \(\text{Cu}^{2+}(aq)\).
• Connection: Connect the copper half-cell to the SHE using:
  • A high-resistance voltmeter (to measure the potential difference, \(E^\ominus\)).
  • A salt bridge (usually filter paper soaked in \(\text{KNO}_3\)), which allows ion movement to complete the circuit but prevents solutions from mixing.
• Measurement: The reading on the voltmeter is the \(E^\ominus\) for the copper half-cell (e.g., \(+0.34\text{ V}\)).
• Half-equation (Reduction): \(\text{Cu}^{2+}(aq) + 2e^- \rightleftharpoons \text{Cu}(s)\)

2.2 Setup for Non-Metal/Non-Metal Ion Half-Cells (e.g., Chlorine) (24.2.3a)

When a gas or non-metal is involved, an inert platinum electrode is used to allow electrical contact.

• Half-cell setup: A Pt electrode dipped into a solution of \(1.0\text{ mol dm}^{-3}\) \(\text{Cl}^-(aq)\), with \(\text{Cl}_2(g)\) bubbled through a glass tube over the electrode at \(101\text{ kPa}\) and \(298\text{ K}\).
• Half-equation (Reduction): \(\text{Cl}_2(g) + 2e^- \rightleftharpoons 2\text{Cl}^-(aq)\)

2.3 Setup for Ion/Ion Half-Cells (e.g., Iron) (24.2.3b)

Sometimes, the half-reaction involves ions of the same element in different oxidation states (e.g., \(\text{Fe}^{3+}\) and \(\text{Fe}^{2+}\)). Since there is no solid metal, an inert platinum electrode is used.

• Half-cell setup: A Pt electrode immersed in an equimolar solution containing both \(1.0\text{ mol dm}^{-3}\) \(\text{Fe}^{3+}(aq)\) and \(1.0\text{ mol dm}^{-3}\) \(\text{Fe}^{2+}(aq)\).
• Function of Pt: It acts as a surface for electron transfer, conducting electrons into or out of the solution without reacting.
• Half-equation (Reduction): \(\text{Fe}^{3+}(aq) + e^- \rightleftharpoons \text{Fe}^{2+}(aq)\)

Did you know? The platinum electrode is often platinised (coated with finely divided Pt) to increase its surface area, ensuring rapid equilibrium at the electrode surface.

Section 3: Standard Cell Potential (\(E^\ominus_{\text{cell}}\))

3.1 Calculating \(E^\ominus_{\text{cell}}\) (24.2.1b, 24.2.4)

A simple cell is made by combining two different half-cells. The Standard Cell Potential (\(E^\ominus_{\text{cell}}\)) is the total potential difference (voltage) generated when two half-cells are connected under standard conditions.

We calculate this by finding the difference between the two standard electrode potentials:

\(E^\ominus_{\text{cell}} = E^\ominus_{\text{reduction}} - E^\ominus_{\text{oxidation}}\)

Alternatively, since all \(E^\ominus\) values are reduction potentials:

\(E^\ominus_{\text{cell}} = E^\ominus_{\text{more positive}} - E^\ominus_{\text{less positive}}\)

Step-by-Step Guide to Cell Calculations:

1. Identify the two reduction half-equations and their \(E^\ominus\) values.
2. The half-reaction with the more positive \(E^\ominus\) will be the reduction reaction (at the cathode).
3. The half-reaction with the less positive \(E^\ominus\) will be forced to reverse and become the oxidation reaction (at the anode).
4. Calculate \(E^\ominus_{\text{cell}}\) using the formula above.

Example: Zinc/Copper Cell

• \(\text{Cu}^{2+} + 2e^- \rightleftharpoons \text{Cu} \quad E^\ominus = +0.34\text{ V}\)
• \(\text{Zn}^{2+} + 2e^- \rightleftharpoons \text{Zn} \quad E^\ominus = -0.76\text{ V}\)

Copper has the more positive \(E^\ominus\), so it undergoes reduction. Zinc is oxidised.

\(E^\ominus_{\text{cell}} = (+0.34\text{ V}) - (-0.76\text{ V}) = +1.10\text{ V}\)

3.2 Polarity, Electron Flow, and Feasibility (24.2.5)

The sign of \(E^\ominus_{\text{cell}}\) dictates two key aspects:

1. Feasibility (Spontaneity) (24.2.5b):
   If \(E^\ominus_{\text{cell}}\) is positive, the reaction is thermodynamically feasible (spontaneous) under standard conditions.
2. Polarity and Electron Flow (24.2.5a):
   • The Anode (oxidation) is the negative electrode. Electrons flow away from the anode through the external circuit.
   • The Cathode (reduction) is the positive electrode. Electrons flow towards the cathode.

   Mnemonic: ANode is NEGative (in a voltaic cell). RED CAT (Reduction occurs at the Cathode).

3.3 Predicting Reactivity and Kinetic Limitations (24.2.6)

We can use \(E^\ominus\) values to predict the relative strength of chemicals as redox agents:

• The species with the most positive \(E^\ominus\) (on the LHS of its half-equation) is the strongest oxidising agent.
• The species with the most negative \(E^\ominus\) (on the RHS of its half-equation) is the strongest reducing agent.

Kinetic limitation caveat: Even if \(E^\ominus_{\text{cell}} > 0\), a reaction might not be observed under standard conditions because of a high activation energy (the rate of reaction is extremely slow).

Section 4: Constructing Redox Equations (24.2.7)

To get the overall redox equation for a feasible cell, we combine the two half-equations:

Example: \(\text{Zn}/\text{Cu}\) cell (\(E^\ominus_{\text{cell}} = +1.10\text{ V}\))

1. Reduction (Cathode): \(\text{Cu}^{2+}(aq) + 2e^- \longrightarrow \text{Cu}(s)\)
2. Oxidation (Anode): \(\text{Zn}\) is oxidised, so reverse its reduction half-equation: \(\text{Zn}(s) \longrightarrow \text{Zn}^{2+}(aq) + 2e^-\)
3. Combine: Add the two equations together, ensuring the number of electrons cancels out:

   \(\text{Cu}^{2+}(aq) + \text{Zn}(s) \longrightarrow \text{Cu}(s) + \text{Zn}^{2+}(aq)\)

Section 5: The Effect of Concentration – The Nernst Equation

The standard potential \(E^\ominus\) applies only under standard conditions (\(1.0\text{ mol dm}^{-3}\)). When ion concentrations vary, the actual electrode potential (\(E\)) changes.

5.1 Qualitative Prediction (24.2.8)

Consider a reduction half-reaction: \(\text{Oxidised Species} + z e^- \rightleftharpoons \text{Reduced Species}\)

• Increasing [Oxidised Species]: By Le Chatelier’s Principle, the equilibrium shifts to the right (favouring reduction). This makes \(E\) more positive.
• Increasing [Reduced Species]: The equilibrium shifts to the left (favouring oxidation). This makes \(E\) less positive (more negative).

5.2 The Nernst Equation (24.2.9)

The Nernst equation calculates the electrode potential \(E\) at \(298\text{ K}\) under non-standard concentrations:

\(E = E^\ominus + \frac{0.059}{z} \log \frac{[\text{oxidised species}]}{[\text{reduced species}]}\)

Where:

• \(E\) is the non-standard electrode potential (\(\text{V}\)).
• \(E^\ominus\) is the standard electrode potential (\(\text{V}\)).
• \(z\) is the number of electrons transferred in the half-reaction.
• \([\text{oxidised species}]\) and \([\text{reduced species}]\) are the concentrations of the aqueous species.
• Note: Pure solids (such as \(\text{Cu}(s)\)) have constant concentrations and are omitted (taken as 1).

Application Example 1: Copper Half-Cell

The reduction equation is: \(\text{Cu}^{2+}(aq) + 2e^- \rightleftharpoons \text{Cu}(s)\). Here, \(z=2\).

The Nernst equation becomes:

\(E = E^\ominus + \frac{0.059}{2} \log [\text{Cu}^{2+}(aq)]\)

Application Example 2: Iron Half-Cell

The reduction equation is: \(\text{Fe}^{3+}(aq) + e^- \rightleftharpoons \text{Fe}^{2+}(aq)\). Here, \(z=1\).

The Nernst equation becomes:

\(E = E^\ominus + \frac{0.059}{1} \log \frac{[\text{Fe}^{3+}(aq)]}{[\text{Fe}^{2+}(aq)]}\)

Section 6: Relating Cell Potential to Gibbs Free Energy (\(\Delta G^\ominus\)) (24.2.10)

The electrical energy generated by a reversible electrochemical cell relates directly to the standard Gibbs free energy change:

\(\Delta G^\ominus = -nE^\ominus_{\text{cell}}F\)

Where:

• \(\Delta G^\ominus\) is the standard Gibbs free energy change (\(\text{J mol}^{-1}\)).
• \(n\) is the amount of charge transferred in moles of electrons for the balanced reaction.
• \(E^\ominus_{\text{cell}}\) is the standard cell potential (\(\text{V}\)).
• \(F\) is the Faraday constant (\(96,500\text{ C mol}^{-1}\)).

Feasibility Summary:

• If \(E^\ominus_{\text{cell}}\) is positive, then \(\Delta G^\ominus\) is negative, indicating a thermodynamically feasible reaction.
• If \(E^\ominus_{\text{cell}}\) is negative, then \(\Delta G^\ominus\) is positive, indicating a non-spontaneous reaction.