Part V: Electrochemistry | Chapter 1

Electrochemical Cells

A rigorous treatment of galvanic and electrolytic cells — from the Nernst equation and standard electrode potentials to concentration cells, temperature dependence of EMF, and modern applications in batteries and fuel cells.

1. Introduction to Electrochemical Cells

An electrochemical cell is a device that converts chemical energy into electrical energy (or vice versa) through redox reactions occurring at electrode–solution interfaces. The two fundamental types are:

Galvanic (Voltaic) Cells

Spontaneous redox reactions produce electrical work. The cell potential is positive:$E_\text{cell} > 0$ and $\Delta G < 0$. The anode (oxidation) is negative and the cathode (reduction) is positive.

Example: The Daniell cell — $\text{Zn}(s)\,|\,\text{Zn}^{2+}(aq)\,||\,\text{Cu}^{2+}(aq)\,|\,\text{Cu}(s)$

Electrolytic Cells

An external voltage drives a non-spontaneous reaction. Here $E_\text{cell} < 0$and $\Delta G > 0$. The anode (oxidation) is positive and the cathode (reduction) is negative.

Example: Electrolysis of water — applying >1.23 V to decompose$\text{H}_2\text{O}$ into $\text{H}_2$ and $\text{O}_2$.

In any electrochemical cell, the overall reaction can be split into two half-reactions: one at the anode (oxidation) and one at the cathode (reduction). By convention, the cell is written with the anode on the left and the cathode on the right, separated by a double vertical line representing the salt bridge:

$$\text{Anode}\;|\;\text{Anode solution}\;||\;\text{Cathode solution}\;|\;\text{Cathode}$$

The electromotive force (EMF) of the cell is defined as the maximum potential difference between the two electrodes under reversible (zero current) conditions. This connects directly to the Gibbs energy change of the cell reaction:

$$\boxed{\Delta G = -nFE}$$

where $n$ is the number of electrons transferred, $F = 96485\;\text{C mol}^{-1}$ is the Faraday constant, and $E$ is the cell EMF.

Key Conventions

Oxidation occurs at the anode; reduction at the cathode.
In a galvanic cell, electrons flow spontaneously from anode to cathode through the external circuit.
The salt bridge maintains electrical neutrality by allowing ion migration between compartments.
Standard conditions: all species at unit activity ($a = 1$), gases at 1 bar, temperature at 298.15 K.
Cell EMF is an intensive property — it does not depend on the stoichiometric coefficients (only $n$ does).

2. Derivation: The Nernst Equation

The Nernst equation relates the cell EMF to the activities (concentrations) of reactants and products. It is one of the most important equations in electrochemistry, connecting thermodynamics to measurable electrode potentials.

2.1 Starting Point: Chemical Potential and Gibbs Energy

For a general chemical reaction at constant temperature and pressure, the Gibbs energy of reaction is:

$$\Delta G = \Delta G^\circ + RT\ln Q$$

where $Q$ is the reaction quotient and $\Delta G^\circ$ is the standard Gibbs energy change.

For a redox reaction with $n$ electrons transferred, we use the fundamental relation$\Delta G = -nFE$ and $\Delta G^\circ = -nFE^\circ$. Substituting:

$$-nFE = -nFE^\circ + RT\ln Q$$

Divide both sides by $-nF$:

$$E = E^\circ - \frac{RT}{nF}\ln Q$$

2.2 The General Nernst Equation

$$\boxed{E = E^\circ - \frac{RT}{nF}\ln Q}$$

The Nernst equation: the cell EMF as a function of the reaction quotient.

For a general cell reaction $a\text{A} + b\text{B} \rightarrow c\text{C} + d\text{D}$, the reaction quotient is:

$$Q = \frac{a_\text{C}^c \cdot a_\text{D}^d}{a_\text{A}^a \cdot a_\text{B}^b}$$

2.3 The 25°C Form

At 298.15 K, converting from natural logarithm to base-10 logarithm ($\ln Q = 2.303\log Q$):

$$E = E^\circ - \frac{(8.314)(298.15)}{n(96485)} \times 2.303 \log Q$$

$$E = E^\circ - \frac{0.05916}{n}\log Q$$

$$\boxed{E = E^\circ - \frac{0.0592\;\text{V}}{n}\log Q \quad (T = 298\;\text{K})}$$

The Nernst equation at 25°C — the most commonly used form.

2.4 Calculating Cell EMF from Half-Cell Potentials

The cell EMF is computed from the standard reduction potentials of the two half-reactions:

$$E^\circ_\text{cell} = E^\circ_\text{cathode} - E^\circ_\text{anode}$$

Both potentials are written as reduction potentials. The anode potential is subtracted because the reaction is reversed (oxidation) at the anode.

Worked Example: Daniell Cell

Cathode: $\text{Cu}^{2+}(aq) + 2e^- \rightarrow \text{Cu}(s)$, $E^\circ = +0.337\;\text{V}$

Anode: $\text{Zn}^{2+}(aq) + 2e^- \rightarrow \text{Zn}(s)$, $E^\circ = -0.763\;\text{V}$

$E^\circ_\text{cell} = 0.337 - (-0.763) = +1.100\;\text{V}$

With the Nernst equation at non-standard concentrations:

$E = 1.100 - \frac{0.0592}{2}\log\frac{[\text{Zn}^{2+}]}{[\text{Cu}^{2+}]}$

3. Derivation: Standard Electrode Potentials and the Electrochemical Series

3.1 The Standard Hydrogen Electrode (SHE)

Absolute single-electrode potentials cannot be measured directly — only differences between two electrodes are observable. By international convention, the standard hydrogen electrode (SHE) is assigned a potential of exactly zero at all temperatures:

$$E^\circ(\text{H}^+/\text{H}_2) \equiv 0\;\text{V} \quad \text{(at all } T\text{)}$$

The SHE consists of a Pt electrode in contact with $\text{H}_2$ gas at 1 bar and $\text{H}^+$ ions at unit activity ($a_{\text{H}^+} = 1$).

The standard reduction potential of any half-reaction is then measured against the SHE. If the metal electrode is more positive than SHE, it is a better oxidizing agent; if more negative, a better reducing agent.

3.2 Deriving the Cell Potential

Consider two half-cells with reduction potentials $E^\circ_1$ and $E^\circ_2$, where $E^\circ_1 > E^\circ_2$. The more positive electrode acts as cathode (reduction) and the more negative as anode (oxidation). The standard cell potential is:

$$E^\circ_\text{cell} = E^\circ_\text{cathode} - E^\circ_\text{anode}$$

Since $E^\circ_\text{cathode} > E^\circ_\text{anode}$, we always have $E^\circ_\text{cell} > 0$ for spontaneous cells.

3.3 Predicting Spontaneity

The connection between cell EMF and thermodynamic spontaneity is direct. From $\Delta G^\circ = -nFE^\circ$:

$E^\circ_\text{cell} > 0$

$\Delta G^\circ < 0$

Spontaneous (galvanic)

$E^\circ_\text{cell} = 0$

$\Delta G^\circ = 0$

Equilibrium

$E^\circ_\text{cell} < 0$

$\Delta G^\circ > 0$

Non-spontaneous (electrolytic)

3.4 Deriving the Equilibrium Constant from Cell Potential

At equilibrium, $\Delta G = 0$ and $Q = K$, so the Nernst equation gives $E = 0$:

$$0 = E^\circ - \frac{RT}{nF}\ln K$$

Rearranging:

$$\ln K = \frac{nFE^\circ}{RT}$$

$$\boxed{K = \exp\!\left(\frac{nFE^\circ}{RT}\right)}$$

The equilibrium constant derived from the standard cell potential.

This relation shows that even a modest cell potential corresponds to an enormous equilibrium constant. For the Daniell cell with $E^\circ = 1.100\;\text{V}$ and $n = 2$:

$$\ln K = \frac{2 \times 96485 \times 1.100}{8.314 \times 298.15} = 85.6$$

$$K = e^{85.6} \approx 1.5 \times 10^{37}$$

This enormous value confirms that the Daniell cell reaction goes essentially to completion.

3.5 The Complete Thermodynamic Connection

$$\Delta G^\circ = -nFE^\circ = -RT\ln K$$

The three fundamental quantities — $\Delta G^\circ$, $E^\circ$, and $K$ — are all interrelated.

Selected Standard Reduction Potentials (25°C)

Half-Reaction	$E^\circ$ (V)
$\text{F}_2 + 2e^- \rightarrow 2\text{F}^-$	+2.87
$\text{Au}^{3+} + 3e^- \rightarrow \text{Au}$	+1.50
$\text{Ag}^+ + e^- \rightarrow \text{Ag}$	+0.799
$\text{Cu}^{2+} + 2e^- \rightarrow \text{Cu}$	+0.337
$2\text{H}^+ + 2e^- \rightarrow \text{H}_2$	0.000
$\text{Ni}^{2+} + 2e^- \rightarrow \text{Ni}$	−0.257
$\text{Fe}^{2+} + 2e^- \rightarrow \text{Fe}$	−0.447
$\text{Zn}^{2+} + 2e^- \rightarrow \text{Zn}$	−0.763
$\text{Li}^+ + e^- \rightarrow \text{Li}$	−3.04

4. Derivation: Concentration Cells

A concentration cell consists of two identical electrodes immersed in solutions of the same electrolyte at different concentrations. Since the electrodes are identical, $E^\circ_\text{cell} = 0$, and the entire driving force comes from the concentration gradient.

4.1 Derivation of the EMF

Consider a cell with a metal M in contact with its ions $\text{M}^{n+}$ at two different activities $a_1$ (dilute, anode) and $a_2$ (concentrated, cathode):

$$\text{M}(s)\;|\;\text{M}^{n+}(a_1)\;||\;\text{M}^{n+}(a_2)\;|\;\text{M}(s)$$

Anode (oxidation): $\text{M}(s) \rightarrow \text{M}^{n+}(a_1) + ne^-$

Cathode (reduction): $\text{M}^{n+}(a_2) + ne^- \rightarrow \text{M}(s)$

Overall: $\text{M}^{n+}(a_2) \rightarrow \text{M}^{n+}(a_1)$

Applying the Nernst equation with $E^\circ = 0$ and $Q = a_1/a_2$:

$$E = 0 - \frac{RT}{nF}\ln\frac{a_1}{a_2} = \frac{RT}{nF}\ln\frac{a_2}{a_1}$$

$$\boxed{E = \frac{RT}{nF}\ln\frac{a_2}{a_1}}$$

EMF of a concentration cell. The cell operates spontaneously when $a_2 > a_1$, driving the system toward equal concentrations.

4.2 Determining Activity Coefficients

Concentration cells provide a precise method for measuring mean ionic activity coefficients$\gamma_\pm$. For a 1:1 electrolyte like HCl, we replace activities with $a = \gamma_\pm m / m^\circ$:

$$E = \frac{RT}{F}\ln\frac{\gamma_{\pm,2}\,m_2}{\gamma_{\pm,1}\,m_1}$$

For a cell with one solution at known activity (reference), measuring E gives $\gamma_\pm$ for the other.

By using a series of concentrations and extrapolating to infinite dilution (where $\gamma_\pm \to 1$), one can determine activity coefficients across a range of molalities with high precision.

4.3 Liquid Junction Potentials

When two electrolyte solutions of different composition or concentration are in direct contact, a liquid junction potential $E_j$ develops due to the different mobilities of cations and anions. For a simple 1:1 electrolyte with transport numbers$t_+$ and $t_-$:

$$E_j = (t_+ - t_-)\frac{RT}{F}\ln\frac{a_1}{a_2}$$

This contribution is minimized by using a salt bridge with equitransferent ions (e.g., KCl, where $t_+ \approx t_-$).

The total measured EMF of a cell with a liquid junction is:

$$E_\text{measured} = E_\text{cell} + E_j$$

5. Derivation: Temperature Dependence of EMF

The temperature coefficient of the cell EMF provides direct access to the entropy and enthalpy of the cell reaction, connecting electrochemical measurements to calorimetry.

5.1 Deriving the Entropy from EMF

Starting from $\Delta G = -nFE$, we take the temperature derivative at constant pressure:

$$\left(\frac{\partial \Delta G}{\partial T}\right)_P = -nF\left(\frac{\partial E}{\partial T}\right)_P$$

Recall from thermodynamics that $\left(\frac{\partial \Delta G}{\partial T}\right)_P = -\Delta S$:

$$-\Delta S = -nF\left(\frac{\partial E}{\partial T}\right)_P$$

$$\boxed{\Delta S = nF\left(\frac{\partial E}{\partial T}\right)_P}$$

The reaction entropy is directly proportional to the temperature coefficient of the EMF.

5.2 Deriving the Enthalpy from EMF

Using $\Delta G = \Delta H - T\Delta S$ and substituting our expressions:

$$-nFE = \Delta H - T\cdot nF\left(\frac{\partial E}{\partial T}\right)_P$$

Solving for $\Delta H$:

$$\Delta H = -nFE + nFT\left(\frac{\partial E}{\partial T}\right)_P$$

$$\boxed{\Delta H = -nF\!\left[E - T\left(\frac{\partial E}{\partial T}\right)_P\right]}$$

The Gibbs–Helmholtz equation applied to electrochemical cells.

5.3 Connecting Electrochemical and Calorimetric Measurements

The heat exchanged by the cell at constant pressure under reversible conditions is:

$$q_\text{rev} = T\Delta S = nFT\left(\frac{\partial E}{\partial T}\right)_P$$

This reveals three distinct cases based on the sign of the temperature coefficient:

$\left(\frac{\partial E}{\partial T}\right)_P > 0$

$\Delta S > 0$: The cell absorbs heat from surroundings. The electrical work exceeds $|\Delta H|$.

$\left(\frac{\partial E}{\partial T}\right)_P = 0$

$\Delta S = 0$: No heat exchange. $\Delta G = \Delta H$ and all enthalpy converts to electrical work.

$\left(\frac{\partial E}{\partial T}\right)_P < 0$

$\Delta S < 0$: The cell releases heat. Only part of $|\Delta H|$ converts to electrical work.

Worked Example: Temperature Dependence of a Cell

For the cell $\text{Zn}\,|\,\text{ZnCl}_2(aq)\,|\,\text{AgCl}\,|\,\text{Ag}$ at 25°C:

$E = 1.015\;\text{V}$, $(\partial E/\partial T)_P = -4.02 \times 10^{-4}\;\text{V K}^{-1}$, $n = 2$

$\Delta G = -2(96485)(1.015) = -195.9\;\text{kJ mol}^{-1}$

$\Delta S = 2(96485)(-4.02 \times 10^{-4}) = -77.6\;\text{J mol}^{-1}\text{K}^{-1}$

$\Delta H = -195.9 + 298.15(-0.0776) = -219.1\;\text{kJ mol}^{-1}$

This value agrees closely with calorimetric measurements, validating the electrochemical approach.

6. Applications of Electrochemical Cells

6.1 Batteries

Lead-Acid Battery

The workhorse of automobile starting systems. The overall reaction is:

$\text{Pb} + \text{PbO}_2 + 2\text{H}_2\text{SO}_4 \rightleftharpoons 2\text{PbSO}_4 + 2\text{H}_2\text{O}$

$E^\circ \approx 2.05\;\text{V}$ per cell; six cells give 12.3 V. Rechargeable because both electrode products ($\text{PbSO}_4$) adhere to the electrode surfaces.

Lithium-Ion Battery

Dominant in portable electronics and electric vehicles. Li$^+$ ions shuttle between graphite anode and metal oxide cathode (e.g., $\text{LiCoO}_2$):

$\text{LiCoO}_2 + \text{C}_6 \rightleftharpoons \text{Li}_{1-x}\text{CoO}_2 + \text{Li}_x\text{C}_6$

Operating voltage ~3.7 V. High energy density (~250 Wh/kg). The large negative reduction potential of Li makes it ideal for high-voltage cells.

6.2 Fuel Cells

A fuel cell converts chemical energy of a fuel (typically $\text{H}_2$) directly to electricity without combustion. The hydrogen–oxygen fuel cell operates as:

Anode: $\text{H}_2 \rightarrow 2\text{H}^+ + 2e^-$

Cathode: $\frac{1}{2}\text{O}_2 + 2\text{H}^+ + 2e^- \rightarrow \text{H}_2\text{O}$

Overall: $\text{H}_2 + \frac{1}{2}\text{O}_2 \rightarrow \text{H}_2\text{O}$, $E^\circ = 1.229\;\text{V}$

The theoretical maximum efficiency of a fuel cell is $\eta = \Delta G / \Delta H$, which is 83% for hydrogen at 25°C — significantly higher than the Carnot limit for heat engines at comparable temperatures.

6.3 Corrosion

Corrosion is an unwanted electrochemical process. Iron rusting is a galvanic cell in miniature: anodic regions (where Fe oxidizes to $\text{Fe}^{2+}$) and cathodic regions (where $\text{O}_2$ is reduced) form on the metal surface. Using the electrochemical series, any metal with $E^\circ < E^\circ(\text{O}_2/\text{H}_2\text{O})$is thermodynamically susceptible to corrosion in aerated water.

Cathodic protection prevents corrosion by connecting the metal to a more active (more negative $E^\circ$) “sacrificial anode” such as zinc or magnesium, which corrodes preferentially.

6.4 pH Measurement

The glass electrode for pH measurement is fundamentally an electrochemical cell. The potential across a thin glass membrane responds to the activity of $\text{H}^+$ ions via the Nernst equation:

$$E = E^\circ_\text{glass} + \frac{RT}{F}\ln a_{\text{H}^+} = E^\circ_\text{glass} - \frac{2.303\,RT}{F}\,\text{pH}$$

At 25°C, the slope is −59.16 mV per pH unit — the “Nernstian response.”

7. Historical Context

Alessandro Volta (1800)

Invented the voltaic pile — the first true battery — by stacking alternating discs of zinc and copper separated by brine-soaked cloth. This device produced a steady electric current for the first time, disproving Galvani's theory of “animal electricity” and founding the field of electrochemistry. The unit of electromotive force (volt) is named in his honor.

John Frederic Daniell (1836)

Developed the Daniell cell, which separated the two half-reactions into distinct compartments connected by a porous barrier. This eliminated the hydrogen gas buildup that plagued voltaic piles and provided a stable, long-lasting source of current for telegraphy.

Walther Nernst (1889)

Derived the equation that bears his name as part of his habilitation thesis at the University of Leipzig, connecting the EMF of an electrochemical cell to the concentrations of reacting species. Nernst also contributed the third law of thermodynamics and received the 1920 Nobel Prize in Chemistry.

Modern Battery Technology (1970s–present)

John Goodenough identified $\text{LiCoO}_2$ as a cathode material (1980), Akira Yoshino built the first practical Li-ion prototype (1985), and Stanley Whittingham pioneered intercalation chemistry. All three shared the 2019 Nobel Prize in Chemistry. Today, Li-ion cells power everything from smartphones to electric vehicles, with solid-state batteries as the next frontier.

8. Python Simulation: Nernst Equation Visualization

The following simulation plots cell potential versus concentration ratio for several electrochemical cells using the Nernst equation. It also explores the temperature dependence of cell EMF, showing how the driving force changes with both composition and thermal conditions.

Python

script.py150 lines

#!/usr/bin/env python3
"""
nernst_equation_visualization.py
Plot cell potential vs concentration ratio for electrochemical cells,
including temperature dependence analysis.
Uses only numpy and matplotlib (no scipy).
"""
import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

# --- Physical constants ---
R = 8.314462        # J/(mol*K)
F = 96485.332       # C/mol

# --- Define electrochemical cells ---
cells = {
    'Daniell (Zn|Cu)': {'E0': 1.100, 'n': 2, 'color': '#facc15'},
    'Ag|Cu': {'E0': 0.462, 'n': 2, 'color': '#22d3ee'},
    'Fe|Cu': {'E0': 0.784, 'n': 2, 'color': '#fb923c'},
    'Zn|Ag': {'E0': 1.562, 'n': 2, 'color': '#a78bfa'},
    'Ni|Cu': {'E0': 0.594, 'n': 2, 'color': '#4ade80'},
}

# --- Concentration ratio range (Q = [product]/[reactant]) ---
log_Q = np.linspace(-6, 6, 500)
Q = 10.0 ** log_Q

# --- Figure setup: dark theme, 2x2 panels ---
fig, axes = plt.subplots(2, 2, figsize=(14, 11))
fig.patch.set_facecolor('#0f172a')
fig.suptitle('Electrochemical Cells: Nernst Equation Analysis',
             color='#fde68a', fontsize=16, fontweight='bold', y=0.98)

for ax in axes.flat:
    ax.set_facecolor('#1e293b')
    ax.tick_params(colors='#cbd5e1', labelsize=9)
    for spine in ax.spines.values():
        spine.set_color('#334155')
    ax.grid(True, color='#334155', alpha=0.4, linewidth=0.5)

# --- Panel 1: E vs log(Q) for several cells at 298 K ---
ax = axes[0, 0]
T = 298.15
for name, info in cells.items():
    E0 = info['E0']
    n = info['n']
    E = E0 - (R * T) / (n * F) * np.log(Q)
    ax.plot(log_Q, E, color=info['color'], linewidth=2, label=f"{name} (E0={E0:.3f} V)")

ax.axhline(y=0, color='#ef4444', linestyle='--', alpha=0.5, linewidth=1)
ax.set_xlabel('log Q', color='#cbd5e1', fontsize=10)
ax.set_ylabel('Cell Potential E (V)', color='#cbd5e1', fontsize=10)
ax.set_title('Cell Potential vs log(Q) at 298 K', color='#fde68a', fontweight='bold', fontsize=11)
ax.legend(fontsize=7, facecolor='#1e293b', edgecolor='#334155', labelcolor='#cbd5e1', loc='upper right')

# --- Panel 2: E vs Q (linear scale, Daniell cell) at several temperatures ---
ax = axes[0, 1]
temperatures = [278.15, 298.15, 318.15, 338.15, 358.15]
temp_colors = ['#60a5fa', '#22d3ee', '#4ade80', '#facc15', '#fb923c']
Q_linear = np.linspace(0.001, 100, 500)

E0_daniell = 1.100
n_daniell = 2
dEdT = -4.0e-4  # typical temperature coefficient (V/K)

for T_val, tc in zip(temperatures, temp_colors):
    # Approximate E0 temperature dependence: E0(T) ~ E0(298) + dEdT*(T-298)
    E0_T = E0_daniell + dEdT * (T_val - 298.15)
    E = E0_T - (R * T_val) / (n_daniell * F) * np.log(Q_linear)
    label = f"T = {T_val - 273.15:.0f} C"
    ax.plot(Q_linear, E, color=tc, linewidth=2, label=label)

ax.axhline(y=0, color='#ef4444', linestyle='--', alpha=0.5, linewidth=1)
ax.set_xlabel('Reaction Quotient Q', color='#cbd5e1', fontsize=10)
ax.set_ylabel('Cell Potential E (V)', color='#cbd5e1', fontsize=10)
ax.set_title('Daniell Cell: E vs Q at Various T', color='#fde68a', fontweight='bold', fontsize=11)
ax.legend(fontsize=8, facecolor='#1e293b', edgecolor='#334155', labelcolor='#cbd5e1')
ax.set_xlim(0, 100)

# --- Panel 3: Temperature dependence of E0, Delta G, Delta S, Delta H ---
ax = axes[1, 0]
T_range = np.linspace(250, 400, 300)
E0_ref = 1.100
dEdT_values = [-4.0e-4, 0.0, 3.0e-4]
dEdT_labels = ['dE/dT = -0.4 mV/K', 'dE/dT = 0', 'dE/dT = +0.3 mV/K']
dEdT_colors = ['#fb923c', '#94a3b8', '#22d3ee']

for dE, lab, col in zip(dEdT_values, dEdT_labels, dEdT_colors):
    E0_T = E0_ref + dE * (T_range - 298.15)
    ax.plot(T_range, E0_T, color=col, linewidth=2, label=lab)

ax.set_xlabel('Temperature (K)', color='#cbd5e1', fontsize=10)
ax.set_ylabel('E0(T) (V)', color='#cbd5e1', fontsize=10)
ax.set_title('Temperature Dependence of E0', color='#fde68a', fontweight='bold', fontsize=11)
ax.legend(fontsize=8, facecolor='#1e293b', edgecolor='#334155', labelcolor='#cbd5e1')

# --- Panel 4: Thermodynamic quantities vs temperature ---
ax = axes[1, 1]
n_val = 2
dEdT_ex = -4.0e-4
E0_T_arr = E0_ref + dEdT_ex * (T_range - 298.15)
DeltaG = -n_val * F * E0_T_arr / 1000.0  # kJ/mol
DeltaS = n_val * F * dEdT_ex  # J/(mol*K), constant
DeltaH = DeltaG + T_range * DeltaS / 1000.0  # kJ/mol

ax.plot(T_range, DeltaG, color='#facc15', linewidth=2, label='Delta G (kJ/mol)')
ax.plot(T_range, DeltaH, color='#fb923c', linewidth=2, label='Delta H (kJ/mol)')
ax2 = ax.twinx()
ax2.plot(T_range, np.full_like(T_range, DeltaS), color='#22d3ee', linewidth=2,
         linestyle='--', label='Delta S (J/mol/K)')
ax2.set_ylabel('Delta S (J/mol/K)', color='#22d3ee', fontsize=10)
ax2.tick_params(colors='#22d3ee', labelsize=9)
ax2.spines['right'].set_color('#22d3ee')

ax.set_xlabel('Temperature (K)', color='#cbd5e1', fontsize=10)
ax.set_ylabel('Energy (kJ/mol)', color='#cbd5e1', fontsize=10)
ax.set_title('Thermodynamic Quantities vs T', color='#fde68a', fontweight='bold', fontsize=11)

# Combine legends
lines1, labels1 = ax.get_legend_handles_labels()
lines2, labels2 = ax2.get_legend_handles_labels()
ax.legend(lines1 + lines2, labels1 + labels2, fontsize=8,
          facecolor='#1e293b', edgecolor='#334155', labelcolor='#cbd5e1', loc='lower left')

plt.tight_layout(rect=[0, 0, 1, 0.95])
plt.savefig('output.png', dpi=150, facecolor='#0f172a', bbox_inches='tight')
print("IMAGE:output.png")

# --- Print numerical summary ---
print("\n=== Nernst Equation Summary ===")
print(f"{'Cell':<20} {'E0 (V)':>10} {'n':>5} {'K (298K)':>15} {'DG0 (kJ/mol)':>15}")
print("-" * 70)
for name, info in cells.items():
    E0 = info['E0']
    n = info['n']
    K = np.exp(n * F * E0 / (R * 298.15))
    DG0 = -n * F * E0 / 1000.0
    print(f"{name:<20} {E0:>10.3f} {n:>5d} {K:>15.3e} {DG0:>15.1f}")

print("\n=== Temperature Dependence (Daniell Cell) ===")
print(f"dE/dT = {dEdT_ex*1000:.2f} mV/K")
print(f"Delta S = {n_val * F * dEdT_ex:.1f} J/(mol*K)")
for T_val in [278.15, 298.15, 318.15, 338.15]:
    E0_T = E0_ref + dEdT_ex * (T_val - 298.15)
    DG = -n_val * F * E0_T / 1000.0
    DH = DG + T_val * n_val * F * dEdT_ex / 1000.0
    print(f"  T={T_val:.1f} K: E0={E0_T:.4f} V, DG={DG:.1f} kJ/mol, DH={DH:.1f} kJ/mol")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

9. Fortran Simulation: Cell Potentials and Thermodynamics

The following Fortran program computes standard cell potentials, equilibrium constants, and thermodynamic quantities ($\Delta G^\circ$, $\Delta S$, $\Delta H$) for a set of electrochemical reactions. It demonstrates Fortran's suitability for systematic numerical calculations in physical chemistry.

Python

script.py264 lines

import subprocess, sys, os, csv, io
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt
import numpy as np

# --- Embedded Fortran source ---
fortran_source = r"""
program electrochemical_cells
  implicit none
  integer, parameter :: dp = selected_real_kind(15, 307)

! Physical constants
  real(dp), parameter :: R_gas = 8.314462d0       ! J/(mol*K)
  real(dp), parameter :: Faraday = 96485.332d0    ! C/mol

! Number of cell reactions to analyze
  integer, parameter :: ncells = 7

! Arrays for cell data
  character(len=30) :: cell_name(ncells)
  real(dp) :: E0_cathode(ncells), E0_anode(ncells)
  integer  :: n_elec(ncells)
  real(dp) :: dEdT(ncells)   ! temperature coefficient (V/K)

! Computed quantities
  real(dp) :: E0_cell, DeltaG0, lnK, K_eq, DeltaS, DeltaH
  real(dp) :: T_ref
  integer  :: i

! Temperature sweep variables
  integer, parameter :: nT = 200
  real(dp) :: T_min, T_max, dT, T_val
  real(dp) :: E0_T, DG_T, DH_T
  integer  :: j

T_ref = 298.15d0
  T_min = 250.0d0
  T_max = 400.0d0
  dT = (T_max - T_min) / dble(nT - 1)

! --- Define cells ---
  ! Cell 1: Daniell cell (Zn|Cu)
  cell_name(1)   = 'Zn|Cu (Daniell)'
  E0_cathode(1)  = 0.337d0    ! Cu2+/Cu
  E0_anode(1)    = -0.763d0   ! Zn2+/Zn
  n_elec(1)      = 2
  dEdT(1)        = -4.02d-4

! Cell 2: Fe|Cu
  cell_name(2)   = 'Fe|Cu'
  E0_cathode(2)  = 0.337d0    ! Cu2+/Cu
  E0_anode(2)    = -0.447d0   ! Fe2+/Fe
  n_elec(2)      = 2
  dEdT(2)        = -2.1d-4

! Cell 3: Zn|Ag
  cell_name(3)   = 'Zn|Ag'
  E0_cathode(3)  = 0.799d0    ! Ag+/Ag
  E0_anode(3)    = -0.763d0   ! Zn2+/Zn
  n_elec(3)      = 2
  dEdT(3)        = -3.5d-4

! Cell 4: Ni|Cu
  cell_name(4)   = 'Ni|Cu'
  E0_cathode(4)  = 0.337d0    ! Cu2+/Cu
  E0_anode(4)    = -0.257d0   ! Ni2+/Ni
  n_elec(4)      = 2
  dEdT(4)        = -1.8d-4

! Cell 5: H2|O2 fuel cell
  cell_name(5)   = 'H2|O2 (fuel cell)'
  E0_cathode(5)  = 1.229d0    ! O2/H2O
  E0_anode(5)    = 0.000d0    ! H+/H2
  n_elec(5)      = 2
  dEdT(5)        = -8.46d-4

! Cell 6: Pb|PbO2 (lead-acid)
  cell_name(6)   = 'Pb|PbO2 (lead-acid)'
  E0_cathode(6)  = 1.685d0    ! PbO2/PbSO4
  E0_anode(6)    = -0.356d0   ! PbSO4/Pb
  n_elec(6)      = 2
  dEdT(6)        = -1.7d-4

! Cell 7: Li|CoO2 (approx Li-ion)
  cell_name(7)   = 'Li|CoO2 (Li-ion)'
  E0_cathode(7)  = 0.700d0    ! approximate
  E0_anode(7)    = -3.040d0   ! Li+/Li
  n_elec(7)      = 1
  dEdT(7)        = -2.0d-4

! --- Print header ---
  write(*,'(A)') '=================================================================='
  write(*,'(A)') '  ELECTROCHEMICAL CELLS: THERMODYNAMIC ANALYSIS (Fortran)'
  write(*,'(A)') '=================================================================='
  write(*,'(A,F8.2,A)') '  Reference temperature: ', T_ref, ' K'
  write(*,'(A,F12.3,A)') '  Faraday constant:     ', Faraday, ' C/mol'
  write(*,'(A,F12.6,A)') '  Gas constant:         ', R_gas, ' J/(mol*K)'
  write(*,'(A)') '=================================================================='
  write(*,'(A)') ''

! --- Compute and print results for each cell ---
  do i = 1, ncells
    E0_cell = E0_cathode(i) - E0_anode(i)
    DeltaG0 = -dble(n_elec(i)) * Faraday * E0_cell / 1000.0d0  ! kJ/mol
    lnK = dble(n_elec(i)) * Faraday * E0_cell / (R_gas * T_ref)
    K_eq = exp(min(lnK, 700.0d0))  ! cap to avoid overflow
    DeltaS = dble(n_elec(i)) * Faraday * dEdT(i)
    DeltaH = DeltaG0 * 1000.0d0 + T_ref * DeltaS
    DeltaH = DeltaH / 1000.0d0  ! kJ/mol

write(*,'(A,A)') '  Cell: ', trim(cell_name(i))
    write(*,'(A,F10.4,A)') '    E0(cathode) = ', E0_cathode(i), ' V'
    write(*,'(A,F10.4,A)') '    E0(anode)   = ', E0_anode(i), ' V'
    write(*,'(A,F10.4,A)') '    E0(cell)    = ', E0_cell, ' V'
    write(*,'(A,I5)')       '    n           = ', n_elec(i)
    write(*,'(A,F10.2,A)') '    Delta G0    = ', DeltaG0, ' kJ/mol'
    write(*,'(A,ES12.4)')  '    ln K        = ', lnK
    if (lnK < 700.0d0) then
      write(*,'(A,ES12.4)')  '    K           = ', K_eq
    else
      write(*,'(A)')         '    K           = > 10^304 (overflow)'
    end if
    write(*,'(A,F10.2,A)') '    Delta S     = ', DeltaS, ' J/(mol*K)'
    write(*,'(A,F10.2,A)') '    Delta H     = ', DeltaH, ' kJ/mol'
    write(*,'(A,ES10.2,A)') '    dE/dT       = ', dEdT(i), ' V/K'
    write(*,'(A)') ''
  end do

! --- Temperature sweep data for plotting ---
  write(*,'(A)') 'DATA_START'
  write(*,'(A)') 'T,E0_1,DG_1,DH_1,E0_2,DG_2,DH_2,E0_3,DG_3,DH_3,E0_4,DG_4,DH_4'

do j = 1, nT
    T_val = T_min + dble(j - 1) * dT
    write(*,'(F8.2)', advance='no') T_val

do i = 1, 4
      E0_cell = E0_cathode(i) - E0_anode(i)
      E0_T = E0_cell + dEdT(i) * (T_val - T_ref)
      DG_T = -dble(n_elec(i)) * Faraday * E0_T / 1000.0d0
      DH_T = (DG_T * 1000.0d0 + T_val * dble(n_elec(i)) * Faraday * dEdT(i)) / 1000.0d0

write(*,'(A,F10.5,A,F10.3,A,F10.3)', advance='no') &
        ',', E0_T, ',', DG_T, ',', DH_T
    end do

write(*,'(A)') ''
  end do

write(*,'(A)') 'DATA_END'
  write(*,'(A)') '=================================================================='

end program electrochemical_cells
"""

# Write, compile, and run the Fortran code
with open('electrochem.f90', 'w') as f:
    f.write(fortran_source)

comp = subprocess.run(['gfortran', '-O2', '-o', './electrochem', 'electrochem.f90'],
                      capture_output=True, text=True)
if comp.returncode != 0:
    print("Compilation error:")
    print(comp.stderr)
    sys.exit(1)

result = subprocess.run(['./electrochem'], capture_output=True, text=True, timeout=30)
stdout = result.stdout

# Parse output: print text, collect DATA
in_data = False
data_lines = []
for line in stdout.splitlines():
    stripped = line.strip()
    if stripped == 'DATA_START':
        in_data = True
        continue
    elif stripped == 'DATA_END':
        in_data = False
        continue
    if in_data:
        data_lines.append(stripped)
    else:
        print(line)

# Parse CSV data
header = data_lines[0].split(',')
rows = []
for line in data_lines[1:]:
    parts = line.split(',')
    try:
        rows.append([float(x.strip()) for x in parts])
    except ValueError:
        continue

data = np.array(rows)
T_arr = data[:, 0]

cell_names_plot = ['Zn|Cu (Daniell)', 'Fe|Cu', 'Zn|Ag', 'Ni|Cu']
colors_plot = ['#facc15', '#fb923c', '#a78bfa', '#4ade80']

# --- Plot results ---
fig, axes = plt.subplots(2, 2, figsize=(14, 11))
fig.patch.set_facecolor('#0f172a')
fig.suptitle('Electrochemical Cells: Fortran Thermodynamic Analysis',
             color='#fde68a', fontsize=16, fontweight='bold', y=0.98)

# Panel 1: E0 vs T
ax = axes[0, 0]
for idx, (name, col) in enumerate(zip(cell_names_plot, colors_plot)):
    E0_col = 1 + idx * 3  # columns: E0, DG, DH for each cell
    ax.plot(T_arr, data[:, E0_col], color=col, linewidth=2, label=name)
ax.set_xlabel('Temperature (K)', color='#cbd5e1', fontsize=10)
ax.set_ylabel('E0(T) (V)', color='#cbd5e1', fontsize=10)
ax.set_title('Standard Cell Potential vs Temperature', color='#fde68a', fontweight='bold', fontsize=11)
ax.legend(fontsize=8, facecolor='#1e293b', edgecolor='#334155', labelcolor='#cbd5e1')

# Panel 2: Delta G vs T
ax = axes[0, 1]
for idx, (name, col) in enumerate(zip(cell_names_plot, colors_plot)):
    DG_col = 2 + idx * 3
    ax.plot(T_arr, data[:, DG_col], color=col, linewidth=2, label=name)
ax.set_xlabel('Temperature (K)', color='#cbd5e1', fontsize=10)
ax.set_ylabel('Delta G0 (kJ/mol)', color='#cbd5e1', fontsize=10)
ax.set_title('Standard Gibbs Energy vs Temperature', color='#fde68a', fontweight='bold', fontsize=11)
ax.legend(fontsize=8, facecolor='#1e293b', edgecolor='#334155', labelcolor='#cbd5e1')

# Panel 3: Delta H vs T
ax = axes[1, 0]
for idx, (name, col) in enumerate(zip(cell_names_plot, colors_plot)):
    DH_col = 3 + idx * 3
    ax.plot(T_arr, data[:, DH_col], color=col, linewidth=2, label=name)
ax.set_xlabel('Temperature (K)', color='#cbd5e1', fontsize=10)
ax.set_ylabel('Delta H (kJ/mol)', color='#cbd5e1', fontsize=10)
ax.set_title('Reaction Enthalpy vs Temperature', color='#fde68a', fontweight='bold', fontsize=11)
ax.legend(fontsize=8, facecolor='#1e293b', edgecolor='#334155', labelcolor='#cbd5e1')

# Panel 4: Equilibrium constant vs T for Daniell cell
ax = axes[1, 1]
R_gas = 8.314462
F_val = 96485.332
for idx, (name, col) in enumerate(zip(cell_names_plot, colors_plot)):
    E0_col = 1 + idx * 3
    n_val = 2
    lnK = n_val * F_val * data[:, E0_col] / (R_gas * T_arr)
    log10K = lnK / np.log(10.0)
    ax.plot(T_arr, log10K, color=col, linewidth=2, label=name)
ax.set_xlabel('Temperature (K)', color='#cbd5e1', fontsize=10)
ax.set_ylabel('log10(K)', color='#cbd5e1', fontsize=10)
ax.set_title('Equilibrium Constant vs Temperature', color='#fde68a', fontweight='bold', fontsize=11)
ax.legend(fontsize=8, facecolor='#1e293b', edgecolor='#334155', labelcolor='#cbd5e1')

plt.tight_layout(rect=[0, 0, 1, 0.95])
plt.savefig('output.png', dpi=150, facecolor='#0f172a', bbox_inches='tight')
print("IMAGE:output.png")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

References

Atkins, P. & de Paula, J. Atkins' Physical Chemistry, 11th ed. (Oxford, 2018), Chapters 6–7.
Bard, A.J. & Faulkner, L.R. Electrochemical Methods: Fundamentals and Applications, 2nd ed. (Wiley, 2001). — The standard graduate reference for electrochemistry.
Nernst, W. (1889). “Die elektromotorische Wirksamkeit der Ionen,” Zeitschrift für physikalische Chemie 4, 129–181. — The original derivation of the Nernst equation.
Volta, A. (1800). Letter to Sir Joseph Banks, Philosophical Transactions of the Royal Society 90, 403–431. — The announcement of the voltaic pile.
Goodenough, J.B. & Park, K.-S. (2013). “The Li-Ion Rechargeable Battery: A Perspective,” Journal of the American Chemical Society 135, 1167–1176.
Hamann, C.H., Hamnett, A. & Vielstich, W. Electrochemistry, 2nd ed. (Wiley-VCH, 2007). — Comprehensive coverage of electrode potentials and thermodynamics.

← Part V Overview Back to Physical Chemistry →

Half-Reaction	\(E^\circ\) (V)
\(\text{F}_2 + 2e^- \rightarrow 2\text{F}^-\)	+2.87
\(\text{Au}^{3+} + 3e^- \rightarrow \text{Au}\)	+1.50
\(\text{Ag}^+ + e^- \rightarrow \text{Ag}\)	+0.799
\(\text{Cu}^{2+} + 2e^- \rightarrow \text{Cu}\)	+0.337
\(2\text{H}^+ + 2e^- \rightarrow \text{H}_2\)	0.000
\(\text{Ni}^{2+} + 2e^- \rightarrow \text{Ni}\)	−0.257
\(\text{Fe}^{2+} + 2e^- \rightarrow \text{Fe}\)	−0.447
\(\text{Zn}^{2+} + 2e^- \rightarrow \text{Zn}\)	−0.763
\(\text{Li}^+ + e^- \rightarrow \text{Li}\)	−3.04