Comprehensive Study Guide for Electrochemistry: Cells, Potentials, and Faraday's Laws, and the Nernst Equation

Fundamental Concepts of Electrochemistry and Electrodes

Electrochemistry involves the study of electricity in relation to chemical changes. A central component in this study is the electrode, which is used to facilitate the flow of electric current. Electrodes are categorized into two types based on the chemical processes that occur at their surface. The anode is the electrode where the oxidation process occurs, characterized by the donation of electrons ($e^-$ donating). Conversely, the cathode is the electrode where the reduction process occurs, characterized by the gaining of electrons ($e^-$ gaining).

A cell is defined as an apparatus formed by two electrodes fixed into an electrolyte, which can be in a molten state or an aqueous solution. In some configurations, two electrodes are fixed into two different electrolytes, which may also be molten or aqueous solutions of the electrolyte. These electrolytes typically exist as molten electrolytes or electrolytic aqueous solutions.

Types of Cells: Electrolytic and Electrochemical

Cells are broadly classified into two categories: electrolytic cells and electrochemical cells. An electrolytic cell is a device in which the electrolysis of a molten electrolyte or its aqueous solution occurs. In this system, electrical energy is converted into chemical energy. For example, during the electrolysis of molten sodium chloride ($NaCl$), the reaction is expressed as 2NaCl→2Na++2Cl−2NaCl \rightarrow 2Na^+ + 2Cl^-. When electricity is applied to molten aluminum oxide ($Al_2O_3$), it dissociates into ions: 2Al2O3→4Al3++6O2−2Al_2O_3 \rightarrow 4Al^{3+} + 6O^{2-}.

In the electrolysis process of molten $Al_2O_3$, electrons are donated at the anode and gained at the cathode. At the anode, the reaction is 6O2−→3O2+12e−6O^{2-} \rightarrow 3O_2 + 12e^-, illustrating electron donation and the liberation of oxygen gas. At the cathode, the reaction is 4Al3++12e−→4Al4Al^{3+} + 12e^- \rightarrow 4Al, illustrating reduction and the deposition of aluminum metal. Another example is the electrolysis of molten $NaCl$. At the anode, 2Cl−→Cl2+2e−2Cl^- \rightarrow Cl_2 + 2e^-, and at the cathode, 2Na++2e−→2Na2Na^+ + 2e^- \rightarrow 2Na. The result of such processes is that the anode liberates gas while the cathode deposits metal.

Faraday's Laws of Electrolysis

Faraday's first law of electrolysis states that the amount of formed product (the deposited amount, $W$) is directly proportional to the flow of charge ($Q$) through the electrolyte. Mathematically, this is expressed as W∝QW \propto Q. Since charge is the product of current ($I$) and time ($t$), Q=ItQ = It, the formula becomes W=ZItW = ZIt, where $Z$ is the electrochemical equivalent. The electrochemical equivalent $Z$ is defined as the amount of product formed during electrolysis per unit of charge flow. The value of $Z$ can be calculated using the equivalent mass ($E$) and the Faraday constant ($96500\,C$): Z=E96500Z = \frac{E}{96500}. Thus, the total mass deposited is W=EIt96500W = \frac{EIt}{96500} or W=QE96500W = \frac{QE}{96500}.

Equivalent mass ($E$) is determined by the molecular mass divided by the valency factor ($n$), expressed as E=Mol. MassnE = \frac{\text{Mol. Mass}}{n}. For example, for copper ($Cu$) from $CuSO_4$ in an aqueous solution, Cu2++2e−→CuCu^{2+} + 2e^- \rightarrow Cu, so $n = 2$ and E=63.52=31.75E = \frac{63.5}{2} = 31.75. For silver ($Ag$), Ag++e−→AgAg^+ + e^- \rightarrow Ag, so $n = 1$ and E=1081=108E = \frac{108}{1} = 108. The Faraday constant ($F$) represents the charge of one mole of electrons. Given Avogadro's number 6.022×10236.022 \times 10^{23} and the charge of a single electron 1.6×10−19 Coulomb1.6 \times 10^{-19}\,\text{Coulomb}, the total charge is Q=6.022×1023×1.6×10−19=96487 CoulombQ = 6.022 \times 10^{23} \times 1.6 \times 10^{-19} = 96487\,\text{Coulomb}, which is approximated to 96500 C96500\,C (1.0 Faraday).

Faraday's second law of electrolysis applies when different electrolytes in aqueous solutions are connected in a series combination. The law states that the amount of different substances released by the same quantity of electricity passing through them is proportional to their chemical equivalent masses. This is expressed as W1E1=W2E2=W3E3\frac{W_1}{E_1} = \frac{W_2}{E_2} = \frac{W_3}{E_3}, where $W$ represents the mass and $E$ represents the equivalent mass. In such series circuits, the gram equivalents released will be similar.

Electrochemical Cells and the Daniel Cell

An electrochemical cell converts chemical energy into electrical energy. These are also known as Galvanic or Voltaic cells, named after Luigi Galvani and Alessandro Volta. A classic example is the Daniel Cell, which is a redox cell. In a Daniel Cell, zinc ($Zn$) and copper ($Cu$) electrodes are used. The anodic half-cell contains a $Zn$ rod in a $ZnSO_4(aq)$ solution, where oxidation occurs: Zn(s)→Zn2+(aq)+2e−Zn(s) \rightarrow Zn^{2+}(aq) + 2e^-. The cathodic half-cell contains a $Cu$ rod in a $CuSO_4(aq)$ solution, where reduction occurs: Cu2+(aq)+2e−→Cu(s)Cu^{2+}(aq) + 2e^- \rightarrow Cu(s). The complete redox reaction is Zn(s)+Cu2+(aq)→Zn2+(aq)+Cu(s)Zn(s) + Cu^{2+}(aq) \rightarrow Zn^{2+}(aq) + Cu(s).

A salt bridge is used to connect the two half-cells. It typically contains electrolytes like $NH_4NO_3$, $KNO_3$, or $K_2SO_4$. The salt bridge serves two primary purposes: to complete the electrical circuit and to maintain electrical neutrality in the solutions. As the reaction proceeds, the mass of the anode rod decreases due to $Zn$ dissolving into the solution, while the mass of the cathode rod increases as $Cu$ is deposited. The cell representation follows the notation: Zn(s)∣Zn2+(aq)(C1)∣∣Cu2+(aq)(C2)∣Cu(s)Zn(s) | Zn^{2+}(aq) (C_1) || Cu^{2+}(aq) (C_2) | Cu(s), where $||$ represents the salt bridge and $C_1, C_2$ represent the concentrations of the electrolyte solutions.

Electrode Potential and Electromotive Force (EMF)

Electrode potential ($E$) is the potential difference established between a metal electrode and its surrounding electrolyte solution. It is classified into Oxidation Potential ($E_{ox}$) and Reduction Potential ($E_{red}$). These are related by the formula Eox=−EredE_{ox} = -E_{red}. The Electromotive Force (EMF) or $E_{cell}$ is the algebraic sum of the electric potentials of the respective electrodes: Ecell=[Eox]A+[Ered]CE_{cell} = [E_{ox}]_A + [E_{red}]_C. Using standard reduction potentials, this is often expressed as Ecell=[Ered]C−[Ered]AE_{cell} = [E_{red}]_C - [E_{red}]_A.

The Standard Hydrogen Electrode (S.H.E.) or Normal Hydrogen Electrode (N.H.E.) is used as a reference electrode. Its standard electrode potential is defined as 0.0 V0.0\,V at $1.0\,mol/dm^3$ concentration and $298\,K$. At the anode, the reaction is H2→2H++2e−H_2 \rightarrow 2H^+ + 2e^-, and at the cathode, it is 2H++2e−→H22H^+ + 2e^- \rightarrow H_2. The potential for these reactions is EH+/H2=+0.0 VE_{H^+/H_2} = +0.0\,V.

Thermodynamics and the Nernst Equation

There is a direct relationship between the Gibbs free energy (ΔG\Delta G) and the EMF of a cell. The total work done by the cell or the change in Gibbs energy is given by ΔG=−nFEcell\Delta G = -nFE_{cell}. For standard conditions, ΔG∘=−nFEcell∘\Delta G^\circ = -nFE_{cell}^\circ. For a process to be spontaneous, ΔG\Delta G must be negative, which implies that $E_{cell}$ must be positive ($E_{cell} = +V$).

The Nernst Equation relates the reduction potential of an electrode (or the cell EMF) to the concentration of the species involved. For a reduction reaction Mn+(aq)+ne−→M(s)M^{n+}(aq) + ne^- \rightarrow M(s), the Nernst Equation is Ered=Ered∘−RTnFln⁡[M][Mn+]E_{red} = E_{red}^\circ - \frac{RT}{nF} \ln\frac{[M]}{[M^{n+}]}. Since the concentration of a pure solid $[M]$ is $1$, and substituting standard values (R=8.314 J/K⋅molR = 8.314\,J/K\cdot mol, T=298 KT = 298\,K, F=96500 CF = 96500\,C), the equation simplifies to Ered=Ered∘−0.059nlog⁡101[Mn+]E_{red} = E_{red}^\circ - \frac{0.059}{n} \log_{10}\frac{1}{[M^{n+}]}. For a full cell reaction aA+bB→cC+dDaA + bB \rightarrow cC + dD, the equation is Ecell=Ecell∘−0.059nlog⁡10QE_{cell} = E_{cell}^\circ - \frac{0.059}{n} \log_{10} Q, where $Q$ is the reaction quotient defined as Q=[C]c[D]d[A]a[B]bQ = \frac{[C]^c [D]^d}{[A]^a [B]^b}.

For example, in the Daniel Cell involving $Zn$ and $Cu$, the equation is Ecell=Ecell∘−0.0592log⁡10[Zn2+][Cu2+]E_{cell} = E_{cell}^\circ - \frac{0.059}{2} \log_{10} \frac{[Zn^{2+}]}{[Cu^{2+}]}. Another calculated example for a $Mg/Ag$ cell: Mg(s)+2Ag+(0.0001M)→Mg2+(0.130M)+2Ag(s)Mg(s) + 2Ag^+(0.0001M) \rightarrow Mg^{2+}(0.130M) + 2Ag(s). Given Ecell∘=3.17 VE_{cell}^\circ = 3.17\,V, the EMF is calculated as Ecell=3.17−0.0592log⁡100.130(0.0001)2=3.17−0.0295×log⁡10(1.30×107)=3.17−0.0295(0.180+7)=2.96 VE_{cell} = 3.17 - \frac{0.059}{2} \log_{10} \frac{0.130}{(0.0001)^2} = 3.17 - 0.0295 \times \log_{10}(1.30 \times 10^7) = 3.17 - 0.0295(0.180 + 7) = 2.96\,V.

Equilibrium Constant and the Electrochemical Series

At equilibrium, the $E_{cell}$ of a system is zero ($0$). Substituting this into the Nernst Equation, we find the relationship with the equilibrium constant ($K_c$): 0=Ecell∘−0.059nlog⁡10Kc0 = E_{cell}^\circ - \frac{0.059}{n} \log_{10} K_c, which leads to log⁡10Kc=nEcell∘0.059\log_{10} K_c = \frac{nE_{cell}^\circ}{0.059}. This allows for the calculation of $K_c$ using the formula Kc=10nEcell∘0.059K_c = 10^{\frac{nE_{cell}^\circ}{0.059}}. Additionally, the relationship with Gibbs energy is ΔG∘=−RTln⁡Kc=−2.303RTlog⁡10Kc\Delta G^\circ = -RT \ln K_c = -2.303 RT \log_{10} K_c.

The Electrochemical Series (E.C.S.) is formed by arranging elements or ions in order of their increasing or decreasing standard reduction potentials (Ered∘E_{red}^\circ). Species with high reduction potentials have high electron-gaining ability, making them strong oxidizing agents (e.g., F2+2e−→2F−F_2 + 2e^- \rightarrow 2F^-, $E^ atural = +2.87\,V$). Species with low (highly negative) reduction potentials have high electron-donating ability, making them strong reducing agents (e.g., Li++e−→LiLi^+ + e^- \rightarrow Li, $E^ atural = -3.05\,V$).

Applications of the E.C.S. include predicting reducing or oxidizing power, metallic character, and chemical reactivity. Metals with lower reduction potentials are more reactive and can displace metals with higher reduction potentials from their salts. For instance, $Zn$ can displace $Cu$ from $CuSO_4$ because $Zn$ has a lower $E_{red}$ ($-0.76\,V$) than $Cu$ ($+0.34\,V$). Furthermore, metals with a reduction potential lower than hydrogen ($0.0\,V$) can release $H_2$ gas from acids. The stability of metal oxides also depends on the E.C.S.; oxides of strong metals (low $E_{red}$) are very stable and do not decompose upon heating, whereas oxides of weak metals (high $E_{red}$, like $HgO$ or $Ag_2O$) are less stable and decompose easily.