Electrochemistry - Class 12

Consider the following electrochemical cell at $298 \text{ K}$:

$Pt(s) | H_2(g, 1 \text{ bar}) | HA(aq, 0.1 \text{ M}) || Cu^{2+}(aq, 0.01 \text{ M}) | Cu(s)$

Given: Acid dissociation constant of the weak acid $HA$, $K_a = 1.0 \times 10^{-5}$ Standard reduction potential, $E^\circ_{Cu^{2+}/Cu} = +0.34 \text{ V}$ $\frac{2.303 RT}{F} = 0.059 \text{ V}$ Faraday's constant, $F = 96500 \text{ C mol}^{-1}$

Calculate the Gibbs free energy change ($\Delta G$) for the cell reaction.

A galvanic cell is constructed at $298\text{ K}$ using Zirconium and Hafnium electrodes immersed in their respective hydroxide suspensions:

$\ce{Zr(s) | Zr(OH)4(s) | Buffer (pH = 2.0) || Buffer (pH = 1.5) | Hf(OH)4(s) | Hf(s)}$

Due to the lanthanoid contraction, $\ce{Zr}$ and $\ce{Hf}$ exhibit nearly identical atomic/ionic radii and chemical properties. Consequently, assume their standard reduction potentials are equal ($E^\circ_{\ce{Zr^{4+}/Zr}} = E^\circ_{\ce{Hf^{4+}/Hf}}$) and their hydroxide solubility products are identical ($K_{sp} = 10^{-52}$). Assuming $\ce{M^{4+}}$ is the only soluble metal species, calculate the cell potential ($E_{cell}$) at $298\text{ K}$.

(Given: $\frac{2.303 RT}{F} = 0.059\text{ V}$)

A $1.0 \text{ L}$ acidic aqueous solution contains \ce{Sn^{4+}} and \ce{Sn^{2+}} ions. The initial concentration of \ce{Sn^{2+}} is equal to the hydrogen ion concentration of a $0.10 \text{ M}$ aqueous solution of a monobasic weak acid \ce{HA}. For \ce{HA}, the molar conductivity is $3.90 \text{ S cm}^2 \text{ mol}^{-1}$ and the limiting molar conductivity is $390 \text{ S cm}^2 \text{ mol}^{-1}$ at $298 \text{ K}$.

The initial reduction potential of the \ce{Sn^{4+}/Sn^{2+}} couple in the solution is $E^\circ_{\ce{Sn^{4+}/Sn^{2+}}} + 0.0295 \text{ V}$.

A constant current of $1.93 \text{ A}$ is passed through the solution using inert electrodes to selectively reduce \ce{Sn^{4+}} to \ce{Sn^{2+}}$. Assuming no change in volume, what is the time required for the reduction potential of the couple to become$E^\circ_{\ce{Sn^{4+}/Sn^{2+}}} - 0.0295 \text{ V}$?

(Given: $\frac{2.303 RT}{F} = 0.059 \text{ V}$ at $298 \text{ K}$, $1 \text{ F} = 96500 \text{ C mol}^{-1}$)

A galvanic cell is constructed as follows: $\ce{Ag(s) | Ag+(aq, 1 M) || Cl-(aq) | Cl2(g, 1 bar) | Pt(s)}$.

The chloride ions in the cathode compartment are continuously consumed by a pseudo-first-order chemical reaction. During the kinetic study, the cell potential $E_{cell}$ is observed to increase linearly with time $t$.

Let $S$ be the constant rate of increase of the cell potential ($S = \frac{dE_{cell}}{dt}$). The experiment is conducted at two different temperatures, $T_1$ and $T_2$ ($T_2 > T_1$), yielding rates $S_1$ and $S_2$, respectively.

Assuming the pre-exponential factor of the consumption reaction is independent of temperature, which of the following expressions correctly represents the activation energy ($E_a$) of the chloride consumption reaction?

An aqueous solution of $\ce{Pd(NO3)2}$ is treated with excess $\ce{KSCN}$ to form a thermodynamically stable square planar complex ion, $P$. The ligand in $P$ exhibits linkage isomerism but coordinates predominantly through its softer donor atom. The overall formation constant ($\beta_4$) of $P$ is $10^{24}$.

A half-cell is constructed by dipping a Palladium rod into a $1.0\text{ L}$ solution initially containing $0.1\text{ M } \ce{Pd^{2+}}$ and $0.5\text{ M } \ce{SCN^-}$ at $298\text{ K}$.

Given: $E^\circ_{\ce{Pd^{2+}/Pd}} = +0.95\text{ V}$, $\frac{2.303RT}{F} = 0.06\text{ V}$.

Identify the correct IUPAC name of the complex ion $P$ and the reduction potential ($E_{\text{red}}$) of this half-cell.

Consider the thermodynamic factors governing the standard electrode potentials ($E^\circ_{M^{2+}/M}$) of the 3d transition metal series at 298 K. Evaluate the following statements:

I. The general trend towards less negative $E^\circ_{M^{2+}/M}$ values across the 3d series is primarily governed by the steady increase in the sum of the first and second ionization enthalpies ($\Delta_{i}H_1 + \Delta_{i}H_2$). II. Copper exhibits a positive $E^\circ_{M^{2+}/M}$ value ($+0.34\text{ V}$) because the energy required for its atomization and ionization is not adequately compensated by the hydration enthalpy of the \ce{Cu^{2+}_{(aq)}} ion. III. The $E^\circ_{M^{2+}/M}$ value of Nickel is less negative than expected from the general trend due to its exceptionally high enthalpy of atomization ($\Delta_{sub}H$). IV. The anomalously high negative $E^\circ_{M^{2+}/M}$ values for Manganese and Zinc are primarily attributed to their exceptionally high negative hydration enthalpies ($\Delta_{hyd}H$).

Which of the following combinations identifies the CORRECT statements?

Consider the following electrochemical cell operating in a mixed aqueous-organic solvent at $298 \text{ K}$:

$\ce{Zn(s) | ZnSO4 (0.1 M) || CuSO4 (0.1 M) | Cu(s)}$

Due to the lower dielectric constant of the solvent, significant ion-pairing occurs, preventing complete dissociation. The observed Van't Hoff factors ($i$) for the $0.1 \text{ M}$ $\ce{ZnSO4}$ and $0.1 \text{ M}$ $\ce{CuSO4}$ solutions are $1.40$ and $1.04$, respectively.

Given standard reduction potentials: $E^\circ_{\ce{Zn^{2+}/Zn}} = -0.76 \text{ V}$ and $E^\circ_{\ce{Cu^{2+}/Cu}} = +0.34 \text{ V}$.

Assume the activity coefficients of the free, un-paired ions are unity, and that ion-pairing is the sole reason for the deviation of $i$ from the ideal stoichiometric value. (Take $\frac{2.303 RT}{F} = 0.06 \text{ V}$)

What is the cell potential, $E_{\text{cell}}$?

An aqueous solution of $1.0\text{ M } \ce{NaCl}$, buffered at $\text{pH} = 7.0$, is electrolyzed at $298\text{ K}$ using smooth Platinum electrodes. A constant current of $10.0\text{ A}$ is passed for $965\text{ s}$.

Given standard reduction potentials at $298\text{ K}$: $E^\circ_{\ce{Cl2(g)}/\ce{Cl^- (aq)}} = +1.36\text{ V}$ $E^\circ_{\ce{O2(g)}, \ce{H^+ (aq)}/\ce{H2O(l)}} = +1.23\text{ V}$ $E^\circ_{\ce{H^+ (aq)}/\ce{H2(g)}} = 0.00\text{ V}$

Kinetic overpotentials ($\eta$) on Pt under these conditions: $\eta_{\ce{O2}} = 0.65\text{ V}$ $\eta_{\ce{Cl2}} = 0.05\text{ V}$ $\eta_{\ce{H2}} = 0.00\text{ V}$

Assume $\frac{2.303 RT}{F} = 0.06\text{ V}$ and $1\text{ F} = 96500\text{ C mol}^{-1}$. All gases are at $1\text{ bar}$.

Calculate the total minimum electrical energy required to drive the cell and produce the major anodic product. (Assume 100% current efficiency).

The standard reduction potentials ($E^\circ$) for the $\ce{MCl6^{2-}}/\ce{MCl4^{2-}}$ redox couples of Palladium (Pd, 4d) and Platinum (Pt, 5d) at 298 K differ by $0.61 \text{ V}$.

An aqueous solution is prepared by mixing equimolar amounts of $\ce{K2PdCl4}$, $\ce{K2PtCl4}$, $\ce{K2PdCl6}$, and $\ce{K2PtCl6}$ at 298 K. A spontaneous redox reaction occurs.

Based on the anomalous stability of higher oxidation states in 5d elements relative to 4d elements (a consequence of the lanthanoid contraction), determine the standard Gibbs free energy change ($\Delta G^\circ$) for the spontaneous reaction, and the number of $d$-electrons present in the central metal ion of the product complex containing the 5d element. (Given: $F = 96500 \text{ C/mol}$. Assume no side reactions occur.)

Consider the two-step synthesis involving the diazotization of aniline (Reaction I) to form intermediate $P$, followed by its electrophilic coupling with 3-methylphenol in a mildly alkaline medium (pH 9-10) to yield the major product $Q$. Reaction I is strictly carried out at $273-278\text{ K}$. Which of the following statements correctly identifies the structure of $Q$ and the fundamental physicochemical rationale for the strict thermal constraints of Reaction I?

The complex salt \ce{K2[Ni(EDTA)]} is dissolved in a large excess of an aqueous \ce{HCl} solution at $298 \text{ K}$ ($\text{pH} = 1.5$). Under these highly acidic conditions, the hexadentate EDTA ligand changes its denticity, and the complex exists exclusively as the protonated monoanion \ce{[Ni(HEDTA)]^-}. The protons required for this transformation are abstracted directly from the \ce{HCl} solution.

The apparent limiting molar conductivity of the salt in this medium is defined as: $\Lambda_{m, app}^\circ = \lim_{c \to 0} \frac{\kappa_{solution} - \kappa_{HCl}}{c}$ where $c$ is the molar concentration of the added salt.

Given the limiting ionic molar conductivities ($\lambda^\circ$) at $298 \text{ K}$ in $\text{S cm}^2 \text{ mol}^{-1}$: $\lambda^\circ(\ce{K^+}) = 73.5$ $\lambda^\circ(\ce{H^+}) = 349.6$ $\lambda^\circ(\ce{[Ni(EDTA)]^{2-}}) = 95.0$ $\lambda^\circ(\ce{[Ni(HEDTA)]^-}) = 45.0$

Calculate $\Lambda_{m, app}^\circ$ for \ce{K2[Ni(EDTA)]} in this medium.

A researcher attempts to determine the limiting molar conductivity ($\Lambda_m^\circ$) of the actinoid salt $\ce{UO2Cl}$ by preparing a $1.0 \times 10^{-3}\text{ M}$ solution in a $1.0\text{ M}$ $\ce{HClO4}$ buffer at $298\text{ K}$ (to prevent hydrolysis). However, the Debye-Hückel-Onsager extrapolation fails because $\ce{UO2+}$ undergoes the following disproportionation equilibrium:

$\ce{2UO2+_{(aq)} + 4H+_{(aq)} <=> U^{4+}_{(aq)} + UO2^{2+}_{(aq)} + 2H2O_{(l)}}$

Given the standard reduction potentials $E^\circ(\ce{UO2^{2+}}/\ce{UO2+}) = +0.162\text{ V}$ and $E^\circ(\ce{UO2+}/\ce{U^{4+}}) = +0.280\text{ V}$, calculate the percentage of $\ce{UO2+}$ that has disproportionated in this buffered solution, which fundamentally alters the charge carrier composition.

(Assume ideal behavior and $\frac{2.303 RT}{F} = 0.059\text{ V}$)

An aqueous solution of a weak monoprotic acid \ce{HA} (molar mass $= 120 \text{ g mol}^{-1}$) has a mass concentration of $0.048 \text{ g cm}^{-3}$ and a density of $1.088 \text{ g cm}^{-3}$ at $298 \text{ K}$. The electrolytic conductivity ($\kappa$) of the solution is $3.2 \times 10^{-3} \text{ S cm}^{-1}$. Given the limiting molar conductivities $\Lambda^\circ_m(\ce{H^+}) = 350 \text{ S cm}^2 \text{ mol}^{-1}$ and $\Lambda^\circ_m(\ce{A^-}) = 50 \text{ S cm}^2 \text{ mol}^{-1}$, calculate the boiling point of this solution at $1 \text{ atm}$. (Assume $K_b$ for water is $0.52 \text{ K kg mol}^{-1}$ and the boiling point of pure water is $373.150 \text{ K}$.)

Consider a $0.05 \text{ M}$ aqueous solution of a weak electrolyte \ce{AB2} (theoretical molar mass = $300 \text{ g mol}^{-1}$) that partially dissociates into \ce{A^{2+}} and \ce{B^-} ions at $298 \text{ K}$. The electrolytic conductivity ($\kappa$) of this solution is $1.2 \times 10^{-3} \text{ S cm}^{-1}$.

The limiting molar conductivities ($\Lambda_m^\infty$) of various strong electrolytes at $298 \text{ K}$ are given as: $\Lambda_m^\infty(\ce{ACl2}) = 260 \text{ S cm}^2 \text{ mol}^{-1}$ $\Lambda_m^\infty(\ce{NaB}) = 110 \text{ S cm}^2 \text{ mol}^{-1}$ $\Lambda_m^\infty(\ce{NaCl}) = 120 \text{ S cm}^2 \text{ mol}^{-1}$

Based on the interplay between electrolytic conductance and colligative properties, what is the observed molar mass of \ce{AB2} in this solution?

The specific acid-catalyzed hydrolysis of ethyl acetate ($\ce{CH3COOC2H5}$) is carried out in an aqueous solution at $298 \text{ K}$. The reaction is catalyzed by a weak acid $\ce{HA}$ ($K_{a} = 1.0 \times 10^{-5}$) with an initial concentration of $0.1 \text{ M}$. The initial concentration of ethyl acetate is $0.2 \text{ M}$.

Given that the rate law for the hydrolysis is $\text{Rate} = k [\ce{Ester}][\ce{H+}]$, and the acid dissociation constant of the product acetic acid ($\ce{CH3COOH}$) at $298 \text{ K}$ is $K_{a} = 2.0 \times 10^{-5}$.

Assuming the volume of the solution remains constant and neglecting the autoionization of water, what is the ratio of the rate of reaction at $t = 0$ to the rate of reaction when the hydrolysis is $75\%$ complete?

Consider the following concentration cell at $298 \text{ K}$:

$Pt(s) | H_2(g, 1 \text{ bar}) | HA(aq, 4 \times 10^{-4} \text{ M}) + NaA(aq, 2 \times 10^{-4} \text{ M}) || HA(aq, 6 \times 10^{-4} \text{ M}) | H_2(g, 1 \text{ bar}) | Pt(s)$

Given:

• The acid dissociation constant of the weak acid $HA$ is $K_a = 10^{-4}$.
• $\frac{2.303 RT}{F} = 0.06 \text{ V}$ at $298 \text{ K}$.
• $\log 2 = 0.30$, $\log 3 = 0.48$.

Calculate the electromotive force ($E_{cell}$) of the cell in millivolts (mV).

An aromatic primary amine $X$ (molecular formula $\ce{C7H9N}$) yields toluene upon diazotization at $273 \text{ K}$ followed by reduction with $\ce{H3PO2}$. A $0.1 \text{ M}$ aqueous solution of $X$ at $298 \text{ K}$ exhibits a pH of $8.7$. Another isomer $Y$ of $X$, which is also a primary aromatic amine, has a $pK_b$ of $9.0$. The $pK_b$ of aniline is $9.4$. Let $\alpha_X$, $\alpha_Y$, and $\alpha_{\text{aniline}}$ be the degrees of dissociation of $X$, $Y$, and aniline respectively, in their $0.1 \text{ M}$ aqueous solutions. Which of the following correctly identifies $X$, the relationship between the degrees of dissociation, and the $pK_b$ of $X$?

An aqueous solution of the coordination compound $\ce{[Co(NH_3)_4Cl_2]Cl}$ is prepared with a molality of $0.1 \text{ m}$. According to Werner's theory, the primary valency undergoes $100\%$ ionization. However, contrary to the assumption of a perfectly rigid coordination sphere, one of the inner-sphere chloride ligands is labile and establishes the following aquation equilibrium at $298 \text{ K}$:

$\ce{[Co(NH_3)_4Cl_2]^+ + H_2O(l) <=> [Co(NH_3)_4(H_2O)Cl]^{2+} + Cl^-}$

The equilibrium constant for this aquation reaction is $K_{eq} = 0.15$ (assume molarity $\approx$ molality). Given the ebullioscopic constant of water $K_b = 0.52 \text{ K kg mol}^{-1}$, what is the observed boiling point elevation ($\Delta T_b$) of the solution?

Two electrolytic cells, X and Y, are connected in series. Cell X contains $500\text{ mL}$ of $0.30\text{ M } \ce{Ce(SO4)2}$ in dilute $\ce{H2SO4}$, and Cell Y contains an aqueous solution of $\ce{CuSO4}$. A constant current is passed through the cells until $1.5875\text{ g}$ of $\ce{Cu}$ is deposited at the cathode of Cell Y. The electrolysis is then stopped. The resulting solution from Cell X is completely titrated with $0.10\text{ M } \ce{FeSO4}$ solution. What is the volume of $\ce{FeSO4}$ solution required to reach the equivalence point?\n\n(Given: Atomic mass of $\ce{Cu} = 63.5\text{ g/mol}$, $E^\circ_{\ce{Ce^{4+}/Ce^{3+}}} = +1.61\text{ V}$, $E^\circ_{\ce{Ce^{3+}/Ce}} = -2.34\text{ V}$, $E^\circ_{\ce{H+/H2}} = 0.00\text{ V}$)

A $1.0 \text{ L}$ aqueous solution of $\ce{H2O2}$ (initial concentration $1.0 \text{ M}$) is subjected to electrolysis at $298 \text{ K}$ using inert electrodes and a constant current of $193 \text{ A}$. Simultaneously, in the presence of a homogeneous catalyst, $\ce{H2O2}$ undergoes a pseudo-first-order disproportionation into $\ce{H2O(l)}$ and $\ce{O2(g)}$ with a rate constant $k = 10^{-3} \text{ s}^{-1}$. The anodic electrolysis reaction strictly oxidizes $\ce{H2O2}$ to $\ce{O2(g)}$ and $\ce{H+(aq)}$. Assuming no volume change and $100\%$ current efficiency, what is the total moles of $\ce{O2(g)}$ evolved from the solution by the time $\ce{H2O2}$ is completely depleted? (Given: $F = 96500 \text{ C mol}^{-1}$)

A $1.00 \text{ L}$ aqueous solution initially containing $0.10 \text{ M}$ \ce{AgNO3} is electrolyzed using inert Platinum electrodes at $298 \text{ K}$. A constant current of $9.65 \text{ A}$ is passed through the solution for $510 \text{ s}$. After the electrolysis is complete, $0.10 \text{ moles}$ of solid \ce{NaCH3COO} is added to the solution. Assume no change in volume occurs upon the addition of the solid and during electrolysis.

Given at $298 \text{ K}$: $K_{sp}(\ce{AgCH3COO}) = 2.5 \times 10^{-3}$ $K_a(\ce{CH3COOH}) = 2.0 \times 10^{-5}$ Faraday's constant $F = 96500 \text{ C mol}^{-1}$ Molar mass of $\ce{AgCH3COO} = 167 \text{ g mol}^{-1}$ Molar mass of $\ce{Ag} = 108 \text{ g mol}^{-1}$

Calculate the mass of the \ce{AgCH3COO} precipitate formed in the solution.

An electrochemical cell consists of $1.0 \text{ L}$ of an aqueous solution of a weak monoprotic acid \ce{HA} ($K_a = 1.0 \times 10^{-4}$) at $298 \text{ K}$. The initial concentration of \ce{HA} is $0.09 \text{ M}$. The cell is equipped with a Silver (\ce{Ag}) anode and a Platinum (\ce{Pt}) cathode. A constant effective DC voltage of $25 \text{ V}$ is applied across the electrodes.

During electrolysis, the following reactions occur: Anode: \ce{Ag(s) + A^{-}(aq) -> AgA(s) + e^{-}} Cathode: \ce{H^{+}(aq) + e^{-} -> 1/2 H2(g)}

The precipitated \ce{AgA(s)} does not interfere with the cell's electrical resistance. The cell constant $l/A$ is exactly $\frac{1}{9.65} \text{ cm}^{-1}$. Given the limiting molar conductivities: $\lambda^\infty(\ce{H^+}) = 350 \text{ S cm}^2 \text{ mol}^{-1}$ and $\lambda^\infty(\ce{A^-}) = 50 \text{ S cm}^2 \text{ mol}^{-1}$.

Assuming the cell operates purely under ohmic control (negligible back EMF and overpotential) and the degree of dissociation $\alpha \ll 1$, calculate the time required to collect $560 \text{ mL}$ of \ce{H2(g)} at STP ($273 \text{ K}, 1 \text{ atm}$).

(Given: $1 \text{ F} = 96500 \text{ C mol}^{-1}$, Molar volume of ideal gas at STP = $22.4 \text{ L mol}^{-1}$)

A solution is prepared by dissolving $15.0\text{ g}$ of a salt \ce{MCl2} (Molar mass = $150\text{ g/mol}$) in $500\text{ g}$ of a solvent $L$ (Molar mass = $50\text{ g/mol}$). The vapor pressure of pure solvent $L$ at $298\text{ K}$ is $50.51\text{ torr}$, and the vapor pressure of the solution is $50.00\text{ torr}$. Assume the volume of the solution is exactly $500\text{ mL}$.

The standard reduction potential $E^\circ(\ce{M^{2+}/M})$ is $-0.75\text{ V}$. The solvent $L$ undergoes reduction at the cathode at a constant potential of $-0.82\text{ V}$ to evolve \ce{H2} gas.

If this solution is electrolyzed at $298\text{ K}$ using inert electrodes, what is the primary cathodic product and the actual reduction potential of the \ce{M^{2+}/M} couple in this solution?

(Given: $\frac{2.303 RT}{F} = 0.06\text{ V}$, $\log 2 = 0.3$)

Consider the following silver concentration cell at $298\text{ K}$:

$\ce{Ag(s) | Ag+(aq, 0.1 M) || Ag+(aq, 0.1 M) | Ag(s)}$

To the anodic compartment, solid $\ce{KCN}$ is added such that the overall analytical concentration of $\ce{CN-}$ becomes $0.3\text{ M}$. Assuming no change in the volume of the solution, calculate the new electromotive force ($E_{\text{cell}}$) of the cell.

[Given: Formation constant ($K_f$) of $\ce{[Ag(CN)2]-}$ is $10^{21}$; $\frac{2.303RT}{F} = 0.059\text{ V}$]

A concentration cell is constructed at 298 K as follows:

\ce{Pt(s) | H2(g, 1 bar) | Glycine (aq, 0.1 M) || Gly.HCl (aq, 0.1 M) | H2(g, 1 bar) | Pt(s)}

where 'Glycine' represents the neutral amino acid and 'Gly.HCl' represents glycine hydrochloride. Given for Glycine: $pK_{a1} = 2.30$, $pK_{a2} = 9.60$. Assume the pH of the 0.1 M Glycine solution is exactly equal to its isoelectric point. Calculate the electromotive force (EMF) of the cell in mV. (Take $\frac{2.303 RT}{F} = 0.060$ V and $\log 2 = 0.30$)

An electrochemical concentration cell is set up at 298 K:

$\ce{Pt(s) | Mn^{2+}(aq, 0.1 M), MnO4^{-}(aq, 0.01 M), H^{+}(aq, pH_a) || MnO4^{-}(aq, 0.1 M), Mn^{2+}(aq, 0.01 M), H^{+}(aq, pH_c) | Pt(s)}$

In the cathodic half-cell, the $\ce{H+}$ concentration is maintained by a $0.1\text{ M}$ solution of a weak monoprotic acid $\ce{HA}$ ($K_a = 10^{-5}$). In the anodic half-cell, the $\ce{H+}$ concentration is governed by a buffer of $\ce{HA}$ and $\ce{A-}$. Initially ($t=0$), the anodic compartment contains $0.1\text{ M HA}$ and $0\text{ M A-}$. A zero-order reaction continuously converts $\ce{HA}$ into $\ce{A-}$ in the anodic compartment at a constant rate of $1.0 \times 10^{-3}\text{ M s}^{-1}$ (assume no volume change).

Calculate the instantaneous rate of change of the cell potential, $\frac{dE_{\text{cell}}}{dt}$ (in $\text{mV s}^{-1}$), at $t = 50\text{ s}$.

(Given: $\frac{2.303 RT}{F} = 0.059\text{ V}$)

A student sets up the following concentration cell at $T$ K to model localized electrochemical corrosion:

$\ce{Cu(s) | Cu+ (aq, 0.002 M) || Cu+ (aq, 0.2 M) | Cu(s)}$

The student assumes that $\ce{Cu+}$ is perfectly stable in aqueous solution and calculates the theoretical electromotive force of the cell as $E_{ideal}$. However, $\ce{Cu+}$ undergoes spontaneous disproportionation. The system is allowed to reach chemical equilibrium in both half-cells before any current is drawn. The actual measured electromotive force of the cell is $E_{real}$.

Given standard reduction potentials at $T$ K: $E^\circ_{\ce{Cu+/Cu}} = 0.52 \text{ V}$ and $E^\circ_{\ce{Cu^{2+}/Cu}} = 0.34 \text{ V}$.

Assume activity coefficients are unity and $\frac{2.303 RT}{F} = 0.06 \text{ V}$. Identify the correct set of values for $(E_{ideal}, E_{real}, \log_{10} K_{eq})$, where $K_{eq}$ is the equilibrium constant for the disproportionation of $\ce{Cu+}$.

A galvanic concentration cell at $298\text{ K}$ is constructed as follows:

$\ce{Ag(s) | Ag+(aq, C_1) || Ag+(aq, C_2) | Ag(s)}$

The concentrations $C_1$ and $C_2$ are determined by the following kinetic data:

1. A bimolecular elementary reaction has a pre-exponential factor $A = 6.022 \times 10^9\text{ L mol}^{-1}\text{ s}^{-1}$ and an activation energy $E_a = 0$. The concentration $C_1$ is numerically equal to its rate constant when expressed in units of $\text{cm}^3\text{ molecule}^{-1}\text{ s}^{-1}$.
2. A separate reaction follows $n^{\text{th}}$ order kinetics. Let $k_L$ be its rate constant in units of $(\text{mol L}^{-1})^{1-n}\text{ s}^{-1}$ and $k_C$ be its rate constant in units of $(\text{mol cm}^{-3})^{1-n}\text{ s}^{-1}$. Given the ratio $k_L / k_C = 10^{-6}$, the concentration $C_2$ is numerically equal to $10^{-n}\text{ M}$.

Given $N_A = 6.022 \times 10^{23}\text{ mol}^{-1}$ and $\frac{2.303 RT}{F} = 0.059\text{ V}$ at $298\text{ K}$, calculate the electromotive force ($E_{\text{cell}}$) of the cell.

A researcher investigates the limiting molar conductivities ($\Lambda^\circ_m$) of the hypothetical unhydrolyzed strong electrolytes $Ti(NO_3)_4\text{(aq)}$, $Zr(NO_3)_4\text{(aq)}$, and $Hf(NO_3)_4\text{(aq)}$ at $298\text{ K}$. The measured $\Lambda^\circ_m$ for $Zr(NO_3)_4\text{(aq)}$ is $486\text{ S cm}^2\text{ mol}^{-1}$. Given that the limiting ionic conductivity of the nitrate ion, $\lambda^\circ_m(NO_3^-)$, is $71.5\text{ S cm}^2\text{ mol}^{-1}$, and considering the effects of the Lanthanoid contraction on hydration enthalpies, which of the following sets most accurately represents the theoretical limiting ionic conductivities ($\lambda^\circ_m$) of the aqueous cations $Ti^{4+}$, $Zr^{4+}$, and $Hf^{4+}$ respectively?

A $0.05 \text{ M}$ aqueous solution of a weak monoprotic acid $HA$ is placed in a conductivity cell with a cell constant of $0.5 \text{ cm}^{-1}$ at $300 \text{ K}$. The measured resistance of the solution is $250 \, \Omega$. Given the limiting molar conductivities $\lambda^\circ(H^+) = 350 \text{ S cm}^2 \text{ mol}^{-1}$ and $\lambda^\circ(A^-) = 50 \text{ S cm}^2 \text{ mol}^{-1}$, calculate the osmotic pressure of this solution at $300 \text{ K}$.

(Assume $RT = 24.6 \text{ L atm mol}^{-1}$)

The transport of alkali metal cations ($M^+$) is evaluated in two distinct solvent systems at $298\text{ K}$:

System $\alpha$: Pure water (analogous to the hydrophilic environment of Vitamin C). System $\beta$: A weakly coordinating, sterically bulky semi-polar solvent (analogous to the hydrophobic lipid environment of Vitamin A), where effective solvation shells cannot form around the cations.

Let $\lambda^\circ_\alpha(M^+)$ and $\lambda^\circ_\beta(M^+)$ represent the limiting ionic molar conductivities in systems $\alpha$ and $\beta$, respectively. Identify the correct relationship for the conductivity trends and the dominant physical parameter governing the mobility in System $\beta$.

A $0.01\text{ M}$ aqueous solution of the violet hydrate isomer of chromium(III) chloride ($\ce{CrCl3 \cdot 6H2O}$) is heated from $298\text{ K}$ to $348\text{ K}$. At $348\text{ K}$, the complex completely isomerizes to its dark green hydrate isomer. The temperature coefficient of equivalent conductivity for all ions is $\alpha = 0.02\text{ K}^{-1}$, such that $\lambda_{eq, T} = \lambda_{eq, 298}[1 + \alpha(T - 298)]$. Assuming the equivalent ionic conductivities ($\lambda_{eq}$) of all cationic and anionic species present are identical at any given temperature, what is the net percentage change in the molar conductivity ($\Lambda_m$) of the solution?

A lead storage battery operates at $298 \text{ K}$ with the following overall discharge reaction:

$Pb(s) + PbO_2(s) + 4H^+(aq) + 2SO_4^{2-}(aq) \rightarrow 2PbSO_4(s) + 2H_2O(l)$

The standard cell potential is $E^\circ_{cell} = 2.05 \text{ V}$. Initially, the bulk electrolyte concentrations are $[H^+] = 1.0 \text{ M}$ and $[SO_4^{2-}] = 1.0 \text{ M}$. During a rapid discharge at a constant current of $I = 2.0 \text{ A}$, severe concentration polarization occurs, causing the local surface concentrations at the electrodes to drop to $[H^+]_{surf} = 0.1 \text{ M}$ and $[SO_4^{2-}]_{surf} = 0.01 \text{ M}$.

Given the activation overpotential $\eta_{act} = 0.10 \text{ V}$ and the internal resistance of the cell $R_{int} = 0.05 \ \Omega$, calculate the actual terminal voltage ($V_{cell}$) of the battery during this discharge.

(Assume $\frac{2.303 RT}{F} = 0.06 \text{ V}$ at $298 \text{ K}$)

An experimental all-iron secondary cell utilizes the \ce{Fe^{3+}/Fe^{2+}} and \ce{Fe^{2+}/Fe} redox couples. To suppress self-discharge at the anode, the bulk electrolyte is buffered at pH = 4.0. At this pH, any \ce{Fe^{3+}} that crosses over to the anode precipitates as \ce{Fe(OH)3(s)}.

During a high-rate discharge, a localized parasitic reaction drops the pH at the anode surface to 1.0. The \ce{Fe(OH)3(s)} dissolves, reaching a local steady-state \ce{Fe^{3+}} concentration of $10^{-2}$ M.

The self-corrosion reaction at the anode is: $\ce{Fe(s) + 2Fe^{3+}(aq) -> 3Fe^{2+}(aq)}$

Given: $E^\circ(\ce{Fe^{3+}/Fe^{2+}}) = +0.77$ V $E^\circ(\ce{Fe^{2+}/Fe}) = -0.44$ V $K_{sp}(\ce{Fe(OH)3}) = 10^{-38}$ The local $[\ce{Fe^{2+}}]$ at the anode remains constant at $10^{-1}$ M. Use $\frac{2.303 RT}{F} = 0.06$ V and $F = 96500$ C/mol.

Calculate the change in the Gibbs free energy of corrosion, $\Delta \Delta G_{corr} = \Delta G_{corr, pH=1} - \Delta G_{corr, pH=4}$, due to this localized pH shift.

A fully charged lead-acid battery contains $621\text{ g}$ of $\ce{Pb}$ (anode) and $478\text{ g}$ of $\ce{PbO2}$ (cathode). Due to structural passivation, only $40\%$ of the $\ce{Pb}$ and $75\%$ of the $\ce{PbO2}$ are electrochemically active. The battery is discharged at a constant current of $19.3\text{ A}$ at $298\text{ K}$ until it is completely dead. Assuming all the formed $\ce{PbSO4}$ adheres strictly to the respective electrodes, calculate the total time of discharge and the final mass of the cathode.\n\n(Given: $F = 96500\text{ C mol}^{-1}$; Molar masses in $\text{g mol}^{-1}$: $\ce{Pb} = 207$, $\ce{S} = 32$, $\ce{O} = 16$)

During the discharge cycle of a secondary battery, a localized pH gradient alters the redox behavior of the electrolyte. To investigate this thermodynamic shift, a technician constructs the following concentration-dependent electrochemical cell at $298 \text{ K}$:

$Pt | M^{2+} (0.1 \text{ M}), M^{3+} (0.01 \text{ M}) || MnO_4^- (0.01 \text{ M}), Mn^{2+} (0.1 \text{ M}), H^+ (pH = x) | Pt$

The technician observes that the cell potential is exactly $0 \text{ V}$, indicating that the $KMnO_4$ indicator fails to spontaneously oxidize $M^{2+}$ under these specific conditions.

Given: $E^\circ_{M^{3+}/M^{2+}} = 1.27 \text{ V}$ $E^\circ_{MnO_4^-/Mn^{2+}} = 1.51 \text{ V}$ $\frac{2.303 RT}{F} = 0.06 \text{ V}$

Calculate the value of $x$ (the pH of the cathodic compartment).

During the rapid exothermic discharge of a secondary lead-acid battery, the local temperature at the cathode increases from $300\text{ K}$ to $350\text{ K}$.

The cathode reduction half-reaction is: $\ce{PbO2(s) + 4H+(aq) + 2e- <=> Pb^{2+}(aq) + 2H2O(l)}$

The generated $\ce{Pb^{2+}}$ immediately establishes equilibrium with the electrolyte: $\ce{PbSO4(s) <=> Pb^{2+}(aq) + SO4^{2-}(aq)}$

Given:

1. $[\ce{H+}] = 1\text{ M}$ and $[\ce{SO4^{2-}}] = 1\text{ M}$ (assume concentrations remain constant).
2. $K_{sp}(\ce{PbSO4})$ at $300\text{ K} = 10^{-8}$
3. $K_{sp}(\ce{PbSO4})$ at $350\text{ K} = 10^{-5}$
4. $\frac{2.303 RT}{F} = 0.06\text{ V}$ at $300\text{ K}$.

Assuming the standard reduction potential ($E^\circ$) for the $\ce{PbO2/Pb^{2+}}$ couple is independent of temperature, calculate the change in the reduction potential of the cathode ($\Delta E_{\text{cathode}}$) due to this temperature increase.

A regenerative $\ce{H2-O2}$ fuel cell system operates at $298\text{ K}$. The cell consists of two compartments separated by a bipolar ion-selective membrane that maintains a strict pH gradient. The anode compartment is buffered at $\text{pH} = 10$ and the cathode compartment is buffered at $\text{pH} = 4$. Gases $\ce{H2}$ and $\ce{O2}$ are continuously supplied and maintained at $1\text{ bar}$ pressure.