D and F Block Elements - Class 12

Consider three consecutive 3d transition metals $M_1$, $M_2$, and $M_3$ (where atomic numbers follow $Z_1 < Z_2 < Z_3$). The standard reduction potential $E^\circ(M^{2+}/M)$ for $M_2$ is anomalously more negative than the expected trend between $M_1$ and $M_3$. $M_1$ and $M_3$ form stable neutral mononuclear carbonyls $M_1(CO)_x$ and $M_3(CO)_y$ that strictly obey the 18-electron rule. $M_2$ forms a stable homoleptic binuclear carbonyl $(M_2)_2(CO)_z$ but no stable neutral mononuclear carbonyl.

Evaluate the following statements: \begin{enumerate} \item[I.] The anomalous $E^\circ(M^{2+}/M)$ of $M_2$ is primarily driven by the exchange energy stabilization of $M_2^{2+}$ and its low enthalpy of atomization. \item[II.] $M_2$ fails to form a stable neutral mononuclear carbonyl because its half-filled d-subshell completely inhibits $\pi$-backdonation. \item[III.] The C-O stretching frequency in the cationic complex $[M_2(CO)_x]^+$ is higher than in $M_1(CO)_x$. \item[IV.] The metal-carbon bond length in $[M_2(CO)_y]^-$ is shorter than in $M_3(CO)_y$. \end{enumerate}

How many of the above statements are CORRECT?

A 3d transition metal $M$ exhibits an anomalously negative standard reduction potential $E^\circ(M^{2+}/M)$ compared to the general trend of the series, attributed to the exchange energy stabilization of its half-filled $d$-subshell. Aniline is treated with an equimolar amount of anhydrous \ce{MCl2} in an inert solvent to form a coordination complex. This complex is then reacted with one equivalent of \ce{Br2} at 298 K. Following standard aqueous workup, the major organic product $P$ is isolated.

Given: $E^\circ(\ce{Br2/Br^-}) = +1.09$ V $E^\circ(\ce{Fe^{3+}/Fe^{2+}}) = +0.77$ V $E^\circ(\ce{Mn^{3+}/Mn^{2+}}) = +1.51$ V

Identify the major organic product $P$.

A metalloenzyme features a rigid tetrahedral binding site that acts as a perfect 'lock and key' for $d^{10}$ metal ions. The standard reduction potential for the aqueous couple \ce{Cu^{2+}{(aq)} / Cu^+{(aq)}} is $+0.15 \text{ V}$ at $300 \text{ K}$.

The binding of \ce{Cu^+_{(aq)}} to the apoenzyme (E) has $\Delta H^\circ = -40 \text{ kJ mol}^{-1}$ and $\Delta S^\circ = -50 \text{ J K}^{-1} \text{ mol}^{-1}$.

The binding of \ce{Cu^{2+}_{(aq)}} to the apoenzyme has $\Delta H^\circ = -70 \text{ kJ mol}^{-1}$. However, because \ce{Cu^{2+}} is a $d^9$ ion, it undergoes Jahn-Teller distortion, forcing a conformational change in the rigid enzyme. This induced fit results in an additional conformational entropy penalty, making its total binding entropy $\Delta S^\circ$ more negative by $36 \text{ J K}^{-1} \text{ mol}^{-1}$ compared to the ideal $d^{10}$ binding.

Calculate the standard reduction potential ($E^\circ$) of the enzyme-bound copper couple (\ce{E-Cu^{2+} / E-Cu^+}) at $300 \text{ K}$. (Assume Faraday's constant $F = 96000 \text{ C mol}^{-1}$)

Consider the thermodynamic stability of high oxidation state compounds of Group 6 and Group 10 transition metals at $298\text{ K}$. Which of the following statements correctly identifies the chemical behavior and its primary underlying thermodynamic cause?

Consider the following three coordination complexes synthesized at 298 K: I. \ce{K3[CoF6]} II. \ce{K2[PtCl4]} III. \ce{K3[RuF6]} Let $\mu_I$, $\mu_{II}$, and $\mu_{III}$ be the spin-only magnetic moments (in Bohr Magnetons) of complexes I, II, and III, respectively. Calculate the value of $(\mu_I^2 + \mu_{II}^2 + \mu_{III}^2)$.

Consider the thermodynamic parameters governing the standard reduction potentials ($E^\circ_{M^{3+}/M}$) of Group 6 transition metals (\ce{Cr}, \ce{Mo}, \ce{W}) at $298\text{ K}$. A bimetallic corrosion cell is constructed by coupling a Tungsten (\ce{W}) component with an Iron (\ce{Fe}) fitting ($E^\circ_{Fe^{2+}/Fe} = -0.44\text{ V}$) in an aerated aqueous environment at $\text{pH} = 7$. Given that the first ionization energy of \ce{W} is comparable to that of \ce{Cr}, which of the following correctly identifies the trend in $E^\circ_{M^{3+}/M}$, the metal acting as the anode in the corrosion cell, and the dominant thermodynamic factor responsible for \ce{W}'s relative nobility?

Evaluate the following four statements regarding the chemistry of d-block and f-block elements. Assign the given numerical value to each statement if it is CORRECT. If it is INCORRECT, assign it a value of 0.

Statement I (Value = 1): Due to the combined effects of lanthanoid and relativistic contractions, the absolute size of the octahedral voids in a 5d transition metal lattice (e.g., \ce{W}) is smaller than that in its 3d counterpart (e.g., \ce{Cr}).

Statement II (Value = 2): Actinoids exhibit a wider range of oxidation states than lanthanoids because the 5f orbitals are less deeply buried than the 4f orbitals, and the energy gap between 5f, 6d, and 7s orbitals is smaller than that between 4f, 5d, and 6s orbitals.

Statement III (Value = 4): In the non-stoichiometric hydride \ce{PdH_x}, the hydrogen atoms occupy the octahedral voids of the FCC palladium lattice, causing a measurable expansion of the unit cell parameter.

Statement IV (Value = 8): If a 4d and a 5d transition metal of the same group crystallize in the same lattice type and have identical atomic radii due to the lanthanoid contraction, the ratio of their theoretical densities is exactly equal to the ratio of their atomic masses.

Calculate the sum of the values of the CORRECT statements.

A 3d transition metal $M$ exhibits a positive standard reduction potential $E^\circ(M^{2+}/M) = +0.34\text{ V}$ at $298\text{ K}$ because its hydration enthalpy fails to compensate for the sum of its atomization and ionization enthalpies. When an aqueous solution of $M^{2+}$ is treated with excess $KCN(aq)$, a soluble coordination complex $\mathbf{X}$ and a toxic gas $\mathbf{Y}$ are formed. Separately, when $M^{2+}(aq)$ is treated with excess $KI(aq)$, a precipitate $\mathbf{Z}$ and a brown solution containing species $\mathbf{W}$ are formed.

Determine the value of the expression: $(p \times r) + s - q$ Where: $p$ = Coordination number of the central metal in complex $\mathbf{X}$ $q$ = Number of unpaired electrons in the central metal of complex $\mathbf{X}$ $r$ = Oxidation state of $M$ in precipitate $\mathbf{Z}$ $s$ = Atomicity (total number of atoms per molecule) of gas $\mathbf{Y}$

A black mineral $\textbf{A}$ undergoes alkaline oxidative fusion to yield a dark green compound $\textbf{B}$. Electrolytic oxidation of $\textbf{B}$ produces an intensely purple-colored compound $\textbf{C}$. Separately, fusion of chromite ore ($\ce{FeCr2O4}$) with $\ce{Na2CO3}$ in air yields a yellow compound $\textbf{D}$.

Both $\textbf{C}$ and $\textbf{D}$ contain central transition metals in their highest oxidation states. Which of the following statements correctly identifies their spin-only magnetic moments ($\mu_{eff}$), the origin of their intense colors, and the relative energy of their color-producing electronic transitions?

Consider the following reaction sequences starting with toluene (\ce{C6H5CH3}):

Sequence I: \ce{C6H5CH3} $\xrightarrow[2. \ \ce{H3O^+}]{1. \ \ce{CrO2Cl2}, \ \ce{CS2}}$ $P$ $\xrightarrow{\ce{HNO3} / \ce{H2SO4}, \ 298 \ \text{K}}$ $P'$ (Major)

Sequence II: \ce{C6H5CH3} $\xrightarrow[2. \ \ce{H3O^+}]{1. \ \ce{WO2Cl2}, \ \ce{CS2}}$ $Q$ $\xrightarrow{\ce{HNO3} / \ce{H2SO4}, \ 298 \ \text{K}}$ $Q'$ (Major)

Assume $Q$ is the recovered organic substrate if no oxidation occurs. Identify the major products $P'$ and $Q'$.

Consider the reaction of 2-iodo-2-methyl-1,1-diphenylpropan-1-ol with 1.0 equivalent of anhydrous \ce{FeCl3} in \ce{CH2Cl2} at 298 K. The reaction proceeds rapidly to yield a major organic product \textbf{Y} and a transition metal complex \textbf{Z}. Given the standard reduction potentials ($E^\circ_{\ce{Fe^{3+}/Fe^{2+}}} = +0.77$ V, $E^\circ_{\ce{I2/I^-}} = +0.54$ V), which of the following correctly identifies \textbf{Y}, the final oxidation state and d-electron configuration of the metal in \textbf{Z}, and the mechanistic rationale?

Consider the following reaction sequence involving Cobalt complexes under specific thermal and solvent constraints:

I. Anhydrous $\ce{CoCl2}$ reacts with excess $\ce{KSCN}$ in acetone to form a deep blue complex $\mathbf{P}$. II. An aqueous solution of $\ce{[Co(NH3)5(H2O)]^3+}$ reacts with $\ce{KSCN}$ at $0^\circ\text{C}$ to yield complex $\mathbf{Q}$ as the major kinetic product. III. Upon gentle heating, complex $\mathbf{Q}$ completely isomerizes to the thermodynamically stable complex $\mathbf{R}$.

Identify the correct set of parameters for complexes $\mathbf{P}$, $\mathbf{Q}$, and $\mathbf{R}$ respectively, in the order: [Hybridization of central metal, Donor atom of the ambidentate ligand].

Consider the hexaaqua complexes of \ce{Cr^{3+}}, \ce{Cr^{2+}}, \ce{Mn^{3+}}, and \ce{Mn^{2+}} in an aqueous medium at $298\text{ K}$. The standard reduction potentials are $E^\circ(\ce{Cr^{3+}}/\ce{Cr^{2+}}) = -0.41\text{ V}$ and $E^\circ(\ce{Mn^{3+}}/\ce{Mn^{2+}}) = +1.57\text{ V}$.

Let $P$ = Magnitude of the Crystal Field Stabilization Energy (CFSE) in units of $\Delta_o$ (ignoring pairing energy) for the $+3$ hexaaqua complex that corresponds to the more negative $E^\circ(M^{3+}/M^{2+})$ value. Let $Q$ = Total number of unpaired electrons in the ground state of the $+3$ hexaaqua complex that corresponds to the highly positive $E^\circ(M^{3+}/M^{2+})$ value. Let $R$ = Number of complexes among the four mentioned above that experience strong Jahn-Teller distortion. Let $S$ = Number of $d$-electrons in the metal ion of the complex that acts as the strongest reducing agent among the four. Let $T$ = Number of $d$-electrons in the metal ion of the complex that acts as the strongest oxidizing agent among the four.

Calculate the value of the expression: $(P \times 10) + (Q \times R) + (S \times T)$.

Manganese(III) ions readily disproportionate in aqueous acidic solutions. However, the presence of fluoride ions strongly stabilizes the +3 oxidation state via the formation of the $\ce{[MnF6]^{3-}}$ complex. Given the following standard reduction potentials at $298\text{ K}$:

$E^\circ(\ce{Mn^{3+}/Mn^{2+}}) = 1.50\text{ V}$ $E^\circ(\ce{MnO2/Mn^{3+}}) = 0.95\text{ V}$

The overall formation constant ($\beta_6$) for $\ce{[MnF6]^{3-}}$ is $10^{15}$. A solution is prepared containing $0.1\text{ M } \ce{[MnF6]^{3-}}$, $0.01\text{ M } \ce{Mn^{2+}}$, and $0.1\text{ M } \ce{F^-}$ in a buffer of $\text{pH} = 0$ with excess solid $\ce{MnO2}$.

Calculate the cell potential ($E_{\text{cell}}$) for the disproportionation of $\ce{[MnF6]^{3-}}$ under these conditions. (Assume $\frac{2.303RT}{F} = 0.06\text{ V}$ at $298\text{ K}$ and that $\ce{Mn^{2+}}$ does not form significant fluoride complexes).

Consider the isoelectronic series of octahedral metal carbonyls: $\ce{[Ti(CO)6]^{2-}}$, $\ce{[V(CO)6]^{-}}$, and $\ce{Cr(CO)6}$. The standard aqueous redox potentials $E^\circ(\ce{M^{3+}/M^{2+}})$ for these metals are highly variable, with Ti and V exhibiting highly negative values, indicating a strong thermodynamic preference for higher oxidation states in water. Despite this, their low-valent carbonyl complexes are stable and exhibit intense colors (e.g., $\ce{[V(CO)6]^{-}}$ is intensely yellow), unlike their pale $\ce{[M(H2O)6]^{2+}}$ counterparts. Which of the following statements correctly identifies the primary origin of this intense color and the correct trend in the C-O stretching frequency ($\nu_{\ce{CO}}$)?

Consider the first-row transition metal iron (Fe). The standard reduction potential for its hexaaqua complex is given as: $\ce{[Fe(H2O)6]^{3+} + e^- \rightleftharpoons [Fe(H2O)6]^{2+}} \quad E^\circ = +0.77 \text{ V}$ Both the $+2$ and $+3$ hexaaqua complexes are high-spin. When excess $\ce{Cl^-}$ is added, the $+2$ ion undergoes ligand substitution and a geometry change to form a high-spin tetrahedral complex: $\ce{[Fe(H2O)6]^{2+} + 4Cl^- \rightleftharpoons [FeCl4]^{2-} + 6H2O}$ Assume the standard free energy change ($\Delta G^\circ$) for this specific substitution is governed entirely by the change in Crystal Field Stabilization Energy (CFSE). Given the crystal field splitting parameters: $\Delta_o \text{ for } \ce{[Fe(H2O)6]^{2+}} = 10,400 \text{ cm}^{-1}$ $\Delta_t \text{ for } \ce{[FeCl4]^{2-}} = 4,000 \text{ cm}^{-1}$ Use the conversion factor $1 \text{ eV} \approx 8,000 \text{ cm}^{-1}$. (Note: $1 \text{ eV}$ per molecule corresponds to $96.5 \text{ kJ/mol}$, meaning $\Delta G^\circ$ in eV is numerically equal to the potential in Volts for a 1-electron transfer). Calculate the standard reduction potential ($E^\circ_{complex}$) for the coupled process: $\ce{[Fe(H2O)6]^{3+} + 4Cl^- + e^- \rightleftharpoons [FeCl4]^{2-} + 6H2O}$

The hypothetical tetrahedral complex $\ce{[RuCl4]^{2-}}$ ($4d^6$) exhibits a high-spin configuration with a crystal field splitting energy $\Delta_t = 10,000 \text{ cm}^{-1}$ and a pairing energy $P = 13,500 \text{ cm}^{-1}$. For the isostructural $\ce{[OsCl4]^{2-}}$ ($5d^6$) complex, the lanthanoid contraction results in nearly identical ionic radii. However, due to the greater radial extension of the 5d orbitals, $\Delta_t$ for the Os complex is exactly 50% larger than that of the Ru complex, while its pairing energy $P$ decreases to $12,000 \text{ cm}^{-1}$. Calculate the expected spin-only magnetic moment ($\mu_s$) and the Crystal Field Stabilization Energy (CFSE, excluding pairing energy) for the $\ce{[OsCl4]^{2-}}$ complex.

Consider the Group 10 elements ($\ce{Ni}$, $\ce{Pd}$, $\ce{Pt}$) at $298\text{ K}$. The standard reduction potentials $E^\circ(\ce{M^{2+}/M})$ are $-0.25\text{ V}$, $+0.92\text{ V}$, and $+1.18\text{ V}$, respectively. The sum of the first two ionization enthalpies ($\Delta_{i}H_1 + \Delta_{i}H_2$) is highest for $\ce{Pt}$ due to the Lanthanoid contraction, whereas the sum of the first four ionization enthalpies ($\sum_{j=1}^{4} \Delta_{i}H_j$) is highest for $\ce{Ni}$.

A solution contains $1.0\text{ M}$ each of $\ce{Ni^{2+}}$, $\ce{Pd^{2+}}$, and $\ce{Pt^{2+}}$ in $1.0\text{ M } \ce{HCl}$. Standard hydrogen gas at $1\text{ bar}$ is bubbled through this solution until thermodynamic equilibrium is reached.

Identify the metal(s) that will precipitate from the solution, and the central metal that forms the most thermodynamically stable hexachloro complex $\ce{[MCl_6]^{2-}}$, respectively.

Consider the homoleptic octahedral fluoro complexes of Group 9 metals in the +3 oxidation state: $\ce{[CoF6]^3-}$, $\ce{[RhF6]^3-}$, and $\ce{[IrF6]^3-}$. Let $\mu_1, \mu_2,$ and $\mu_3$ be their respective spin-only magnetic moments (in BM).

Consider the homoleptic chloro complexes of Group 10 metals in the +2 oxidation state: $\ce{[NiCl4]^2-}$, $\ce{[PdCl4]^2-}$, and $\ce{[PtCl4]^2-}$. Let $n_1, n_2,$ and $n_3$ be the number of unpaired electrons in these complexes, respectively.

Calculate the value of $Z = (\mu_1^2 + \mu_2^2 + \mu_3^2) + (n_1 + n_2 + n_3)$. (Assume spin-only magnetic moments are calculated using the formula $\mu = \sqrt{n(n+2)}$ BM).

Zirconium ($Zr$) and Hafnium ($Hf$) are often considered 'chemical twins' due to the lanthanide contraction, resulting in nearly identical ionic radii for $Zr^{4+}$ ($72$ pm) and $Hf^{4+}$ ($71$ pm). However, their highest oxidation state oxides, $ZrO_2$ and $HfO_2$, exhibit distinct thermodynamic properties at $298$ K and standard pressure. Which of the following statements correctly identifies and explains this difference?

Consider the standard reduction potentials for the $\ce{M^{3+}_{(aq)} / M^{2+}_{(aq)}}$ redox couple of the Group 5 transition metals Niobium (Nb) and Tantalum (Ta) at 298 K. Due to the lanthanoid contraction, their corresponding ionic radii are virtually identical ($r_{\ce{Nb^{n+}}} \approx r_{\ce{Ta^{n+}}}$). Assuming the entropy change ($\Delta S^\circ$) for the reduction half-reaction is identical for both metals, which of the following expressions correctly approximates the difference in their standard reduction potentials, $\Delta E^\circ = E^\circ(\ce{Nb^{3+}/Nb^{2+}}) - E^\circ(\ce{Ta^{3+}/Ta^{2+}})$?

The thiometallate anions \ce{[MoS4]^{2-}} and \ce{[WS4]^{2-}} both exhibit tetrahedral geometry. Due to the lanthanoid contraction, the ionic radii of \ce{Mo^{6+}} and \ce{W^{6+}} are nearly identical ($\approx 0.41$ \AA). Based on the differences in their quantum mechanical electronic environments, identify the correct relationship between their tetrahedral crystal field splitting energies ($\Delta_t$) and the energies of their lowest-energy Ligand-to-Metal Charge Transfer (LMCT) bands ($E_{\text{LMCT}}$).

Equimolar $0.01\text{ m}$ aqueous solutions of $\ce{ZrCl4}$ and $\ce{HfCl4}$ are prepared at $298\text{ K}$. Let $\Delta T_{b(\ce{Zr})}$ and $\Delta T_{b(\ce{Hf})}$ be their respective boiling point elevations. Let $K_{h(\ce{Zr})}$ and $K_{h(\ce{Hf})}$ be the equilibrium constants for their primary hydrolysis step: $\ce{M^{4+}(aq) + H2O(l) \rightleftharpoons M(OH)^{3+}(aq) + H+(aq)}$. Assuming dilute ideal behavior, which of the following sets of relations correctly compares these systems?

Consider the hexaaquacopper(II) complex ion, \ce{[Cu(H2O)6]^{2+}}, in an aqueous medium at 298 K. The experimentally observed effective magnetic moment ($\mu_{eff}$) is approximately 1.95 BM, which is significantly higher than the calculated spin-only value ($\mu_{so} = 1.73$ BM). Which of the following statements correctly identifies the structural geometry of the complex and the primary quantum mechanical origin of this magnetic deviation?

An intense purple-colored transition metal oxoanion $X$ is synthesized by the electrolytic oxidation of a green-colored oxoanion $Y$ in an alkaline medium at 298 K. The intense color of $X$ is characterized by a high molar extinction coefficient ($\epsilon \approx 2400 \, L \, mol^{-1} \, cm^{-1}$). Which of the following correctly identifies the nature of the electronic transition responsible for the color of $X$ and its spin-only magnetic moment ($\mu_s$)?

Consider the species $\ce{Ce^{3+}}$ and $\ce{Ta^{4+}}$ in their respective octahedral complexes at $298\text{ K}$. Both ions possess a single unpaired electron outside a closed-shell core. While the Lanthanoid contraction significantly affects the atomic radii of 5d transition metals, their magnetic behavior diverges fundamentally from that of the Lanthanoids. Which of the following options correctly provides the theoretical magnetic moment ($\mu_{eff}$) of $\ce{Ce^{3+}}$ (assuming only the ground state multiplet is populated) and the primary reason for the magnetic behavior of $\ce{[TaCl_6]^{2-}}$?

A solution of the diamagnetic square planar complex \textit{trans}-\ce{[Ni(L)2Cl2]} (where \ce{L} is a bulky phosphine ligand) in a non-polar solvent at $298\text{ K}$ establishes a thermodynamic equilibrium with its \textit{cis}-isomer. Due to severe steric hindrance between the bulky \ce{L} ligands, the \textit{cis}-isomer undergoes a complete tetrahedral twist, adopting a high-spin electronic configuration. If the observed effective magnetic moment ($\mu_{\text{eff}}$) of the equilibrium mixture is $1.414\text{ BM}$, what is the mole percentage of the \textit{cis}-isomer in the solution? (Assume spin-only magnetic moments for all species).

Transition metals in their highest oxidation states exhibit unique optical and chemical properties governed by charge transfer and relativistic effects. Which of the following statements regarding these high oxidation state species is \textbf{INCORRECT}?

A galvanic cell is constructed at 298 K: $\ce{Pt | Fe^{2+} (1.1 \times 10^{-3} M), Fe^{3+} (10^{-7} M) || Fe^{3+} (1.1 \times 10^{-3} M), Fe^{2+} (10^{-7} M) | Pt}$ The volumes of both half-cells are 1.0 L. The cell is allowed to discharge spontaneously through a resistor until the cell potential drops to 0.118 V, at which point the circuit is opened.

An aliquot of the solution from the left half-cell (anode compartment) is taken, and solid \ce{KSCN} is added to it such that the equilibrium concentration of free \ce{SCN^-} is 0.09 M. This forms the intensely red complex \ce{[Fe(SCN)]^{2+}} via Ligand-to-Metal Charge Transfer (LMCT).

Given the formation constant of \ce{[Fe(SCN)]^{2+}} is $K_f = 100 \text{ M}^{-1}$ and its molar absorptivity is $\epsilon = 5000 \text{ L mol}^{-1} \text{ cm}^{-1}$ at $\lambda_{max}$.

Calculate the absorbance of this solution in a 1.0 cm cuvette. (Assume no volume change upon addition of \ce{KSCN}, and \ce{Fe^{2+}} does not complex with \ce{SCN^-}. Given: $\frac{2.303 RT}{F} = 0.059$ V)

The kinetics of the outer-sphere electron transfer reaction between tris(2,2'-bipyridine)ruthenium(II) and hexaamminecobalt(III) is given by:

$\ce{[Ru(bpy)3]^{2+} + [Co(NH3)6]^{3+} -> [Ru(bpy)3]^{3+} + [Co(NH3)6]^{2+}}$

The reaction progress is monitored at $300\text{ K}$ by the disappearance of the intense Metal-to-Ligand Charge Transfer (MLCT) band of $\ce{[Ru(bpy)3]^{2+}}$ at $452\text{ nm}$. You are given the following thermodynamic and kinetic parameters: $\bullet$ Standard free energy of reaction, $\Delta G^\circ = -24.0\text{ kJ mol}^{-1}$ $\bullet$ Total reorganization energy, $\lambda = 144.0\text{ kJ mol}^{-1}$ $\bullet$ Theoretical collision frequency, $Z = 2.0 \times 10^{11}\text{ M}^{-1}\text{s}^{-1}$ $\bullet$ Observed bimolecular rate constant, $k_{obs} = 9.0 \times 10^{-2}\text{ M}^{-1}\text{s}^{-1}$ $\bullet$ Universal gas constant, $R = \frac{1}{120}\text{ kJ K}^{-1}\text{mol}^{-1}$

Assume $e^{-10} = 4.5 \times 10^{-5}$. According to Marcus theory, the simple Arrhenius collision-frequency model fails here due to the non-adiabatic nature of the electron transfer (involving a spin-state change in Cobalt). Calculate the transmission coefficient ($\kappa$) and the activation free energy ($\Delta G^\ddagger$ in $\text{kJ mol}^{-1}$) for this reaction.

Consider the following statements regarding the bonding and properties of transition metal complexes and interstitial compounds at $298\\text{ K}$:\n\nI. In the non-stoichiometric interstitial hydride $PdH_{0.56}$, the trapped hydrogen atoms stabilize the lattice via synergic back-bonding between the filled metal $d$-orbitals and the vacant hydrogen $1s$ orbital.\nII. The $C-O$ stretching frequency in $[Ti(CO)_6]^{2-}$ is lower than that in $[V(CO)_6]^-$ due to a greater extent of metal-to-ligand $d\\pi \\rightarrow \\pi^*$ back-donation.\nIII. Interstitial carbides such as $WC$ are chemically inert and extremely hard because the carbon atoms occupy interstitial voids without disrupting the metallic lattice conductivity, lacking the precise coordinate-covalent geometry of synergic bonds.\nIV. The formation of metal carbonyls requires the transition metal to have vacant $d$-orbitals for $\\sigma$-donation and filled $d$-orbitals for $\\pi$-back-donation, a symmetry condition not met by interstitial hydrogen atoms.\n\nHow many of the above statements are correct?

Transition metals Zirconium (\ce{Zr}, 4d) and Hafnium (\ce{Hf}, 5d) crystallize in hexagonal close-packed (HCP) lattices at standard conditions. The shortest interatomic distance in the \ce{Zr} lattice is $320 \text{ pm}$. A materials chemist synthesizes interstitial carbides by introducing Carbon atoms ($r_C = 77 \text{ pm}$) into the octahedral voids of both metals. Based on periodic trends and geometric packing constraints, which of the following accurately describes the theoretical octahedral void radii ($r_{void}$) and the mechanical nature of the resulting \ce{HfC} compared to classical expectations?

A sample of palladium foil is saturated with hydrogen gas to form the interstitial hydride $PdH_{0.60}$. The foil is immersed in an excess of aqueous acidic $KMnO_4$ solution ($pH = 1$) at 298 K. The reduction of $MnO_4^-$ by the interstitial hydrogen occurs via a Langmuir-Hinshelwood surface-catalyzed mechanism, where both species must adsorb on the Pd surface. Assuming the bulk concentration of $MnO_4^-$ is extremely high, what is the apparent reaction order with respect to bulk $[MnO_4^-]$, and how many moles of $KMnO_4$ are reduced per mole of Pd?

An aqueous solution of $0.1 \text{ M } H_2SO_4$ is electrolyzed at $273 \text{ K}$ and $1 \text{ atm}$ using a Platinum anode and a Palladium ($Pd$) cathode. The mass of the Palladium cathode is $10.64 \text{ g}$. A constant current of $9.65 \text{ A}$ is passed through the cell for $2000 \text{ s}$. During the electrolysis, the Palladium cathode absorbs atomic hydrogen to form an interstitial hydride with the limiting composition $PdH_{0.60}$ before any hydrogen gas is evolved. Assuming $100\%$ current efficiency for the reduction of $H^+$ and that the cathode reaches its maximum hydrogen absorption capacity, what is the volume of $H_2(g)$ evolved at the cathode?\n\n(Given: Atomic mass of $Pd = 106.4 \text{ g/mol}$, $1 \text{ F} = 96500 \text{ C/mol}$, Molar volume of an ideal gas at $273 \text{ K}$ and $1 \text{ atm} = 22.4 \text{ L/mol}$)

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

$\ce{Pt(s) | Br_2(l) | Br^{-}(0.1 M) || MnO_4^{-}(0.1 M), Mn^{2+}(0.001 M), H^{+}(pH = x) | Pt(s)}$

Given the standard reduction potentials: $E^\circ_{\ce{MnO_4^{-}/Mn^{2+}}} = 1.51\text{ V}$ $E^\circ_{\ce{Br_2/Br^{-}}} = 1.09\text{ V}$

Assume $\frac{2.303 RT}{F} = 0.06\text{ V}$. If the cell potential is exactly zero, what is the value of $x$?

Consider the hydrocarbon $\mathbf{X}$, 1-methylcyclohex-1-ene. It undergoes two distinct oxidative pathways under different conditions:

Pathway I: $\mathbf{X} \xrightarrow{KMnO_4, \ H^+, \ \Delta} \mathbf{P}$

Pathway II: $\mathbf{X} \xrightarrow{KMnO_4, \ OH^-, \ 273 \ K} \mathbf{Q} \xrightarrow{HIO_4} \mathbf{R}$

Identify the major organic products $\mathbf{P}$ and $\mathbf{R}$.

A purple-colored diamagnetic transition metal complex $\mathbf{A}$ is heated with concentrated $KOH$ to form a green-colored compound $\mathbf{B}$ along with the evolution of $O_2$ gas. Upon altering the pH to a neutral medium, $\mathbf{B}$ undergoes disproportionation to regenerate $\mathbf{A}$ along with a dark brown solid $\mathbf{C}$. When $\mathbf{C}$ is heated with concentrated $H_2SO_4$ and $NaCl$, a greenish-yellow gas is evolved, and a soluble manganese salt $\mathbf{D}$ is formed.

Assuming $\mathbf{D}$ exists as a hexaaqua complex in an aqueous solution, what is the spin-only magnetic moment ($\mu_{eff}$) and the Crystal Field Stabilization Energy (CFSE) of the central metal ion in $\mathbf{D}$?

Consider the following thermodynamic data at 298 K: (i) $\ce{Cr2O7^{2-} + 14H+ + 6e- -> 2Cr^{3+} + 7H2O} \quad E^\circ = 1.33 \text{ V}$ (ii) $\ce{2CrO4^{2-} + 2H+ <=> Cr2O7^{2-} + H2O} \quad K = 1.0 \times 10^{14}$ (iii) Solubility product of $\ce{Cr(OH)3(s)}$, $K_{sp} = 1.0 \times 10^{-30}$ (iv) Ionic product of water, $K_w = 1.0 \times 10^{-14}$

Assuming $\frac{2.303 RT}{F} = 0.06 \text{ V}$, calculate the standard reduction potential ($E^\circ_{\text{basic}}$) for the following half-reaction in basic medium: $\ce{CrO4^{2-} + 4H2O + 3e- -> Cr(OH)3(s) + 5OH-}$

Consider the following parameters for Chromium and its compounds:

$P =$ Number of equivalent terminal \ce{Cr-O} bonds in the dichromate ion (\ce{Cr2O7^{2-}}). $Q =$ Highest oxidation state of Chromium in a characterized binary fluoride. $R =$ Number of $d\pi-p\pi$ bonds in a monomeric \ce{CrO3} molecule. $S =$ Oxidation state of Chromium in the deep blue complex formed when acidified \ce{K2Cr2O7} reacts with \ce{H2O2} in the presence of ether.

Calculate the value of $(P \times Q) - (R \times S)$.

Unlike Tollens' reagent (which oxidizes both aldoses and ketoses) and Fehling's solution (which requires isomerizable sugars in alkaline media), Cr(VI) in an acidic buffer (pH = 3.0) at 298 K selectively oxidizes the $\beta$-pyranose anomer of D-glucose to D-glucono-$\delta$-lactone. The active oxidant is exclusively the monomeric acid chromate ion, \ce{HCrO4-}.

In a kinetic experiment, D-glucose is dissolved to a total concentration of $0.10 \text{ M}$. It undergoes rapid mutarotation to reach an equilibrium between $\alpha$ and $\beta$ anomers with $K_{eq} = \frac{[\beta]}{[\alpha]} = 4.0$.

The total analytical concentration of Cr(VI) added is $0.30 \text{ M}$. The dimerization constant for the equilibrium $2\ce{HCrO4-} \rightleftharpoons \ce{Cr2O7^{2-}} + \ce{H2O}$ is $K_d = 50 \text{ M}^{-1}$.

The rate law for the consumption of the active oxidant is given by: $-\frac{d[\ce{HCrO4-}]}{dt} = k[\beta\text{-anomer}][\ce{HCrO4-}]$ where $k = 1.2 \times 10^{-2} \text{ M}^{-1}\text{s}^{-1}$.

Assuming the mutarotation equilibrium is maintained, what is the initial rate of formation of D-glucono-$\delta$-lactone?

Consider the pH-dependent equilibrium between the yellow oxoanion $\ce{CrO4^2-}$ and the orange binuclear oxoanion $\ce{Cr2O7^2-}$ in aqueous solution at 298 K. Let $\Delta E_1$ and $\Delta E_2$ be the energies of the lowest-energy allowed electronic transitions responsible for the colors of $\ce{CrO4^2-}$ and $\ce{Cr2O7^2-}$, respectively. Identify the correct set of parameters for this system in the following order: [Origin of Color, Energy Relationship, Geometry of Cr in $\ce{Cr2O7^2-}$, $\ce{Cr-O-Cr}$ bond angle nature].

A $0.025 \text{ mol kg}^{-1}$ aqueous solution of potassium dichromate (\ce{K2Cr2O7}) is prepared at $298 \text{ K}$. In aqueous solution, the dichromate ion undergoes hydrolysis according to the equilibrium: $\ce{Cr2O7^{2-}(aq) + H2O(l) <=> 2HCrO4^{-}(aq)}$ The thermodynamic equilibrium constant for this reaction is $K_{eq} = 0.1152$. At this concentration, the activity coefficients for the ions are $\gamma_{\ce{HCrO4^-}} = 0.80$ and $\gamma_{\ce{Cr2O7^{2-}}} = 0.50$. Assuming the activity of neutral water is unity, \ce{K2Cr2O7} acts as a strong electrolyte before hydrolysis, and the solution behaves ideally with respect to colligative properties once equilibrium is established (osmotic coefficient $\phi = 1$), what is the apparent Van't Hoff factor ($i$) for \ce{K2Cr2O7} in this solution?

Consider the chemical consequences of the lanthanoid contraction on the f-block and post-lanthanoid transition elements. Which of the following statements is \textbf{INCORRECT}?

Consider the thermodynamic and equilibrium properties of lanthanoid hydroxides, $\ce{Ln(OH)3}$, and substituted phenols at $298\text{ K}$. The solubility product ($K_{sp}$) of $\ce{La(OH)3}$ is approximately $2 \times 10^{-20}$, and the $pK_a$ of phenol is $10.0$. Which of the following statements correctly distinguishes between their intrinsic electronic properties and effective aqueous behavior?

The actinoid series exhibits complex redox chemistry due to the relativistic stabilization of the 7s orbital and the varying energy of the 5f orbitals. Consider the Latimer diagram for Americium (Am) in aqueous acidic medium ($\ce{pH} = 0$) at $298\text{ K}$: $\ce{AmO2^{2+}(aq) ->[+1.60\text{ V}] AmO2+(aq) ->[+1.04\text{ V}] Am^{4+}(aq) ->[+2.60\text{ V}] Am^{3+}(aq) ->[-2.07\text{ V}] Am(s)}$ Unlike early actinoids (e.g., U, Np) where the $+4$ state is highly stable, the increasing effective nuclear charge in Am significantly lowers the 5f orbital energy, making $\ce{Am^{4+}}$ a powerful oxidant. A chemist investigates the thermodynamic stability of the $\ce{AmO2+(aq)}$ ion. It undergoes spontaneous disproportionation to yield $\ce{AmO2^{2+}(aq)}$ and the most thermodynamically stable lower oxidation state under these conditions. Calculate the standard Gibbs free energy change ($\Delta G^\circ$) for the balanced disproportionation reaction involving $3\text{ moles}$ of $\ce{AmO2+(aq)}$. (Given: $F = 96500\text{ C mol}^{-1}$)

The magnetic properties of actinoid complexes deviate significantly from those of 3d transition metals due to strong spin-orbit coupling, which prevents the quenching of orbital angular momentum. Calculate the theoretical effective magnetic moment ($\mu_{eff}$) of the \ce{Pu^{3+}} ion in aqueous solution at 298 K. Assume the magnetic behavior is dictated entirely by its ground state term symbol (Russell-Saunders coupling).

(Given: Atomic number of Pu = 94, $\sqrt{35} \approx 5.92$, $\sqrt{65} \approx 8.06$)

Actinoids in the $+4$ oxidation state exhibit coordination chemistry that significantly deviates from $3d$ transition metals. When an aqueous solution of $Th^{4+}$ is treated with one equivalent of $EDTA^{4-}$ at $298\text{ K}$, a stable, neutral mononuclear complex $[Th(EDTA)(H_2O)_n]$ is isolated. Which of the following correctly identifies the number of coordinated water molecules ($n$), the total coordination number (CN) of Thorium, and the primary governing principle for this structure?

A 50.0 mL aqueous solution contains $0.02$ M $\ce{U(C2O4)2}$ and $0.03$ M $\ce{FeC2O4}$ dissolved in dilute $\ce{H2SO4}$. This solution is titrated with $0.05$ M $\ce{KMnO4}$.

At 298 K, a rapid titration captures only the instantaneous outer-sphere electron transfer reactions, requiring $V_1$ mL of $\ce{KMnO4}$ to reach the first persistent pale pink endpoint.

At 350 K, a prolonged titration ensures complete thermodynamic oxidation of all oxidizable species, requiring $V_2$ mL of $\ce{KMnO4}$.

Given that the oxidations of $\ce{U^{4+}}$ to $\ce{UO2^{2+}}$ and $\ce{C2O4^{2-}}$ to $\ce{CO2}$ involve inner-sphere mechanisms and bond cleavages that are kinetically frozen during the rapid 298 K titration, calculate the value of $(V_2 - V_1)$.

A non-aqueous electrolytic cell at $298 \text{ K}$ contains a solution of Uranium salts where the actinoid exists as a mixture of $U^{3+}$ and $U^{4+}$ ions in a $1:1$ molar ratio. A constant current of $19.3 \text{ A}$ is passed through the cell for $5000 \text{ s}$ to deposit solid Uranium ($\ce{U_{(s)}}$) at the cathode. Due to ion-pairing and solvent interactions in the non-aqueous medium, the current efficiency for the cathodic deposition is $75\%$. Assuming both $U^{3+}$ and $U^{4+}$ are reduced to $\ce{U_{(s)}}$ simultaneously at rates proportional to their bulk mole fractions, calculate the mass of Uranium deposited. \n\n(Given: Faraday's constant $F = 96500 \text{ C mol}^{-1}$, Atomic mass of $U = 238 \text{ g mol}^{-1}$)

Consider the following statements regarding the chemistry of f-block elements and their consequences:

I. Mischmetal is an alloy comprising $\sim 50\%$ $\ce{Ce}$, $\sim 25\%$ $\ce{La}$, along with $\ce{Nd}$, $\ce{Pr}$, and $\sim 5\%$ $\ce{Fe}$, widely used in $\ce{Mg}$-based alloys for lighter flints. II. The post-lanthanoid transition metals $\ce{Zr}$ and $\ce{Hf}$ possess nearly identical atomic radii ($160\text{ pm}$ and $159\text{ pm}$ respectively) primarily due to the poor shielding effect of $4f$ electrons. III. The basicity of lanthanoid hydroxides, $\ce{Ln(OH)3}$, increases from $\ce{La(OH)3}$ to $\ce{Lu(OH)3}$ due to the increase in covalent character of the $\ce{Ln-OH}$ bond. IV. The separation of $\ce{Zr(IV)}$ and $\ce{Hf(IV)}$ is highly challenging because their identical principal quantum numbers for the valence d-orbitals lead to identical hydration enthalpies. V. The third ionization enthalpy ($IE_3$) of $\ce{Eu}$ is anomalously high because the third electron must be removed from a highly stable, half-filled $4f^7$ subshell.

Determine the total number of correct statements.