Some Basic Concepts of Chemistry - Class 11 Part 1

An aqueous solution is prepared by dissolving $25$ g of \ce{K3[Fe(CN)6]} (Molar mass $= 329$ g mol$^{-1}$) in $1.0 \times 10^2$ g of water. The experimental freezing point of this solution is measured to be $-3.6^\circ\text{C}$.

Assuming the Van't Hoff factor ($i$) remains constant over the temperature range, calculate the experimental Van't Hoff factor ($i$) to the correct number of significant figures, and the expected boiling point of the solution at $1$ atm.

(Given: $K_f = 1.86$ K kg mol$^{-1}$, $K_b = 0.512$ K kg mol$^{-1}$, normal boiling point of water is exactly $100^\circ\text{C}$)

A $3.000 \text{ g}$ sample containing a mixture of \ce{Fe3O4}, \ce{FeC2O4}, and inert impurities is dissolved in dilute \ce{H2SO4} to prepare exactly $250.0 \text{ mL}$ of stock solution. A $20.00 \text{ mL}$ aliquot of this solution is heated to $70^\circ\text{C}$ and requires $42.35 \text{ mL}$ of $0.02000 \text{ M}$ \ce{KMnO4} for complete titration. Another $20.00 \text{ mL}$ aliquot is treated with Zn dust and dilute \ce{H2SO4} to completely reduce all iron to \ce{Fe^{2+}}. After filtering off the unreacted zinc, the solution is heated to $70^\circ\text{C}$ and requires $42.74 \text{ mL}$ of the same \ce{KMnO4} solution for complete titration. Calculate the mass of \ce{Fe3O4} in the original sample. (Molar mass of \ce{Fe3O4} = $231.53 \text{ g/mol}$). Apply strict rules of significant figures at all stages of your calculation.

An analytical chemist is preparing a reaction mixture at $350 \text{ K}$. Exactly $25.00 \text{ mL}$ of an organic solvent is transferred at $350 \text{ K}$ using a temperature-calibrated Class A volumetric pipette. According to the laboratory's standard operating procedure, the mass must be calculated using the experimentally measured density at the working temperature, which is $1.2 \text{ g/mL}$. For reference, the literature density of the solvent at $298 \text{ K}$ is $1.275 \text{ g/mL}$, and its coefficient of thermal volume expansion is $\gamma = 1.10 \times 10^{-3} \text{ K}^{-1}$.

Calculate the mass of the solvent transferred at $350 \text{ K}$, reported to the correct number of significant figures.

A standardized aqueous solution of \ce{NaF} has a measured concentration of $0.012$ M. At 298 K, the solubility of \ce{BaF2 (s)} in this solution is analytically determined to be $2.2 \times 10^{-3}$ M. Assuming ideal behavior, calculate the solubility product constant ($K_{sp}$) of \ce{BaF2} at 298 K, reported to the correct number of significant figures.

Hydrogen sulfide (\ce{H2S}) is a toxic gas. Its Henry's law constant ($K_H$) in water at $298 \text{ K}$ is $1.5 \times 10^8 \text{ Pa}$. If the partial pressure of \ce{H2S} over an aqueous solution is $5.20 \times 10^2 \text{ Torr}$, determine the molality of \ce{H2S} in the solution. (Assume ideal behavior and report your answer adhering strictly to the rules of significant figures).

A zero-order reaction $\ce{A(aq) -> Products(aq)}$ occurs in a liquid solvent at $298 \text{ K}$. The initial molality of reactant A is $2.5 \text{ m}$. After $60 \text{ min}$, the molality of A decreases to $0.7 \text{ m}$. The density of the solution remains constant at $1.25 \text{ g mL}^{-1}$ throughout the reaction, and all products remain dissolved in the solution. Given that the molar mass of A is $200 \text{ g mol}^{-1}$, calculate the rate constant $k$ of the reaction.

At $300 \text{ K}$, the standard Gibbs free energy change for the gas-phase reaction: $\ce{N2(g) + 3H2(g) <=> 2NH3(g)}$ is $\Delta G^\circ = -11.49 \text{ kJ mol}^{-1}$ (Standard state: $P^\circ = 1 \text{ bar}$). Let $Y$ be the numerical value of the concentration-based equilibrium constant $K_c$ (Standard state: $c^\circ = 1 \text{ mol L}^{-1}$) for this reaction at $300 \text{ K}$.

Additionally, at $300 \text{ K}$, the ionic product of water is $K_w = 1.0 \times 10^{-14}$. Let $\Delta G^\circ_w$ be the standard Gibbs free energy change for the autoionization reaction: $\ce{H2O(l) <=> H+(aq) + OH-(aq)}$

Which of the following correctly gives the values of $Y$ and $\Delta G^\circ_w$? (Given: $R = 8.314 \text{ J K}^{-1} \text{ mol}^{-1}$, $\ln 10 = 2.303$, $1 \text{ bar} = 10^5 \text{ Pa}$, $1 \text{ atm} = 101325 \text{ Pa}$, density of water $\approx 1 \text{ g mL}^{-1}$)

An aqueous solution is prepared by dissolving $200\text{ g}$ of \ce{Fe2(SO4)3} (Molar mass $= 400\text{ g mol}^{-1}$) in $1000\text{ g}$ of water. The freezing point of this solution is found to be $-3.255^\circ\text{C}$. If the density of the solution at $300\text{ K}$ is $1.44\text{ g mL}^{-1}$, what is the osmotic pressure of the solution at $300\text{ K}$?\n\n(Given: $K_f\text{ of water} = 1.86\text{ K kg mol}^{-1}$, $R = 0.082\text{ L atm K}^{-1}\text{ mol}^{-1}$. Assume the Van't Hoff factor remains constant across the temperature range.)

At $298\text{ K}$, gaseous Boron trifluoride ($\ce{BF3}$) reacts with Ammonia ($\ce{NH3}$) to form the Lewis acid-base adduct $\ce{F3B\cdot NH3(s)}$. Infrared spectroscopy reveals that the dominant B-F symmetric stretching frequency shifts from $\sim 1450\text{ cm}^{-1}$ in free $\ce{BF3}$ to $\sim 1100\text{ cm}^{-1}$ in the adduct. Assuming the reduced mass ($\mu$) of the B-F oscillator remains approximately constant, which of the following correctly correlates the dimensional shift in the B-F bond stretching force constant ($k_{BF}$) with the underlying electronic and structural changes?

The spectroscopic bond dissociation energy ($D_0$) of a homonuclear diatomic molecule \ce{X2(g)} at $0\text{ K}$ is $240.00\text{ kJ mol}^{-1}$. At $300\text{ K}$, the vibrational contribution to the internal energy ($U_{vib}$) of \ce{X2(g)} is $0.735\text{ kJ mol}^{-1}$. Assuming ideal gas behavior for all species, calculate the standard bond enthalpy ($\Delta_{bond}H^\circ$) of \ce{X2(g)} at $300\text{ K}$.

[Given: Universal gas constant $R = 8.3\text{ J K}^{-1}\text{ mol}^{-1}$]

An infinite pool of an equimolar mixture of $\ce{^{14}NH3}$ and $\ce{^{15}NH3}$ undergoes catalytic oxidation. The rate-determining step exhibits a primary kinetic isotope effect (KIE) such that the rate constants ratio is $k_{14}/k_{15} = 4.0$. The reaction is quenched at $<1\%$ conversion. The resulting nitrogen oxides are quantitatively isolated and completely converted into pure $\ce{N2O4(g)}$.

A sample of this synthesized $\ce{N2O4(g)}$ is placed in an evacuated, rigid container and heated to $350\text{ K}$. The system reaches thermodynamic equilibrium:

$\ce{N2O4(g) <=> 2NO2(g)}$

At equilibrium, the total pressure is $1.0\text{ atm}$ and the observed vapor density of the gaseous mixture is $30.8$.

Calculate the equilibrium constant $K_p$ for this dissociation at $350\text{ K}$.

(Given: Exact isotopic masses $\ce{^{14}N} = 14.0\text{ u}$, $\ce{^{15}N} = 15.0\text{ u}$, $\ce{O} = 16.0\text{ u}$. Assume ideal gas behavior and vapor density is relative to $M = 2.0\text{ u}$.)

A synthetic sample of chromium is prepared by mixing the isotopes \ce{^{50}Cr} and \ce{^{53}Cr} in a $1:2$ molar ratio. This isotopic mixture is used to synthesize the blue-colored, ether-soluble complex Chromium pentoxide (\ce{CrO5}). A $1.32\text{ g}$ sample of this complex is added directly to an acidified $0.5\text{ M}$ aqueous solution of \ce{FeSO4}, where it is completely reduced.

Identify the true oxidation state of Chromium in the \ce{CrO5} complex and the exact volume of the \ce{FeSO4} solution required for complete reduction.

A $3.0 \text{ M}$ solution of a non-volatile, non-electrolyte solute (molar mass $= 60 \text{ g mol}^{-1}$) is prepared in a special isotopic water solvent at $312 \text{ K}$. The solvent is a mixture of $\ce{H2O}$ and $\ce{D2O}$ such that the overall atomic ratio of $\ce{H:D}$ in the solvent is $1:3$. The density of the resulting solution is $1.155 \text{ g mL}^{-1}$. If the vapor pressure of the pure isotopic solvent mixture at $312 \text{ K}$ is $53.00 \text{ torr}$, what is the vapor pressure of the solution? (Assume ideal behavior and that the isotopic composition of the vapor phase does not significantly alter the bulk solvent composition).

A $100 \text{ mL}$ aqueous solution containing $0.1 \text{ M } \ce{K2Cr2O7}$ and excess $\ce{H2SO4}$ is treated with $V \text{ mL}$ of $0.4 \text{ M } \ce{H2O2}$ at $273 \text{ K}$ in the presence of diethyl ether. The organic layer, containing the blue-colored complex $\ce{CrO5}$, is completely separated. The extracted $\ce{CrO5}$ is then titrated against a $0.2 \text{ M}$ acidified $\ce{KMnO4}$ solution. It requires exactly $50 \text{ mL}$ of the $\ce{KMnO4}$ solution to completely oxidize the peroxy linkages of $\ce{CrO5}$ to $\ce{O2(g)}$, leaving chromium in the $+6$ oxidation state as $\ce{Cr2O7^{2-}}$.

Assuming ideal stoichiometry and $100\%$ extraction efficiency, determine the actual value of $V$. Furthermore, what would be the calculated value of $V$ if one incorrectly assumes that the average oxidation state of Cr in $\ce{CrO5}$ dictates its $n$-factor during the titration?

A $16.2\text{ g}$ equimolar mixture of ethyl ethanoate and diethyl ether is heated with $19.2\text{ g}$ of anhydrous \ce{HI} at $373\text{ K}$ in a sealed tube. The reaction proceeds under kinetic control. Assuming no further reaction of the initial cleavage products occurs under these limited \ce{HI} conditions, what is the total combined mass of all halogenated organic products and alcohols present in the final reaction mixture?

(Given molar masses in $\text{g/mol}$: $\text{H} = 1$, $\text{C} = 12$, $\text{O} = 16$, $\text{I} = 127$)

A $1.325 \text{ g}$ sample of impure anhydrous sodium carbonate ($\ce{Na2CO3}$) is dissolved in water and treated with $50.0 \text{ mL}$ of $0.50 \text{ M}$ $\ce{HCl}$. The solution is boiled to completely expel dissolved $\ce{CO2}$. The excess acid is then back-titrated with $0.20 \text{ M}$ aqueous ammonia ($\ce{NH3}$). The equivalence point of the back-titration is reached after adding $25.0 \text{ mL}$ of the $\ce{NH3}$ solution.

Given: $K_b(\ce{NH3}) = 2.0 \times 10^{-5}$, $K_w = 1.0 \times 10^{-14}$, $\log 5 = 0.70$. The total volume of the solution at the equivalence point is exactly $100.0 \text{ mL}$.

Determine the percentage purity of the $\ce{Na2CO3}$ sample and identify the most suitable indicator for the back-titration from the following options: Methyl Red ($pK_{In} = 5.1$), Bromothymol Blue ($pK_{In} = 7.1$), Phenolphthalein ($pK_{In} = 9.4$).

A $1.84 \text{ g}$ sample of pure Tungsten ($\ce{W}$, atomic mass $= 184 \text{ g/mol}$) is reacted with $1.064 \text{ g}$ of Fluorine gas ($\ce{F2}$, molar mass $= 38 \text{ g/mol}$) in a sealed, evacuated vessel at $600 \text{ K}$. Tungsten exhibits variable oxidation states, primarily forming $\ce{WF5}$ and $\ce{WF6}$ depending on the stoichiometric ratio of the reactants. Assuming the reaction proceeds to completion such that no elemental tungsten or fluorine remains, what is the mole ratio of $\ce{WF6}$ to $\ce{WF5}$ in the final product mixture?

A $1.2 \text{ M}$ aqueous solution of a highly soluble organic weak monoprotic acid ($HA$, molar mass = $150 \text{ g mol}^{-1}$) has a density of $1.08 \text{ g mL}^{-1}$ at $298 \text{ K}$. When this solution is cooled, it begins to freeze at $-2.604^\circ\text{C}$. Given the cryoscopic constant of water $K_f = 1.86 \text{ K kg mol}^{-1}$, what is the degree of dissociation ($\alpha$) of the acid at its freezing point?

Consider the acid-catalyzed hydrolysis of an ester \ce{A} in a highly concentrated aqueous solution at $298\text{ K}$: $\ce{A(aq) + H2O(l) ->[H^+] B(aq) + C(aq)}$ The initial reaction mixture contains $1.0\text{ mol}$ of \ce{A} and $4.0\text{ mol}$ of \ce{H2O}. The partial molar volumes of the components, which are assumed to remain constant over the course of the reaction, are given as: $\bar{V}_{\ce{A}} = 128\text{ mL/mol}$, $\bar{V}_{\ce{H2O}} = 18\text{ mL/mol}$, $\bar{V}_{\ce{B}} = 26\text{ mL/mol}$, and $\bar{V}_{\ce{C}} = 20\text{ mL/mol}$.

The true rate law for the reaction is $\text{Rate} = k[\ce{A}][\ce{H2O}]$. In standard dilute solutions, this is approximated as a pseudo-first-order reaction with an apparent rate constant $k_{\text{obs}} = k[\ce{H2O}]$. Due to the high concentration and significant volume contraction upon reaction, $[\ce{H2O}]$ is not constant. Calculate the exact ratio of the instantaneous pseudo-first-order rate constant at $50\%$ completion of \ce{A} to its initial value, i.e., $\frac{k_{\text{obs}}(50\%)}{k_{\text{obs}}(0\%)}$.

A neutral mixed-valence transition metal cluster has the empirical formula \ce{M2L5} (where \ce{L} is a monoanionic ligand), implying an average fractional oxidation state of $+2.5$ for the metal \ce{M}. To resolve this fractional state, the cluster exists in benzene as a discrete, undissociated hexanuclear species containing \ce{M^{2+}} and \ce{M^{3+}} ions in a $1:1$ ratio.

A solution is prepared containing exactly $1.0\text{ mole}$ of the empirical formula unit \ce{M2L5} per $1.950\text{ kg}$ of benzene (\ce{C6H6}, molar mass = $78\text{ g mol}^{-1}$).

Given that the vapor pressure of pure benzene at $298\text{ K}$ is $152.0\text{ Torr}$, calculate the vapor pressure of the solution at $298\text{ K}$. (Assume ideal behavior).

A deep blue solid $X$ is precipitated when an aqueous solution of iron(III) chloride is treated with potassium hexacyanoferrate(II) at 298 K. Let $n_{inner}$ and $n_{outer}$ be the number of unpaired electrons present on a single iron atom in the inner coordination sphere and outer ionization sphere of $X$, respectively. What is the value of $|n_{outer} - n_{inner}|$ and the average oxidation state of iron in the anhydrous empirical formula of $X$?

A gaseous hydrocarbon (A) has an empirical formula of \ce{CH2}. At $300\text{ K}$ and $100\text{ kPa}$, the density of the gas is $3.36\text{ kg m}^{-3}$. Compound (A) exhibits geometrical isomerism and undergoes reductive ozonolysis to yield only one type of organic product (B). What is the molar mass of (A) and the number of $\alpha$-hydrogens in (B)? (Given: $R = 8.314\text{ J K}^{-1}\text{ mol}^{-1}$)

A $2.29 \text{ g}$ sample of a pure oxide of manganese, \ce{Mn_xO_y}, is fused with excess \ce{KOH} in the presence of \ce{O2} gas. The oxide is completely converted to a green melt, consuming exactly $560 \text{ mL}$ of \ce{O2} gas at STP ($22.4 \text{ L/mol}$).

The green melt is extracted with water and acidified with dilute \ce{H2SO4}, causing complete disproportionation of the green species into a black precipitate and a purple solution.

After filtering the black precipitate, the purple filtrate is adjusted to a faintly alkaline pH ($pH \approx 8$) and treated with excess \ce{KI}. This results in the complete reduction of the purple species, forming more of the black precipitate and a soluble iodine-containing species.

Identify the empirical formula of the original oxide \ce{Mn_xO_y} and calculate the exact number of moles of the soluble iodine-containing species formed.

(Assume atomic masses: Mn = $55 \text{ u}$, O = $16 \text{ u}$)

An organic compound $X$ has the empirical formula \ce{C5H11O} and a molar mass of $174 \text{ g/mol}$. The $^1\text{H}$ NMR spectrum of $X$ exhibits exactly three singlet signals with an integration ratio of 6:3:2. A chemist attempts to synthesize $X$ via the Williamson ether synthesis by reacting the corresponding symmetrical dibromide $A$ with excess sodium methoxide in methanol at $330 \text{ K}$. However, due to mechanistic constraints, the expected substitution fails completely. Instead, a highly stable conjugated hydrocarbon $Y$ is isolated as the major product. What is the IUPAC name of $Y$?

A neutral transition metal carbonyl complex of Platinum ($\ce{Pt}$, atomic mass $= 195 \text{ u}$) is isolated. Elemental analysis yields the following mass composition: $\ce{Pt} = 69.88\%$, $\ce{C} = 12.91\%$, $\ce{O} = 17.21\%$.

Calculate the sum of the atomicity (total number of atoms) of the true molecular formula and the total valence electron count (18-electron rule count) around its central metal atom.

(Assume the complex exists in its simplest stable form consistent with the analytical data and the relativistic d-orbital contraction typical of heavy transition metals).

An aqueous solution contains $0.02\text{ mol}$ of $\ce{K2Cr2O7}$ and $0.04\text{ mol}$ of $\ce{KMnO4}$. The solution is divided into two equal aliquots, A and B.

Aliquot A is strongly acidified with dilute $\ce{H2SO4}$ and an excess of $\ce{KI}$ is added. The liberated $\ce{I2}$ is titrated with a $C\text{ M}$ $\ce{Na2S2O3}$ solution, requiring $V_A\text{ mL}$.

Aliquot B is buffered to a faintly alkaline pH ($pH \approx 9$) and an excess of $\ce{KI}$ is added. Under these conditions, $\ce{I-}$ is oxidized to $\ce{IO3-}$. The unreacted $\ce{KI}$ is removed, and the $\ce{IO3-}$ formed is quantitatively isolated, dissolved in dilute acid, and treated with excess $\ce{KI}$ to liberate $\ce{I2}$. This liberated $\ce{I2}$ is titrated with the same $C\text{ M}$ $\ce{Na2S2O3}$ solution, requiring $V_B\text{ mL}$.

Assuming complete reactions, what is the ratio $V_A / V_B$?

A sample containing $x$ moles of a non-stoichiometric iron oxide, \ce{Fe_{0.93}O (s)}, is treated with an aqueous solution containing $y$ moles of sodium hypochlorite, \ce{NaClO (aq)}, in a strongly alkaline medium at 298 K. The iron oxide is completely oxidized to soluble ferrate(VI) ions, \ce{FeO4^{2-} (aq)}, while the hypochlorite is reduced to chloride ions, \ce{Cl^- (aq)}. If the redox reaction goes to completion with no excess reagents, what is the exact ratio of $x : y$?

A piece of iron (\ce{Fe}) is exposed to a drop of water at $298 \text{ K}$, initiating a differential aeration corrosion cell. The periphery of the drop acts as the cathode due to high oxygen concentration, while the center acts as the anode. The area of the cathodic region ($A_c$) is $20$ times the area of the anodic region ($A_a$). If the steady-state cathodic current density ($i_c$) is $9.65 \times 10^{-5} \text{ A cm}^{-2}$, calculate the localized rate of mass loss of iron at the anodic region in $\text{g h}^{-1} \text{ cm}^{-2}$.

(Given: Molar mass of \ce{Fe} = $56 \text{ g mol}^{-1}$, $1 \text{ F} = 96500 \text{ C mol}^{-1}$, assume iron oxidizes to \ce{Fe^{2+}} at the anode)

A $100 \text{ mL}$ solution initially containing $1.0 \times 10^{-3} \text{ M}$ $\ce{Fe(ClO4)3}$ is subjected to electrolysis at a constant current of $9.65 \text{ mA}$ for $500 \text{ s}$. The solution also contains inert $\ce{NaClO4}$ such that the total ionic strength ($I$) of the medium is maintained constant at $0.04 \text{ M}$. After the electrolysis is stopped, the potential of a Platinum electrode dipped in this solution is measured against a Standard Hydrogen Electrode (SHE) at $298 \text{ K}$.

Assuming the only reaction during electrolysis is the reduction of $\ce{Fe^{3+}}$ to $\ce{Fe^{2+}}$, calculate the measured cell potential.

[Given: $E^\circ_{\ce{Fe^{3+}/Fe^{2+}}} = 0.770 \text{ V}$; Faraday's constant $F = 96500 \text{ C mol}^{-1}$; $\frac{2.303 RT}{F} = 0.060 \text{ V}$ at $298 \text{ K}$. Debye-Hückel Limiting Law: $\log \gamma_i = -A z_i^2 \sqrt{I}$, where $A = 0.50 \text{ M}^{-1/2}$ for water at $298 \text{ K}$.]

A $1.20 \text{ g}$ sample of pure iron pyrites (\ce{FeS2}) is roasted in excess oxygen. The entire \ce{SO2} gas produced is collected and dissolved in water. To this solution, $20 \text{ mL}$ of $0.2 \text{ M}$ acidified \ce{K2Cr2O7} is added, which completely reacts with a portion of the \ce{SO2}. Subsequently, $0.012 \text{ moles}$ of \ce{H2S} gas is bubbled through the remaining solution, resulting in the precipitation of elemental sulfur (\ce{S}). Assuming no other side reactions, determine the mass of elemental sulfur precipitated and the mass of the excess reagent remaining in the final step.\n\n(Molar masses in $\text{g mol}^{-1}$: \ce{FeS2} = 120, \ce{SO2} = 64, \ce{H2S} = 34, \ce{S} = 32)

Consider the following three unbalanced redox reactions involving mixed-valence oxides in an acidic medium:

I. $\ce{Fe3O4(s) + MnO4^{-}(aq) + H+(aq) -> Fe^{3+}(aq) + Mn^{2+}(aq) + H2O(l)}$ II. $\ce{Pb3O4(s) + H2O2(aq) + H+(aq) -> Pb^{2+}(aq) + O2(g) + H2O(l)}$ III. $\ce{Mn3O4(s) + C2O4^{2-}(aq) + H+(aq) -> Mn^{2+}(aq) + CO2(g) + H2O(l)}$

Let $n_1, n_2,$ and $n_3$ be the $n$-factors (valency factors) of $\ce{Fe3O4}$, $\ce{Pb3O4}$, and $\ce{Mn3O4}$ respectively in the above reactions. Let $f_1, f_2,$ and $f_3$ be the exact fraction of metal atoms in each respective oxide formula unit that undergo a change in their oxidation state during the reactions.

Calculate the value of the expression: $\Omega = \frac{n_1}{f_1} + \frac{n_2}{f_2} + \frac{n_3}{f_3}$

An unknown solid element $M$ reacts with oxygen at $298\text{ K}$ to form a stable oxide $X$. Quantitative analysis reveals that $10.45\text{ g}$ of $M$ combines exactly with $1.20\text{ g}$ of oxygen to form $X$. A theoretical higher oxide $Y$ of element $M$ is proposed, which obeys the Law of Multiple Proportions. The ratio of the mass of oxygen combining with $1\text{ g}$ of $M$ in oxide $Y$ to that in oxide $X$ is exactly $5:3$.

Given that $M$ is a p-block element, which of the following statements correctly identifies the acid-base nature of oxide $X$ and the thermodynamic reality of oxide $Y$ at $298\text{ K}$?

A metal $M$ (Atomic mass = $115 \text{ g mol}^{-1}$) forms two chlorides, $A$ and $B$. Elemental analysis reveals that $1.00 \text{ g}$ of $M$ combines with $0.308 \text{ g}$ of chlorine in compound $A$, and with $0.924 \text{ g}$ of chlorine in compound $B$.

$0.30 \text{ moles}$ of solid $A$ is dissolved in $1 \text{ L}$ of water at $298 \text{ K}$. Upon dissolution, the metal ion undergoes complete disproportionation into solid metal $M$ and the higher oxidation state metal ion found in $B$.

The resulting aqueous mixture is filtered, and the filtrate is electrolyzed using a current of $9.65 \text{ A}$ for $1000 \text{ s}$. Due to concurrent $H_2$ gas evolution at the cathode, the current efficiency for the deposition of $M$ is only $60\%$.

Calculate the total mass of solid metal $M$ recovered from both the initial dissolution process and the subsequent electrolysis. (Given: $F = 96500 \text{ C mol}^{-1}$, Atomic mass of $Cl = 35.5 \text{ g mol}^{-1}$)

A highly reactive intermediate \ce{X} (molar mass = $24 \text{ g mol}^{-1}$) is generated photochemically in a solvent \ce{M} at $298 \text{ K}$. It reacts with dissolved \ce{O2} to form an excited adduct \ce{XO2^}, which then undergoes two competitive processes: \begin{align} \ce{XO2^* (solv)} &\xrightarrow{k_1} \ce{XO (solv)} + \ce{O (solv)} \quad \text{(Unimolecular dissociation)} \ \ce{XO2^* (solv)} + \ce{M (l)} &\xrightarrow{k_2} \ce{XO2 (solv)} + \ce{M^* (l)} \quad \text{(Collisional deactivation)} \end{align*} The oxygen atoms (\ce{O}) generated in the first step immediately recombine to form \ce{O2} and do not react with \ce{X}. Given the rate constants: $k_1 = 1.5 \times 10^9 \text{ s}^{-1}$ $k_2 = 5.0 \times 10^7 \text{ M}^{-1}\text{s}^{-1}$ The concentration of the solvent is $[\ce{M}] = 10 \text{ M}$. A student incorrectly assumes that the average mass of oxygen combined with \ce{X} in the bulk mixture must be the arithmetic mean of the masses dictated by the Law of Multiple Proportions for \ce{XO} and \ce{XO2} (i.e., an equimolar mixture). What is the \textbf{actual} total mass of oxygen atoms incorporated into the stable products (\ce{XO} and \ce{XO2}) when $12.0 \text{ g}$ of \ce{X} is completely consumed in this solvent?

A $540 \text{ mL}$ sample of a saturated aqueous solution of \ce{AgNO3} at $298 \text{ K}$ is subjected to electrolysis using inert electrodes. At $298 \text{ K}$, the solubility of \ce{AgNO3} is $170 \text{ g}$ per $100 \text{ g}$ of water, and the density of the saturated solution is $1.50 \text{ g/mL}$. A constant direct current of $19.3 \text{ A}$ is passed through the solution for $5.00 \text{ hours}$. Assuming $100\%$ current efficiency, what is the mass of silver deposited at the cathode? \ (Given: $F = 96500 \text{ C/mol}$, Molar mass of $\ce{Ag} = 108 \text{ g/mol}$, $\ce{AgNO3} = 170 \text{ g/mol}$)

Group 4 transition metals $M$ ($M = \ce{Zr}, \ce{Hf}$) form stable octahedral complex anions $\ce{[MCl6]^{x-}}$ in their highest common oxidation state. The potassium salt of this complex, $\ce{K_x[MCl6]}$, crystallizes in a cubic lattice where the complex anions form a face-centered cubic (FCC) array and $\ce{K+}$ ions occupy the tetrahedral voids. At $298\text{ K}$, the density of the Zirconium salt is $2.80\text{ g cm}^{-3}$. Given the atomic masses (in $\text{g mol}^{-1}$): $\ce{K} = 39.0$, $\ce{Cl} = 35.5$, $\ce{Zr} = 91.0$, $\ce{Hf} = 178.5$. Assuming ideal behavior and accounting for the post-lanthanoid radii equality, what is the theoretical density of the Hafnium salt?

An experiment is conducted to determine the degree of dissociation ($\alpha$) of an isotopically pure salt, $\ce{^6Li^{35}Cl}$, in aqueous solution. Exactly $20.492\text{ g}$ of the salt is dissolved in $1000.0\text{ g}$ of pure water at $298\text{ K}$. The vapor pressure of the resulting solution is measured to be $23.3602\text{ Torr}$, while the vapor pressure of pure water at this temperature is $23.7600\text{ Torr}$. Given the exact isotopic masses: $\ce{^6Li} = 6.015\text{ u}$, $\ce{^{35}Cl} = 34.969\text{ u}$, $\ce{^1H} = 1.008\text{ u}$, $\ce{^{16}O} = 15.999\text{ u}$. Calculate the degree of dissociation ($\alpha$) of $\ce{^6Li^{35}Cl}$.

A saturated solution of \ce{K2SO4} is prepared by dissolving exactly $17.426 \text{ g}$ of the salt in $1.000 \text{ kg}$ of pure water at $298 \text{ K}$. To this solution, $7.455 \text{ g}$ of solid \ce{KCl} is added. Assuming that the addition of \ce{KCl} does not change the mass of the solvent and that the solution behaves ideally, calculate the freezing point depression ($\Delta T_f$) of the final solution at equilibrium.

Given: $K_f$ of water = $1.86 \text{ K kg mol}^{-1}$ Exact atomic masses (in amu): $\text{K} = 39.10$, $\text{S} = 32.06$, $\text{O} = 16.00$, $\text{Cl} = 35.45$

An unknown, symmetrically substituted N,N-dimethylaniline derivative, $\mathbf{X}$, is subjected to precise combustion analysis. Combustion of exactly $2.3340$ g of $\mathbf{X}$ produces $7.0414$ g of \ce{CO2} and $2.4320$ g of \ce{H2O}. (Use exact atomic weights: C = 12.011, H = 1.008, N = 14.007, O = 15.999). Aqueous basicity measurements reveal that $\mathbf{X}$ is a significantly stronger base than standard N,N-dimethylaniline ($\Delta pK_b > 2$). Which of the following correctly identifies $\mathbf{X}$ and the primary electronic/steric reason for its anomalous basicity?

A highly precise thermogravimetric analysis is performed on $2.5000 \text{ g}$ of a pure, hydrated coordination complex of an unknown first-row transition metal $M$. The complex has the empirical formula $K_a[M(C_2O_4)_b] \cdot xH_2O$. Upon heating to $150^\circ\text{C}$ to constant weight, the sample loses exactly $0.2727 \text{ g}$ of water. The anhydrous residue is then exhaustively reduced, yielding exactly $0.2976 \text{ g}$ of pure metal $M$. Given the exact atomic masses (in $\text{g/mol}$): $\text{K} = 39.10$, $\text{C} = 12.01$, $\text{O} = 16.00$, $\text{H} = 1.008$, and assuming the ligands act as strong field ligands in this specific geometry, what is the spin-only magnetic moment ($\mu$) and the hybridization of the central metal ion $M$ in the original complex? \newline \textit{(Note: Use exact atomic masses for first-row transition metals to identify $M$ and its oxidation state. Common rounded masses: $\text{Cr}=52, \text{Mn}=55, \text{Fe}=56, \text{Co}=59, \text{Ni}=59, \text{Cu}=64$)}

An unknown mono-substituted aniline derivative $X$ features a bulky substituent at the ortho position. High-resolution mass spectrometry (HRMS) determines the exact molar mass of $X$ to be $149.227 \text{ g/mol}$. (Use precise atomic masses: $C=12.01, H=1.008, N=14.007, O=16.00$). Given that the $pK_b$ of unsubstituted aniline is $9.40$ at $298 \text{ K}$, which of the following is the most chemically accurate $pK_b$ value for $X$, considering the dominant electronic and steric effects?

The C-O stretching frequency ($\bar{\nu}$) is related to the bond force constant ($k$) by the harmonic oscillator model, where $k$ is directly proportional to the Effective Bond Order (EBO). The stretching frequency of free \ce{CO} is $2143 \text{ cm}^{-1}$, and its EBO is defined as exactly $3.00$.

Consider the following species: (I) Free \ce{CO} (II) \ce{CO3^{2-}} (III) \ce{[V(CO)6]^-} ($\bar{\nu}_{\ce{CO}} = 1860 \text{ cm}^{-1}$) (IV) \ce{H3B <- CO} ($\bar{\nu}_{\ce{CO}} = 2165 \text{ cm}^{-1}$)

Assuming the reduced mass of the C-O oscillator remains constant, calculate the approximate EBO of the C-O bond in species (III) and (IV) respectively. Furthermore, determine the correct increasing order of C-O bond lengths ($d_{\ce{C-O}}$) for the four species.

A neutral, homoleptic carbonyl complex of a first-row transition metal $M$ (Atomic mass $\approx 51 \text{ u}$) contains $32.90\%$ carbon by mass. A solution prepared by dissolving $2.19 \text{ g}$ of this complex in $100 \text{ g}$ of benzene freezes at $5.018^\circ\text{C}$. The freezing point of pure benzene is $5.530^\circ\text{C}$ and its cryoscopic constant is $K_f = 5.12 \text{ K kg mol}^{-1}$.

Assuming ideal behavior in solution, determine the molecular formula of the complex and the Effective Atomic Number (EAN) of the metal center.

A solid mixture of $\ce{KMnO4}$ and $\ce{K2Cr2O7}$ is dissolved in water to prepare $200\text{ mL}$ of a stock solution.

Aliquot 1 ($100\text{ mL}$) of this solution is strongly acidified with dilute $\ce{H2SO4}$ and treated with an excess of aqueous $\ce{KI}$. The liberated iodine requires $160\text{ mL}$ of $0.1\text{ M } \ce{Na2S2O3}$ solution for complete titration.

Aliquot 2 ($100\text{ mL}$) is buffered to $\text{pH} = 9$. It is then titrated directly with $0.1\text{ M } \ce{Na2S2O3}$ solution, requiring $7.5\text{ mL}$ to reach the potentiometric end point.

Calculate the mass of $\ce{KMnO4}$ and $\ce{K2Cr2O7}$ in the original solid mixture. (Given molar masses: $\ce{KMnO4} = 158\text{ g/mol}$, $\ce{K2Cr2O7} = 294\text{ g/mol}$)

A $50.0\text{ mL}$ sample of $0.20\text{ M}$ aqueous solution of a weak monoacidic base, \ce{BOH} ($K_b = 2.0 \times 10^{-5}$), is titrated against $0.10\text{ M}$ \ce{HCl}(aq) at $298\text{ K}$. An indicator \ce{HIn} is used to detect the endpoint. The indicator transitions from its basic color to its acidic color, and the endpoint is visually detected when the concentration of the acidic form is exactly 10 times that of the basic form. Assuming ideal behavior, what must be the exact $pK_{In}$ of the indicator to perfectly coincide with the equivalence point of this titration?

(Given: $\log 3 = 0.48$, $K_w = 1.0 \times 10^{-14}$)

An organic weak acid $\ce{HA}$ (Molar mass = $120 \text{ g mol}^{-1}$) is dissolved in benzene. At $298 \text{ K}$, the molarity of the solution is $0.80 \text{ M}$ and its density is $0.896 \text{ g mL}^{-1}$. In benzene, $\ce{HA}$ undergoes dimerization: $2\ce{HA} \rightleftharpoons \ce{(HA)2}$. The dimerization equilibrium constant on the molality scale at the freezing point of the solution is $K_m = 10.0 \text{ kg mol}^{-1}$. Given the freezing point of pure benzene is $278.50 \text{ K}$ and its cryoscopic constant $K_f = 5.12 \text{ K kg mol}^{-1}$, calculate the freezing point of the solution.

A buried iron pipe undergoes corrosion via a differential aeration cell in a moist soil environment at $298\text{ K}$. The pipe surface develops a localized anodic pit of area $A_a = 10\text{ cm}^2$ surrounded by a large cathodic region of area $A_c = 1000\text{ cm}^2$. The reduction of dissolved oxygen, $\ce{O2(g) + 2H2O(l) + 4e^- -> 4OH^-(aq)}$, occurs uniformly over the cathodic region with a current density of $1.93\,\mu\text{A cm}^{-2}$. Assuming the dissolution of iron, $\ce{Fe(s) -> Fe^{2+}(aq) + 2e^-}$, is the sole anodic process and the system operates under steady-state conditions, what is the total mass of iron lost from the anodic pit over a period of $10^5\text{ s}$? (Given: Atomic weight of $\ce{Fe} = 56\text{ g mol}^{-1}$)

A $1.00$ L aqueous solution at $298$ K initially contains $0.200$ M of \ce{Fe^{3+}} and $0.050$ M of \ce{Fe^{2+}}, along with a high concentration of \ce{HClO4} to maintain a constant high ionic strength. A constant current of $1.93$ A is passed through the solution for $5000$ seconds, selectively reducing \ce{Fe^{3+}} to \ce{Fe^{2+}} at the cathode with $100\%$ current efficiency. After the electrolysis, the potential of a Platinum electrode dipped in this solution is measured against a Standard Hydrogen Electrode (SHE).

Given: $E^\circ_{\ce{Fe^{3+}/Fe^{2+}}} = 0.770$ V $\frac{2.303 RT}{F} = 0.059$ V Activity coefficients in the final medium: $\gamma_{\ce{Fe^{3+}}} = 0.15$, $\gamma_{\ce{Fe^{2+}}} = 0.40$ $\log 2 = 0.301$, $\log 3 = 0.477$ Assume no change in the volume of the solution.

What is the measured potential of the electrode?

Consider the following three redox processes occurring in an acidic aqueous medium at $298 \text{ K}$:

Process 1: Spontaneous intramolecular redox (decomposition) of Chromium(VI) peroxide (\ce{CrO5}) to yield \ce{Cr^{3+}(aq)}, \ce{H2O(l)}, and \ce{O2(g)}. Process 2: Complete reduction of \ce{CrO5} by an excess of \ce{Fe^{2+}(aq)} ions, yielding \ce{Cr^{3+}(aq)}, \ce{Fe^{3+}(aq)}, and \ce{H2O(l)}. Process 3: Complete reduction of Caro's acid (\ce{H2SO5}) by an excess of \ce{I^{-}(aq)} ions, yielding \ce{SO4^{2-}(aq)}, \ce{I2(s)}, and \ce{H2O(l)}.

Let $n_1$, $n_2$, and $n_3$ be the n-factors (number of equivalents per mole) of the primary reactants (\ce{CrO5} in Processes 1 and 2; \ce{H2SO5} in Process 3). Which of the following correctly represents the values of $n_1$, $n_2$, and $n_3$?

Anhydrous Prussian blue is a mixed-valence coordination compound with the empirical formula $\ce{Fe7(CN)18}$. Let $x$ be the average oxidation state of all iron atoms in the compound. Let $y$ be the total number of unpaired electrons present in one formula unit of the compound. Let $z$ be the magnitude of the Crystal Field Stabilization Energy (CFSE) of the inner-sphere complex anion in terms of $\Delta_o$ (ignoring pairing energy). Calculate the value of $\frac{7 \cdot x \cdot y}{z}$.