Solutions - Class 12

A redox half-cell is prepared at $298\text{ K}$ by mixing $50\text{ mL}$ of $0.40\text{ M } \ce{Fe_2(SO_4)_3}$ and $50\text{ mL}$ of $0.10\text{ M } \ce{FeSO_4}$. In this highly concentrated electrolyte medium, the solution exhibits significant non-ideal behavior. The single-ion activity coefficients are determined to be $\gamma_{\ce{Fe^{3+}}} = 0.01$ and $\gamma_{\ce{Fe^{2+}}} = 0.10$. Given the standard reduction potential $E^\circ_{\ce{Fe^{3+}/Fe^{2+}}} = 0.770\text{ V}$ and $\frac{2.303 RT}{F} = 0.059\text{ V}$, calculate the actual reduction potential of this half-cell. (Given: $\log 2 = 0.301$, $\log 5 = 0.699$)

An aqueous buffer solution at $298\text{ K}$ is prepared by dissolving $0.144\text{ mol}$ of a neutral amino acid zwitterion ($\ce{HA^\pm}$) and $0.144\text{ mol}$ of its sodium salt ($\ce{NaA}$) in water, along with an inert non-electrolyte. The total mass of the resulting solution is exactly $2000\text{ g}$ and the mass of the water solvent is $1.00\text{ kg}$. The density of the solution is $1.250\text{ g mL}^{-1}$.

The thermodynamic acid dissociation constant for the zwitterion is $pK_{a2} = 9.60$. Assuming the Debye-Hückel limiting law applies, what is the $pH$ of this buffer solution?

(Given: Debye-Hückel constant $A = 0.50\text{ L}^{1/2}\text{ mol}^{-1/2}$ at $298\text{ K}$. The limiting law is $\log \gamma_i = -A z_i^2 \sqrt{I_c}$, where $I_c$ is the ionic strength in molarity).

The acid-catalyzed hydrolysis of ethanoic anhydride in a non-ideal dioxane-water mixture follows second-order kinetics (first-order with respect to each reactant):

$\ce{(CH3CO)2O + H2O ->[H^+] 2CH3COOH}$

At $300\text{ K}$, a reaction mixture is prepared with initial concentrations $[\ce{(CH3CO)2O}]_0 = 0.12\text{ M}$ and $[\ce{H2O}]_0 = 1.2\text{ M}$. The initial rate law is expressed as $-\frac{d[\ce{(CH3CO)2O}]}{dt} = k_{\text{obs}}[\ce{(CH3CO)2O}]$, where $k_{\text{obs}} = 3.6 \times 10^{-4}\text{ s}^{-1}$.

An identical mixture (containing the same initial moles of all components) is reacted at $330\text{ K}$. Due to the non-ideal nature of the solvent, heating from $300\text{ K}$ to $330\text{ K}$ causes a $20\%$ increase in the total volume of the solution.

Given that the activation energy for the true second-order rate constant is $E_a = 3300 R \ln(3)\text{ J mol}^{-1}$ (where $R$ is the universal gas constant), what is the initial rate of appearance of ethanoic acid at $330\text{ K}$?

A student attempts to synthesize the Schiff base of 2,2,4,4-tetramethyl-3-pentanone using an aqueous solution of methanamine at 298 K. To maximize the yield, the total analytical concentration of methanamine is increased to 5.0 M. However, the yield of the imine remains negligible at both pH = 2.0 and pH = 4.5. Which of the following statements correctly identifies the primary chemical principles responsible for this failure?

An aqueous system is prepared at $298 \text{ K}$ by mixing exactly $50.0 \text{ moles}$ of $\ce{H2O(l)}$, $0.40 \text{ moles}$ of a completely soluble 1:1 strong electrolyte $\ce{NX}$, and $0.50 \text{ moles}$ of a sparingly soluble 1:1 salt $\ce{MX}$. The final volume of the aqueous solution is exactly $1.0 \text{ L}$. Given that the solubility product of $\ce{MX}$ at $298 \text{ K}$ is $K_{sp} = 5.0 \times 10^{-2}$, what is the exact relative lowering of vapor pressure ($\frac{\Delta P}{P^\circ}$) of the solution? Assume ideal behavior for all dissolved species.

An aqueous solution of ethanol (\ce{C2H5OH}) exhibits a positive deviation from Raoult's Law, forming a minimum-boiling azeotrope at a specific liquid mole fraction $x_E$ and temperature $T_{az}$ under a constant pressure $P$. Let $\Delta H_{vap, sys}$ and $\Delta S_{vap, sys}$ represent the integral enthalpy and entropy of vaporization of the azeotropic mixture. Let $\gamma_E$ and $\gamma_W$ be the activity coefficients of ethanol and water in the liquid phase, and $P_E^\circ$ and $P_W^\circ$ be their respective pure component vapor pressures at $T_{az}$. Assuming the vapor phase behaves ideally, which of the following sets of thermodynamic relations is strictly valid during the reversible, isobaric vaporization of this azeotrope?

At 298 K, 37.0 g of 1-chlorobutane (Molar mass = 92.5 g/mol) is mixed with 36.9 g of anhydrous tetrabutylammonium iodide (TBAI, Molar mass = 369 g/mol). The system undergoes an in-situ Finkelstein-type exchange:

$\ce{C4H9Cl(l) + TBAI(solv) <=> C4H9I(l) + TBACl(solv)}$

The equilibrium constant for this reaction at 298 K is $K_c = 1.0$. Unlike the classic Finkelstein reaction, all ammonium salts remain completely dissolved as non-volatile, undissociated ion pairs in the homogeneous liquid phase. Assuming the final liquid mixture of haloalkanes behaves as an ideal solution, what is the total vapor pressure of the system at equilibrium?

(Given at 298 K: $P^\circ_{\ce{C4H9Cl}} = 100$ Torr, $P^\circ_{\ce{C4H9I}} = 30$ Torr)

Two volatile liquids, \ce{A} and \ce{B}, form a non-ideal solution exhibiting a positive deviation from Raoult's law. At $300\text{ K}$, the vapor pressures of pure \ce{A} and pure \ce{B} are $300\text{ Torr}$ and $100\text{ Torr}$, respectively. A student incorrectly assumes that the mixture behaves ideally and that the vapor pressures of both pure liquids follow a temperature coefficient $Q_{10} = 2$ (i.e., vapor pressure doubles for every $10\text{ K}$ rise in temperature). Experimentally, the mixture forms a minimum-boiling azeotrope at $330\text{ K}$ with a liquid mole fraction of \ce{A}, $x_A = 0.5$. Calculate the magnitude of the difference between the true mole fraction of \ce{A} in the vapor phase ($y_{A, \text{true}}$) at the azeotropic point and the vapor mole fraction of \ce{A} ($y_{A, \text{calc}}$) calculated by the student at $330\text{ K}$ for the same liquid composition.

Two volatile liquids, \ce{A} and \ce{B}, form a minimum boiling azeotrope at $T_{az} = 324 \text{ K}$ and $1 \text{ atm}$. The azeotropic mixture has a mole fraction of \ce{A}, $x_A = 0.6$. The pure liquids \ce{A} and \ce{B} have boiling points $T_A = 340 \text{ K}$ and $T_B = 390 \text{ K}$ at $1 \text{ atm}$, respectively. The enthalpies of vaporization of pure \ce{A} and pure \ce{B} are $30.6 \text{ kJ mol}^{-1}$ and $35.1 \text{ kJ mol}^{-1}$, respectively, and can be assumed to be independent of temperature. The enthalpy of mixing to form $1 \text{ mole}$ of the liquid azeotrope from pure liquids \ce{A} and \ce{B} at $324 \text{ K}$ is $\Delta H_{\text{mix}} = +1.62 \text{ kJ mol}^{-1}$. Assuming the vapor phase behaves as an ideal gas mixture, what is the entropy of vaporization ($\Delta S_{\text{vap}}$) of $1 \text{ mole}$ of the azeotropic mixture at its boiling point?

An equimolar gaseous mixture of four halomethanes, \ce{CH3F}, \ce{CH3Cl}, \ce{CH3Br}, and \ce{CH3I}, is equilibrated with a minimum-boiling azeotropic mixture of ethanol and water at $320\text{ K}$ and $1\text{ atm}$. Let $K_{H(F)}$, $K_{H(Cl)}$, $K_{H(Br)}$, and $K_{H(I)}$ be their respective Henry's Law constants in this solvent system. Given that the dipole moments ($\mu$) of the halomethanes follow the order \ce{CH3Cl} > \ce{CH3F} > \ce{CH3Br} > \ce{CH3I}, which of the following correctly identifies the order of their Henry's Law constants and the dominant physical factor dictating this trend?

An open reactive distillation apparatus is charged with an equimolar liquid mixture of 2,2-dimethylpropanoic acid, 2-methylpropan-2-ol, and 1-butoxybutane. The mixture is heated to $80^\circ\text{C}$ in the presence of a catalytic amount of concentrated \ce{H2SO4}. Let $r_{est}$ be the rate of esterification, $r_{dehyd}$ be the rate of alcohol dehydration, and $r_{cleav}$ be the rate of ether cleavage. It is observed that the liquid-vapor equilibrium composition alters significantly over time due to the rapid evolution of a gas $\textbf{G}$, violating the assumption of a static azeotrope. Which of the following represents the correct order of initial reaction rates and the identity of $\textbf{G}$?

An aqueous solution is prepared by dissolving $0.08\text{ mol}$ of $\ce{KMnO4}$ and $0.03\text{ mol}$ of $\ce{Na2S2O3}$ in $1\text{ kg}$ of pure water. The solution is allowed to react completely at $298\text{ K}$ in a neutral medium. After the reaction is complete, the solid precipitate formed is filtered off. Assuming complete dissociation of all soluble electrolytes and neglecting any change in the mass of water, what is the depression in freezing point ($\Delta T_f$) of the resulting filtrate?

(Given: $K_f$ of water = $1.86\text{ K kg mol}^{-1}$)

A concentration cell is constructed at 298 K as follows:

$\ce{Ag(s) | AgCl(s) | NaCl(aq, m_1) || NaCl(aq, m_2), Na2SO4(aq, m_3) | AgCl(s) | Ag(s)}$

The two half-cell solutions are separated by a salt bridge. It is given that the two solutions are perfectly isotonic. Assume ideal van't Hoff factors for osmotic pressure ($i_{\ce{NaCl}} = 2$, $i_{\ce{Na2SO4}} = 3$) and that the Debye-Hückel limiting law is valid for calculating the activity coefficients of individual ions.

If $m_3 > 0$, which of the following statements correctly describes the cell potential ($E_{\text{cell}}$) and the relationship between the single-ion activity coefficients of chloride ($\gamma_{\ce{Cl^-}, 1}$ and $\gamma_{\ce{Cl^-}, 2}$) in the two half-cells?

Two aqueous solutions are prepared at $298 \text{ K}$ using $1.0 \text{ kg}$ of water each:

Solution A: $0.10 \text{ mol}$ \ce{HCl} Solution B: $0.10 \text{ mol}$ \ce{HCl} and $1.80 \text{ mol}$ Sucrose (\ce{C12H22O11})

The solutions are placed in separate open beakers inside a sealed desiccator and allowed to reach thermodynamic equilibrium via isothermal solvent vapor transfer. Assume ideal behavior for all solutes ($\gamma = 1$) and complete dissociation of \ce{HCl}.

After equilibrium is established, the two solutions are used to construct the following concentration cell with a liquid junction:

$Pt(s) | \ce{H2}(g, 1 \text{ bar}) | \text{Solution B}_{eq} \vdots \text{Solution A}_{eq} | \ce{H2}(g, 1 \text{ bar}) | Pt(s)$

Given the transport number of \ce{Cl^-} is $t_- = 0.20$ (assume constant across the junction) and $\frac{2.303 RT}{F} = 0.060 \text{ V}$ at $298 \text{ K}$. What is the measured electromotive force ($E_{cell}$) of this cell?

An aqueous solution of \ce{K4[Fe(CN)6]} has a molality of $0.40 \text{ mol kg}^{-1}$. To purify the water via reverse osmosis at $300 \text{ K}$, a minimum applied pressure of $39.28 \text{ bar}$ is required to overcome the osmotic pressure. Assume the solution follows the exact thermodynamic relation for real solutions: $\Pi = -\frac{RT}{V_m} \ln(x_{\text{solvent}})$, where $V_m$ is the molar volume of pure water ($0.018 \text{ L mol}^{-1}$). Given: $R = 0.083 \text{ L bar K}^{-1} \text{ mol}^{-1}$, $e^{-0.0284} = 0.9720$, and moles of water in $1 \text{ kg} = 55.55 \text{ mol}$. Calculate the percentage degree of dissociation ($\alpha$) of the complex in this solution.

A volatile gas \ce{G} is dissolved in water at $298\text{ K}$ under a constant partial pressure $P_G = 760\text{ Torr}$. The ideal Henry's law constant for \ce{G} in pure water is $K_H^\circ = 7.6 \times 10^5\text{ Torr}$. To this solution, a non-volatile, fully dissociating strong electrolyte \ce{Ca(NO3)2} is added such that its molality is $2.0576\text{ mol kg}^{-1}$. Due to the 'salting-out' effect, the chemical potential of the solvent decreases, which increases the activity coefficient ($\gamma_G$) of the dissolved gas. The apparent Henry's law constant becomes $K_H = K_H^\circ \cdot \gamma_G$. If $\gamma_G$ is thermodynamically linked to the vapor pressure lowering of the solvent such that $\gamma_G = \frac{P^\circ_{\ce{H2O}}}{P_{\ce{H2O}}}$, calculate the new equilibrium mole fraction of gas \ce{G} ($x_G'$) in the solution. (Assume the amount of dissolved gas is negligible compared to the solvent and \ce{Ca(NO3)2}, and the molar mass of water is $18\text{ g mol}^{-1}$).

A multimeric globular protein ($M_w = 2.56 \times 10^5 \text{ g/mol}$) consists of 8 identical non-covalently linked subunits. At low concentrations, it exists entirely in its native folded octameric state. However, when $256 \text{ g}$ of this protein is dissolved in $900 \text{ g}$ of water at $298 \text{ K}$, the high concentration induces complete denaturation. The protein fully dissociates into its 8 unfolded monomeric peptide chains. Furthermore, due to the exposure of previously buried hydrophobic and hydrophilic residues, each unfolded monomeric chain strongly binds $250$ molecules of water in its primary hydration shell, effectively removing them from the bulk solvent (these bound water molecules do not contribute to the vapor pressure). Given that the vapor pressure of pure water at $298 \text{ K}$ is $24.00 \text{ Torr}$, calculate the vapor pressure lowering ($\Delta P$) of this solution. (Assume the unfolded chains act as ideal independent kinetic particles and the molar mass of water is $18.0 \text{ g/mol}$).

A $34.2 \text{ g}$ sample of sucrose ($\ce{C12H22O11}$) is dissolved in $100.0 \text{ g}$ of water. A trace amount of an acid catalyst is added, and the solution is heated until the inversion of sucrose is complete. The resulting monosaccharides undergo mutarotation and reach their respective anomeric equilibria. For D-glucose, the equilibrium mixture contains $36\%$ $\alpha$-anomer and $64\%$ $\beta$-anomer. For D-fructose, the equilibrium mixture contains $70\%$ $\beta$-pyranose, $20\%$ $\beta$-furanose, and $10\%$ $\alpha$-furanose. Assuming the trace acid contributes negligibly to the molality, what is the boiling point elevation ($\Delta T_b$) of the final solution? ($K_b$ for water = $0.52 \text{ K kg mol}^{-1}$, Molar mass of sucrose = $342 \text{ g mol}^{-1}$, Molar mass of water = $18 \text{ g mol}^{-1}$)

An organic base \ce{B} (molar mass = $100 \text{ g mol}^{-1}$) is studied in two different solvents to understand its solvation thermodynamics.

Experiment 1: $10.0 \text{ g}$ of \ce{B} is dissolved in $100 \text{ g}$ of benzene ($K_b = 2.53 \text{ K kg mol}^{-1}$). The boiling point of the solution is elevated by $1.518 \text{ K}$. In benzene, \ce{B} undergoes partial dimerization due to intermolecular hydrogen bonding.

Experiment 2: $10.0 \text{ g}$ of \ce{B} is dissolved in $100 \text{ g}$ of water ($K_b = 0.52 \text{ K kg mol}^{-1}$). The boiling point elevation is observed to be $0.5252 \text{ K}$. In water, \ce{B} acts as a weak monobasic amine.

Experiment 3: $0.05 \text{ mol}$ of anhydrous \ce{HCl} gas is dissolved in the aqueous solution from Experiment 2 (assume no change in the mass of the solvent and no volume change).

Assuming the molarity of the aqueous solution is approximately equal to its molality, calculate the dimerization equilibrium constant ($K_d$) of \ce{B} in benzene, the base ionization constant ($K_b$) in water, and the new boiling point elevation ($\Delta T_b'$) of the solution in Experiment 3.

A $2.665 \text{ g}$ sample of a mixture containing two hydrate isomers of $\ce{CrCl3 \cdot 6H2O}$, Isomer X (pale green) and Isomer Y (dark green), is dissolved in $100 \text{ g}$ of water at $298 \text{ K}$. The boiling point of this solution is found to be $100.1456^\circ\text{C}$ at $1 \text{ atm}$. If the entire solution is treated with excess $\ce{AgNO3(aq)}$, what mass of $\ce{AgCl(s)}$ will precipitate?

(Assume the boiling point of pure water is $100.000^\circ\text{C}$ at $1 \text{ atm}$, $100\%$ ionization of the complex salts, ideal behavior, and no aquation of the inner coordination sphere. Given: $K_b(\ce{H2O}) = 0.52 \text{ K kg mol}^{-1}$, Molar mass of $\ce{CrCl3 \cdot 6H2O} = 266.5 \text{ g mol}^{-1}$, Molar mass of $\ce{AgCl} = 143.5 \text{ g mol}^{-1}$)

When $2.44 \text{ g}$ of benzoic acid ($\ce{C6H5COOH}$) is dissolved in $100 \text{ g}$ of benzene ($\ce{C6H6}$), the freezing point of the solution is depressed by $0.6144 \text{ K}$. Assuming benzoic acid undergoes dimerization in benzene, calculate the equilibrium constant $K_m$ (in $\text{kg mol}^{-1}$) for the dimerization process.

(Given: $K_f$ for benzene $= 5.12 \text{ K kg mol}^{-1}$, Molar mass of benzoic acid $= 122 \text{ g mol}^{-1}$)

An alkene $X$ (\ce{3-methylbut-1-ene}) is subjected to hydroboration-oxidation using \ce{B2H6}/THF followed by alkaline \ce{H2O2} to yield a major organic product $Y$. Exactly $7.0\text{ g}$ of $X$ is completely converted to $Y$, and the entire yield of $Y$ is dissolved in $250\text{ g}$ of benzene. The freezing point of the resulting solution is depressed by $1.6384\text{ K}$. Assuming $Y$ undergoes partial dimerization in benzene due to intermolecular hydrogen bonding, identify the IUPAC name of $Y$ and its percentage of dimerization. (Given: $K_f$ for benzene = $5.12\text{ K kg mol}^{-1}$, atomic masses: C=12, H=1, O=16)

A $33.3 \text{ g}$ sample of a non-reducing tetrasaccharide ($\ce{C24H42O21}$, molar mass = $666 \text{ g mol}^{-1}$) is dissolved in $100.0 \text{ g}$ of water. The solution is subjected to complete hydrolysis using a catalytic amount of an insoluble solid acid resin, which is subsequently filtered off. The solution is then allowed to stand at $298 \text{ K}$ until mutarotation equilibrium is reached.

The specific rotation of pure $\alpha$-D-glucose is $+112^\circ$ and pure $\beta$-D-glucose is $+19^\circ$. The final equilibrium mixture exhibits a specific rotation of $+52.6^\circ$.

Assuming ideal behavior, what is the freezing point depression ($\Delta T_f$) of the final equilibrium solution?

(Given: $K_f$ of water = $1.86 \text{ K kg mol}^{-1}$, Molar mass of water = $18 \text{ g mol}^{-1}$)

Pentaamminechloridocobalt(III) chloride, \ce{[Co(NH3)5Cl]Cl2}, undergoes a pseudo-first-order aquation reaction in water to form pentaammineaquacobalt(III) chloride, \ce{[Co(NH3)5(H2O)]Cl3}. A solution is prepared by dissolving $0.10 \text{ mol}$ of \ce{[Co(NH3)5Cl]Cl2} and $0.05 \text{ mol}$ of \ce{NaCl} in $1.0 \text{ kg}$ of water at $298 \text{ K}$. The half-life of the aquation reaction at this temperature is $69.3 \text{ min}$. Assuming ideal behavior and complete ionization of all ionic species, what is the theoretical freezing point of the solution corresponding to its composition after $138.6 \text{ min}$? (Given: $K_f$ for water is $1.86 \text{ K kg mol}^{-1}$)

A biochemist is investigating the osmotic pressure ($\pi$) of a tetrameric protein solution at $300\text{ K}$. In its native folded state, the protein behaves as an ideal solute with a molar mass $M_{native} = 1.0 \times 10^5\text{ g mol}^{-1}$. Upon the addition of a chemical denaturant, the protein completely dissociates into four identical unfolded monomeric chains. The osmotic pressure is measured against a dialysate containing the exact same concentration of denaturant, ensuring only the protein contributes to the measured $\pi$.

Due to the massive increase in the hydrodynamic radius and excluded volume upon unfolding, the denatured monomers exhibit non-ideal behavior. The osmotic pressure of such macromolecular solutions follows the virial expansion: $\frac{\pi}{C} = RT \left( \frac{1}{M} + BC \right)$ where $C$ is the protein concentration in $\text{g L}^{-1}$, $M$ is the molar mass of the kinetic solute particle, and $B$ is the second virial coefficient. For the denatured state, $B = 3.6 \times 10^{-5}\text{ L mol g}^{-2}$.

Calculate the ratio of the osmotic pressure of the denatured state to the native state ($\frac{\pi_{den}}{\pi_{nat}}$) at a protein concentration of $C = 10\text{ g L}^{-1}$.

An aqueous solution contains $0.10\text{ M}$ of a weak monoacidic base $B$ ($pK_b = 4.52$). The pH of this solution is adjusted by adding a negligible volume of concentrated \ce{H2SO4}. At what pH will this solution become isotonic with a $0.1375\text{ M}$ aqueous solution of glucose at $298\text{ K}$?

(Assume \ce{H2SO4} dissociates completely into \ce{2H+} and \ce{SO4^{2-}}, and the solutions behave ideally. Given: $\log 2 = 0.30$, $\log 3 = 0.48$)

An aqueous solution of the trisaccharide raffinose ($\ce{C18H32O16}$) is placed in an osmotic cell at $300\text{ K}$. In the presence of a dilute acid catalyst, raffinose undergoes a pseudo-first order hydrolysis to yield equimolar amounts of D-glucose, D-fructose, and D-galactose. The initial osmotic pressure of the solution, due solely to raffinose, is $1.50\text{ atm}$. If the pseudo-first order rate constant for the hydrolysis is $k = 2.0 \times 10^{-3}\text{ s}^{-1}$, what is the initial rate of change of the osmotic pressure, $\left( \frac{d\pi}{dt} \right)_{t=0}$, of the solution? (Assume ideal behavior and that the volume of the solution remains constant).

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

$\ce{Ag(s, P_1) | Ag+(aq, 0.1 M, P_1) || Ag+(aq, 0.01 M, P_2) | Ag(s, P_2)}$

The left compartment is maintained at a standard pressure $P_1 = 1\text{ bar}$. To completely halt the flow of electrons (i.e., $E_{\text{cell}} = 0$), a specific hydrostatic pressure $P_2$ is applied to the right compartment.

Given: $RT \ln 10 = 5700\text{ J mol}^{-1}$ Partial molar volume of $\ce{Ag+(aq)}$, $\bar{V}_{\ce{Ag+}} = 67\text{ cm}^3\text{ mol}^{-1}$ Molar volume of $\ce{Ag(s)}$, $V_{\ce{Ag}} = 10\text{ cm}^3\text{ mol}^{-1}$ $1\text{ bar} = 10^5\text{ Pa}$

Assuming the liquids and solids are incompressible, what is the required pressure $P_2$?

Compartments A and B of equal volume are separated by a specialized chiral semipermeable membrane. Initially, Compartment A contains a $0.15\text{ M}$ aqueous solution of fully dissociated racemic 1-phenylethylammonium chloride, denoted as $(\pm)-M^+Cl^-$. Compartment B initially contains pure water. The chiral membrane is perfectly permeable to water, the $(S)-M^+$ enantiomer, and $Cl^-$ ions, but is completely impermeable to the $(R)-M^+$ enantiomer due to stereospecific binding interactions. The system is allowed to reach thermodynamic equilibrium at $300\text{ K}$, with external pressure applied to maintain constant and equal volumes in both compartments. What is the osmotic pressure difference ($\Delta \pi = \pi_A - \pi_B$) required to maintain this equilibrium? (Given: $R = 0.082\text{ L atm K}^{-1}\text{ mol}^{-1}$)

A $0.10 \text{ M}$ aqueous solution of a neutral $\alpha$-amino acid (with $pK_{a1} = 2.00$ and $pK_{a2} = 9.00$) is prepared at $300 \text{ K}$. The osmotic pressure of this initial solution is $2.46 \text{ atm}$, indicating ideal behavior and that the amino acid exists almost entirely as a single-particle zwitterion.

To $1.0 \text{ L}$ of this solution, a small volume of highly concentrated aqueous \ce{HCl} is added to adjust the pH exactly to $2.00$. Assuming the volume of the solution remains unchanged at $1.0 \text{ L}$, what is the new osmotic pressure of the solution at $300 \text{ K}$?

(Given: $R = 0.082 \text{ L atm K}^{-1} \text{mol}^{-1}$, assume all species behave ideally)

A 3d transition metal $M$ is unique in its series for possessing a positive standard reduction potential ($E^\circ_{M^{2+}/M} > 0$). An aqueous solution is prepared by dissolving $0.1\text{ mol}$ of its chloride salt, \ce{MCl2}, in $1\text{ kg}$ of water. To this solution, $0.5\text{ mol}$ of \ce{KCN} is added, resulting in the evolution of a toxic gas that escapes completely, and the formation of a stable, tetrahedral mononuclear complex of $M$. Assuming 100% dissociation of all soluble salts and ideal behavior, what is the depression in freezing point ($\Delta T_f$) of the final solution? ($K_f$ of water = $1.86\text{ K kg mol}^{-1}$)

A $1.0\text{ kg}$ aqueous solution contains a total of $0.10\text{ moles}$ of a mixture of two octahedral \ce{Co(III)} ammine complexes: a yellow complex $P$ and a green complex $Q$. Both complexes are synthesized exclusively from anhydrous \ce{CoCl3} and \ce{NH3}. The boiling point of this solution is $100.1352^\circ\text{C}$ at $1\text{ atm}$.

When excess \ce{AgNO3(aq)} is added to this solution, $W$ grams of \ce{AgCl(s)} precipitate is formed. Assuming $100\%$ ionization of the complexes and ideal behavior, what is the value of $W$?

(Given: $K_b$ for water = $0.52\text{ K kg mol}^{-1}$, Molar mass of \ce{AgCl} = $143.5\text{ g mol}^{-1}$, Boiling point of pure water = $100.0000^\circ\text{C}$)

An amount $W$ of Mohr's salt, $\ce{(NH4)2Fe(SO4)2 \cdot 6H2O}$, is dissolved in $94.6\text{ g}$ of water at $298\text{ K}$. The resulting solution exhibits a boiling point elevation of $1.30\text{ K}$. Assuming complete dissociation of the double salt into its constituent ions, what is the value of $W$?

(Given: $K_b$ for water $= 0.52\text{ K kg mol}^{-1}$, Molar mass of Mohr's salt $= 392\text{ g mol}^{-1}$)

An unknown aromatic amine $X$ (molecular formula \ce{C7H9N}) is dissolved in water to form a $0.10 \text{ M}$ solution at $300 \text{ K}$. The osmotic pressure of this solution is measured to be $2.46 \text{ atm}$ (Use $R = 0.082 \text{ L atm K}^{-1} \text{mol}^{-1}$). When $X$ is treated with \ce{NaNO2 / HCl} at $0-5^\circ\text{C}$ followed by alkaline $\beta$-naphthol, an intense orange-red dye is formed.

A student incorrectly assumes that the nitrogen lone pair in $X$ is fully localized (ignoring its resonance with the aromatic ring) and assigns it a hypothetical base dissociation constant $K_b = \frac{1}{900}$.

Based on this flawed assumption, what is the student's calculated osmotic pressure ($\pi_{\text{calc}}$), and what is the true identity of $X$?

A $2.665 \text{ g}$ sample of a hydrate isomer of $\ce{CrCl3 \cdot 6H2O}$ (Molar mass = $266.5 \text{ g mol}^{-1}$) is dissolved in water and passed through a cation-exchange resin in the $\ce{H+}$ form. The effluent requires $20.0 \text{ mL}$ of $1.0 \text{ M NaOH}$ for complete neutralization. A separate $0.10 \text{ m}$ aqueous solution of this specific isomer exhibits a degree of dissociation $\alpha = 0.85$. Assuming no hydrolysis of the complex cation, what is the freezing point of this $0.10 \text{ m}$ solution? (Given: $K_f$ for water = $1.86 \text{ K kg mol}^{-1}$)

A redox reaction between $\ce{KMnO4}$ and a weak acid reductant $\ce{HA}$ ($K_a = 2.0 \times 10^{-5}$) proceeds in an acidic medium containing $\ce{HCl}$. The proposed mechanism involves the formation of a transient $\ce{Mn(VI)}$ intermediate:

Step 1: $\ce{MnO4- + HA ->[k_1] MnO4^{2-} + HA^{\bullet+}}$ (slow) Step 2: $\ce{MnO4^{2-} + H+ ->[k_2] Mn^{3+} + \text{Products}}$ (fast)

An electrochemical half-cell is prepared at $298 \text{ K}$ containing $0.05 \text{ M } \ce{KMnO4}$, $0.05 \text{ M } \ce{Mn^{2+}}$, $0.1 \text{ M } \ce{HA}$, and $1.0 \times 10^{-3} \text{ M } \ce{HCl}$.

Given: $k_1 = 4.0 \times 10^{-2} \text{ M}^{-1}\text{s}^{-1}$ $k_2 = 1.0 \times 10^3 \text{ M}^{-1}\text{s}^{-1}$ $E^\circ_{\ce{MnO4- / Mn^{2+}}} = 1.51 \text{ V}$ $\frac{2.303 RT}{F} = 0.06 \text{ V}$, $\log 2 = 0.30$.

Assuming the system is at a steady state for $\ce{MnO4^{2-}}$ and the weak acid equilibrium is established instantaneously, determine the steady-state concentration of $\ce{MnO4^{2-}}$ and the instantaneous reduction potential ($E_{red}$) of the $\ce{MnO4- / Mn^{2+}}$ couple.

A gaseous reactant \ce{A} is continuously bubbled through an aqueous solution containing $0.20 \text{ M}$ of reactant \ce{B} at $298 \text{ K}$. The partial pressure of \ce{A(g)} above the solution is maintained constant at $0.60 \text{ bar}$. The Henry's law constant for \ce{A} in water at $298 \text{ K}$ is given as $K_H = 12.0 \text{ bar L mol}^{-1}$.

The overall reaction is: $\ce{2A(aq) + 3B(aq) -> A2B3(aq)}$

The proposed mechanism for this reaction is: Step 1: \ce{2A(aq) <=> A2(aq)} \quad (Fast equilibrium, $K_c = 4.0 \text{ L mol}^{-1}$) Step 2: \ce{A2(aq) + B(aq) -> A2B(aq)} \quad (Slow, $k_1 = 2.5 \times 10^{-2} \text{ L mol}^{-1} \text{ s}^{-1}$) Step 3: \ce{A2B(aq) + 2B(aq) -> A2B3(aq)} \quad (Fast)

Assuming ideal behavior, what is the initial rate of disappearance of \ce{B}?

An advanced electrochemical cell is designed to study the thermodynamics of 'bubble-free' water electrolysis at $298\text{ K}$. Two inert platinum electrodes are immersed in an aqueous buffer solution of $pH = 2.0$.

During steady-state electrolysis, the products exist entirely as dissolved gases before any bubbles can form. The local concentration of dissolved hydrogen at the cathode is maintained at $1.0 \times 10^{-4}\text{ M}$, and dissolved oxygen at the anode is $1.0 \times 10^{-5}\text{ M}$.

The Henry's Law constants for the gases in water at $298\text{ K}$ are $K_H(\ce{H2}) = 1.0 \times 10^{-3}\text{ M bar}^{-1}$ and $K_H(\ce{O2}) = 1.0 \times 10^{-3}\text{ M bar}^{-1}$.

Given the standard electrode potential $E^\circ_{\ce{O2(g)/H2O}} = 1.23\text{ V}$ and $\frac{2.303 RT}{F} = 0.06\text{ V}$.

Calculate the minimum theoretical applied voltage ($V_{min}$) required to sustain this electrolysis.

A closed vessel of total volume $2.0\text{ L}$ contains $1.0\text{ L}$ of an aqueous solution of a reactive substrate \ce{S} at an initial concentration of $10.0\text{ M}$. The remaining $1.0\text{ L}$ is filled with a reactive gas \ce{G} at an initial partial pressure of $2.0\text{ atm}$. The gas \ce{G} dissolves in the aqueous phase and reacts with \ce{S} according to the elementary bimolecular reaction: $\ce{G(aq) + S(aq) -> Products}$ Given:

• Rate constant $k = 0.1\text{ L mol}^{-1}\text{ min}^{-1}$.
• Henry's law constant for \ce{G} in water is $K_H = 5.55 \times 10^3\text{ atm}$ (defined as $P_G = K_H \chi_G$, where $\chi_G$ is the mole fraction of \ce{G} in the solution).
• The dissolution of \ce{G} is rapid and maintains phase equilibrium at all times.
• The temperature is constant, and $RT = 25\text{ L atm mol}^{-1}$.
• The concentration of water is $55.5\text{ M}$ and remains constant.
• The volume of the solution remains exactly $1.0\text{ L}$.

Calculate the time required for the partial pressure of gas \ce{G} in the vessel to drop to $0.25\text{ atm}$. (Assume the concentration of \ce{S} remains effectively constant).

At $298\text{ K}$, the vapor pressure of pure liquid A is $150\text{ Torr}$ and that of pure liquid B is $400\text{ Torr}$. When a small amount of B is dissolved in A to form a dilute solution, the solvent A follows Raoult's Law, whereas the volatile solute B follows Henry's Law with a Henry's Law constant $K_H = 1900\text{ Torr}$. If the vapor phase in equilibrium with the solution contains B at a mole fraction of $0.4$, what is the total vapor pressure of the system?

A saturated aqueous solution of cerium(III) sulfate, \ce{Ce2(SO4)3}, is prepared at $320 \text{ K}$ in $1 \text{ kg}$ of water. The excess un-dissolved solid is rapidly filtered off at $320 \text{ K}$. The filtrate is then cooled to determine its freezing point. Assuming ideal behavior and complete dissociation of the salt, what is the freezing point depression ($\Delta T_f$) of this filtrate?

Given: Solubility of \ce{Ce2(SO4)3} at $300 \text{ K} = 2.0 \times 10^{-2} \text{ mol kg}^{-1}$ $\Delta H^\circ_{\text{sol}} = -138.28 \text{ kJ mol}^{-1}$ (assume independent of temperature) $K_f \text{ for water} = 1.86 \text{ K kg mol}^{-1}$ $R = 8.314 \text{ J K}^{-1} \text{mol}^{-1}$ $\ln 2 = 0.693$

Solid \ce{Ni(OH)2} is added to an aqueous solution containing $0.31\text{ M}$ of the bidentate ligand ethylenediamine (en) at $298\text{ K}$. The system reaches equilibrium. Given the solubility product of \ce{Ni(OH)2} is $K_{sp} = 4.0 \times 10^{-15}$ and the overall formation constant of the complex \ce{[Ni(en)3]^{2+}} is $\beta_3 = 1.0 \times 10^{18}$, calculate the solubility of \ce{Ni(OH)2} in this solution. (Assume no volume change upon addition of solid and ignore the basicity of ethylenediamine).

An $\alpha$-amino acid has $pK_{a1} = 2.0$ and $pK_{a2} = 9.0$. The intrinsic solubility of its zwitterionic form ($HA^\pm$) in pure water at $300\text{ K}$ is $S_0 = 0.05\text{ M}$. An excess of the solid amino acid is added to water, and the solution is titrated with concentrated $\ce{HNO3}$ until the equilibrium pH of the solution reaches $2.0$. Assuming the volume change upon adding $\ce{HNO3}$ is negligible, what is the osmotic pressure of the resulting saturated solution at $300\text{ K}$? (Given: $R = 0.082\text{ L atm K}^{-1}\text{ mol}^{-1}$)

Solid $\ce{A}$ (Molar mass = $150 \text{ g mol}^{-1}$) is added in excess to a solvent at $298 \text{ K}$. The saturation solubility of $\ce{A}$ in this solvent is $3.0 \text{ g L}^{-1}$. To this saturated solution, reactant $\ce{B}$ is added such that its initial concentration is $0.50 \text{ mol dm}^{-3}$. The overall reaction is: $\ce{3A(aq) + 2B(aq) -> P(aq)}$ The proposed mechanism for this reaction is: Step 1: $\ce{2A(aq) + B(aq) ->[k_1] I(aq)}$ (Slow) Step 2: $\ce{I(aq) + A(aq) + B(aq) ->[k_2] P(aq)}$ (Fast) Given the rate constant for the slow step is $k_1 = 4.0 \times 10^{-2} \text{ dm}^6 \text{ mol}^{-2} \text{ s}^{-1}$, what is the initial rate of disappearance of $\ce{B(aq)}$ in $\text{mol dm}^{-3} \text{ s}^{-1}$?

At $340\text{ K}$, $47.4\text{ g}$ of potash alum ($\ce{KAl(SO4)2 \cdot 12H2O}$) is completely dissolved in $177.0\text{ g}$ of anhydrous chloral ($\ce{CCl3CHO}$). Chloral reacts quantitatively with the released water of crystallization to form chloral hydrate, which acts as the sole liquid solvent. Assuming complete dissociation of the inorganic salt and ideal behavior of the resulting solution, what is the relative lowering of vapor pressure of the system? (Given molar masses in $\text{g/mol}$: $\ce{K}=39$, $\ce{Al}=27$, $\ce{S}=32$, $\ce{O}=16$, $\ce{H}=1$, $\ce{C}=12$, $\ce{Cl}=35.5$)

Due to significant ion-pair formation, a $0.05 \text{ M}$ aqueous solution of the coordination complex pentaamminechloridocobalt(III) chloride exhibits incomplete dissociation at $300 \text{ K}$. The solution is placed in a reverse osmosis chamber separated from pure water by a semipermeable membrane. The resistance of this solution, measured in a conductivity cell with a cell constant of $1.2 \text{ cm}^{-1}$, is $200 \Omega$. Given the limiting molar conductivities: $\lambda^\infty(\ce{[Co(NH_3)_5Cl]^{2+}}) = 160 \text{ S cm}^2 \text{ mol}^{-1}$ and $\lambda^\infty(\ce{Cl^-}) = 70 \text{ S cm}^2 \text{ mol}^{-1}$. Assuming the Arrhenius theory of electrolytic dissociation is valid for the apparent degree of dissociation, what is the minimum external pressure that must be applied to the solution to initiate reverse osmosis? (Use $R = 0.082 \text{ L atm K}^{-1} \text{ mol}^{-1}$)

In an industrial reverse osmosis (RO) plant, an aqueous solution of a reactive pollutant \ce{A} (bulk concentration $C_b = 10 \text{ mol m}^{-3}$) is processed at $298 \text{ K}$. The RO membrane perfectly rejects \ce{A} ($R = 100\%$). Due to concentration polarization, a boundary layer exists at the membrane surface with a mass transfer coefficient $k_m = 1.0 \times 10^{-5} \text{ m s}^{-1}$. The steady-state water flux through the membrane is $J_w = 2.0 \times 10^{-5} \text{ m s}^{-1}$.

At the membrane surface, \ce{A} undergoes a first-order degradation reaction $\ce{A -> B}$ with a surface rate constant $k_s = 1.0 \times 10^{-5} \text{ m s}^{-1}$. This reaction occurs exclusively at the membrane surface.

Assuming steady-state film theory, what is the rate of degradation of \ce{A} per unit membrane area ($r''$)?

(Note: $e$ is the base of the natural logarithm)

A $40 \text{ g L}^{-1}$ aqueous solution of a monomeric globular peptide ($M = 5,000 \text{ g mol}^{-1}$) is subjected to purification via Reverse Osmosis (RO) at $298 \text{ K}$. In its native state, the peptide behaves as an ideal solute. However, under the extreme hydraulic pressure at the RO membrane surface, the peptide undergoes localized denaturation. While the rigid planarity of the peptide bonds ($\omega \approx 180^\circ$) is preserved, the unfolding exposes hydrophobic residues, causing the peptide to quantitatively aggregate into stable pentamers. Furthermore, the extended conformation of the denatured chains interacts non-ideally with the solvent, characterized by a second virial coefficient $A_2 = 4.9 \times 10^{-5} \text{ L mol g}^{-2}$. Given $RT = 25 \text{ L atm mol}^{-1}$, calculate the effective osmotic pressure ($\pi$) exerted by the protein at the membrane surface.

An aqueous solution is prepared by dissolving $W$ grams of a weak monobasic acid \ce{HA} (molar mass $= 120 \text{ g mol}^{-1}$) in water to make $500 \text{ mL}$ of solution at $298 \text{ K}$. This solution is found to be isotonic with a $0.06 \text{ M}$ aqueous solution of glucose at the same temperature. Given the acid dissociation constant $K_a$ of \ce{HA} is $2.5 \times 10^{-3}$ at $298 \text{ K}$, determine the value of $W$.

A globular metalloprotein exists as a tetramer ($M_w = 60,000 \text{ g mol}^{-1}$). In its native folded state, the tetramer completely sequesters $4$ molecules of a non-electrolyte cofactor and $2$ formula units of $\ce{Fe2(SO4)3}$ deep within its hydrophobic core. These sequestered species are tightly bound and do not contribute to the initial osmotic pressure.

An aqueous solution is prepared by dissolving $15.0 \text{ g}$ of this protein in $500 \text{ mL}$ of water at $300 \text{ K}$. Under severe osmotic stress, the protein completely denatures, dissociating into its $4$ monomeric chains and releasing all sequestered species into the aqueous medium. The released $\ce{Fe2(SO4)3}$ undergoes complete ionization.

Assuming the volume remains constant, what is the osmotic pressure of the solution after complete denaturation? (Given: $R = 0.082 \text{ L atm K}^{-1} \text{ mol}^{-1}$)