Equilibrium - Class 11 Part 1

A $1.0\text{ L}$ aqueous solution at $298\text{ K}$ contains $0.10\text{ mol}$ of a salt $BHCl$, where $B$ is a weak monoacidic base with $K_b = 10^{-9}$. To this solution, $0.01\text{ mol}$ of $\ce{KMnO4}$ and $0.025\text{ mol}$ of $\ce{H2O2}$ are added. The ensuing redox reaction proceeds to completion, reducing $\ce{MnO4-}$ to $\ce{Mn^{2+}}$ and oxidizing $\ce{H2O2}$ to $\ce{O2(g)}$. Assuming no change in volume and ignoring the hydrolysis of $\ce{Mn^{2+}}$, what is the final pH of the solution? (Given: $\log 3 = 0.48$, $\log 7 = 0.85$, $K_w = 10^{-14}$)

An aqueous solution of a weak electrolyte \ce{X} (molar mass = $120 \text{ g mol}^{-1}$) has an initial concentration of $2.5 \text{ M}$ and a density of $1.10 \text{ g mL}^{-1}$ at $298 \text{ K}$. The electrolyte dissociates reversibly as: $\ce{X(aq) <=> Y(aq) + Z(aq)}$ The forward reaction rate depends only on $[\ce{X}]$, and the backward reaction rate depends symmetrically on $[\ce{Y}]$ and $[\ce{Z}]$. The rate constants are given as $k_f = 0.64 \text{ L mol}^{-1} \text{s}^{-1}$ and $k_b = 0.01 \text{ L}^3 \text{mol}^{-3} \text{s}^{-1}$.

Assuming \ce{Y} and \ce{Z} do not undergo any further reactions and the density of the solution remains constant, calculate the depression in freezing point ($\Delta T_f$) of the solution at equilibrium. (Given: $K_f$ for water = $1.86 \text{ K kg mol}^{-1}$)

The formation of the Wheland intermediate ($\sigma$-complex) during the electrophilic aromatic substitution of 1,3,5-trimethoxybenzene (TMB) with tropylium tetrafluoroborate ($\ce{C7H7+ BF4-}$) is an equilibrium-controlled process:

$\ce{TMB + C7H7+ <=> \sigma-complex+}$

In 1,2-dichloroethane (DCE, a non-polar solvent), the equilibrium constant is $K_1 = 0.1$ at 298 K. When the solvent is changed to acetonitrile (MeCN, a polar aprotic solvent), the $\sigma$-complex and the transition state ($TS^\ddagger$) are significantly stabilized by solvation. The transfer free energies from DCE to MeCN are $\Delta G_{tr}(\sigma^+) = -22.8 \text{ kJ/mol}$ and $\Delta G_{tr}(TS^\ddagger) = -11.4 \text{ kJ/mol}$.

Assuming the transfer free energies of the neutral TMB and the large, diffuse $\ce{C7H7+}$ cation are negligible ($\approx 0$), calculate the equilibrium constant $K_2$ and the ratio of the forward rate constants ($k_{f2}/k_{f1}$) in MeCN at 298 K.

(Given: $2.303 RT = 5.7 \text{ kJ/mol}$ at 298 K)

Consider the reversible liquid-phase isomerization reaction: $\ce{A(l) \rightleftharpoons B(l)}$.

At $350 \text{ K}$, the liquid mixture of A and B exhibits non-ideal behavior. The system is simultaneously in vapor-liquid equilibrium and chemical equilibrium.

At this temperature, the mixture forms an azeotrope (where the vapor phase mole fractions equal the liquid phase mole fractions) at a liquid phase mole fraction of A, $x_A = 0.40$.

Given that the vapor pressures of pure liquid A and pure liquid B at $350 \text{ K}$ are $250 \text{ torr}$ and $150 \text{ torr}$, respectively, what is the standard Gibbs free energy change ($\Delta G^\circ$) for the isomerization reaction at $350 \text{ K}$?

(Assume the vapor phase behaves ideally and the standard state for each component is the pure liquid at $350 \text{ K}$).

Consider the gas-phase exothermic reaction $X(g) + Y(g) \rightleftharpoons Z(g)$ catalyzed by a solid metal surface at $400\text{ K}$. The reaction proceeds via a Langmuir-Hinshelwood mechanism where $X$ and $Y$ compete for the same active sites. The adsorption of $X$ is highly favored such that its surface coverage $\theta_X \approx 1$, while $Y$ adsorbs weakly.

If the volume of the reaction vessel is suddenly halved at constant temperature before any $Z$ is formed, which of the following correctly describes the changes to the initial forward reaction rate ($r_f$), the thermodynamic equilibrium constant ($K_p$), and the final equilibrium moles of $Z$ ($n_Z$)?

Excess \ce{AgNO2(s)} is added to a large volume of a buffer solution at pH = 3.0. The system is maintained in a closed vessel with a constant partial pressure of \ce{NO(g)} at 1 bar. Given the following standard reduction potentials and constants at 298 K:

$E^\circ(\ce{NO3^- / HNO2}) = 0.94\text{ V}$ $E^\circ(\ce{HNO2 / NO}) = 1.12\text{ V}$ $K_{sp}(\ce{AgNO2}) = 3.0 \times 10^{-3}$ $K_a(\ce{HNO2}) = 1.0 \times 10^{-3}$ $\frac{2.303 RT}{F} = 0.06\text{ V}$

Calculate the equilibrium concentration of \ce{Ag^+} ions in the solution. (Assume activity coefficients are unity and no other complexation occurs).

A crystalline native protein, $P_{N(s)}$, is in equilibrium with its aqueous solution at $298\text{ K}$. At its isoelectric point ($\text{pH} = 6.0$), its intrinsic solubility is $S_0 = 1.0 \times 10^{-4}\text{ mol/L}$. In its native folded state, the dominant ionizable groups governing solubility have $pK_{a1} = 4.0$ (protonation of basic residues) and $pK_{a2} = 8.0$ (deprotonation of acidic residues).

When the solution is acidified to $\text{pH} = 2.0$ using $\ce{HCl}$, the protein undergoes complete and irreversible acid denaturation, precipitating as a new solid phase, $P_{D(s)}$. For this denatured state, the exposure of the hydrophobic core reduces its intrinsic solubility to $S_{0,D} = 1.0 \times 10^{-7}\text{ mol/L}$. Furthermore, the loss of the native tertiary structure and salt bridges shifts the $pK_{a1}$ of the basic residues to $5.0$.

Assuming the activity coefficients are unity, what is the actual equilibrium solubility of the protein at $\text{pH} = 2.0$?

Gaseous bromoethane ($\ce{CH3CH2Br}$) undergoes a heterogeneous Swarts exchange with solid silver fluoride ($\ce{AgF}$) at $298\text{ K}$ in a closed $2.0\text{ L}$ vessel:

$\ce{CH3CH2Br(g) + AgF(s) <=> CH3CH2F(g) + AgBr(s)}$

The reaction occurs entirely on the solid surface. The intrinsic rate laws for the forward and reverse reactions are $r_f = k_f [\ce{CH3CH2Br}][\ce{AgF}]_{eff}$ and $r_b = k_b [\ce{CH3CH2F}][\ce{AgBr}]_{eff}$. The vessel contains a mixed powder bed such that the effective surface site concentrations remain constant at $[\ce{AgF}]_{eff} = 3.0\text{ mol L}^{-1}$ and $[\ce{AgBr}]_{eff} = 1.0\text{ mol L}^{-1}$.

Given the intrinsic rate constants $k_f = 2.0 \times 10^{-2}\text{ L mol}^{-1}\text{ s}^{-1}$ and $k_b = 1.0 \times 10^{-2}\text{ L mol}^{-1}\text{ s}^{-1}$, if $1.4\text{ moles}$ of $\ce{CH3CH2Br(g)}$ are initially introduced into the vessel, determine the true thermodynamic equilibrium constant ($K_c$) and the equilibrium concentration of $\ce{CH3CH2F(g)}$.

Solid ammonium carbamate, $\ce{NH2COONH4(s)}$, undergoes reversible dissociation into ammonia and carbon dioxide gases. When excess $\ce{NH2COONH4(s)}$ is introduced into an evacuated rigid vessel at $300\text{ K}$, the equilibrium total pressure is found to be $0.30\text{ atm}$.

In a separate experiment, $1.00\text{ mole}$ of $\ce{NH2COONH4(s)}$ is placed in an evacuated $24.63\text{ L}$ vessel at $300\text{ K}$ and allowed to reach equilibrium.

Given $R = 0.0821\text{ L atm K}^{-1}\text{ mol}^{-1}$ and assuming ideal gas behavior, what is the mass of the unreacted $\ce{NH2COONH4(s)}$ at equilibrium in the second experiment? (Molar mass of $\ce{NH2COONH4} = 78\text{ g mol}^{-1}$)

The standard Gibbs free energy change for the dissolution and dissociation of a gaseous weak acid $\ce{HA(g)}$ in water at $320\text{ K}$ is $\Delta G^\circ = 12.25\text{ kJ mol}^{-1}$. $\ce{HA(g) <=> H+(aq) + A-(aq)}$ The standard states are defined as $P^\circ = 1\text{ bar}$ for the gas and $m^\circ = 1\text{ mol kg}^{-1}$ for the aqueous species. At $320\text{ K}$, the density of pure water is $0.95\text{ kg L}^{-1}$.

If the equilibrium pressure of $\ce{HA(g)}$ is maintained at $1\text{ atm}$ ($1\text{ atm} = 1.013\text{ bar}$), what is the equilibrium concentration of $\ce{H+(aq)}$ in molarity ($\text{mol L}^{-1}$)?

Assume ideal behavior for the gas and the solution, and that the solution is sufficiently dilute such that its density equals that of pure water. (Given: $R = 8.314\text{ J K}^{-1}\text{ mol}^{-1}$, $e^{-4.604} = 0.01$)

The gas-phase reaction \ce{A(g) + 2B(g) <=> C(g)} has an equilibrium constant $K_c = 2500 \text{ M}^{-2}$ at $600 \text{ K}$. The activation energy for the forward uncatalyzed reaction is $80 \text{ kJ mol}^{-1}$.

To improve the rate of product formation, a solid catalyst is introduced, which alters the reaction pathway to the following mechanism: Step 1: \ce{A(g) + B(g) <=> I(g)} (Fast equilibrium) Step 2: \ce{I(g) + B(g) <=> C(g)} (Slow, rate-determining step)

The catalyst lowers the activation energy of the forward reaction by $11.5 \text{ kJ mol}^{-1}$. Assuming ideal gas behavior and that the concentration of the intermediate \ce{I(g)} is negligible at macroscopic equilibrium, what is the value of the equilibrium constant $K_p$ for the catalyzed reaction at $600 \text{ K}$?

(Given: $R \approx \frac{1}{12} \text{ L bar K}^{-1} \text{ mol}^{-1}$ for pressure calculations, $R = 8.314 \text{ J K}^{-1} \text{ mol}^{-1}$ for energy calculations, and $e^{2.3} \approx 10$)

Consider the formation of tetracarbonylnickel(0) in the Mond process at temperature $T$:

$\ce{Ni(s) + 4CO(g) <=> Ni(CO)4(g)}$

Due to strong intramolecular $\pi$-backbonding (synergic effect), the $\ce{Ni(CO)4}$ molecule possesses a highly polarizable electron cloud, leading to significant intermolecular Van der Waals attractions. Assume $\ce{CO(g)}$ behaves as an ideal gas. For $\ce{Ni(CO)4(g)}$, the repulsive volume parameter is negligible ($b \approx 0$), but the attractive parameter '$a$' is significant.

Let the partial pressure of $\ce{Ni(CO)4(g)}$ at equilibrium be $p$. The compressibility factor of $\ce{Ni(CO)4(g)}$ is $Z = 1 - x$, and its fugacity coefficient is $\phi = e^{-x}$, where $x = \frac{ap}{(RT)^2}$.

If $K_f$ is the true thermodynamic equilibrium constant defined in terms of fugacity, which of the following correctly expresses the apparent concentration-based equilibrium constant $K_c$?

Consider the following gas-phase equilibrium established in a rigid, closed vessel of volume $V$ at $500\text{ K}$: $\ce{PCl5(g) \rightleftharpoons PCl3(g) + Cl2(g)}$ Argon gas is isothermally introduced into the vessel until the total pressure of the system is exactly doubled. Which of the following statements correctly describes the effect on the equilibrium position, the value of the equilibrium constant $K_p$, and the ratio $K_p/K_c$?

Consider the elementary reversible dimerization of nitrogen dioxide in a cylinder fitted with a frictionless, movable piston at $298\text{ K}$ (assume ideal gas behavior): $2\ce{NO2(g)} \rightleftharpoons \ce{N2O4(g)}$ The system is initially at chemical equilibrium with forward rate $r_f$ and backward rate $r_b$. An inert gas ($\ce{He}$) is injected isothermally and isobarically, increasing the total volume of the system by a factor of $\alpha$ ($\alpha > 1$). Let $r_f'$ and $r_b'$ be the instantaneous rates immediately after the injection (prior to any chemical relaxation). Which of the following correctly represents the ratio $\frac{r_f'}{r_b'}$ and the direction of the subsequent thermodynamic shift?

A $1.0\text{ L}$ aqueous solution is prepared by dissolving $12.5\text{ mmol}$ of a diprotic acid $\ce{H2A}$ ($K_{a1} = 1.0 \times 10^{-2}$, $K_{a2} = 1.0 \times 10^{-3}$) and $30.0\text{ mmol}$ of its monosodium salt $\ce{NaHA}$ at $298\text{ K}$. Assuming ideal behavior (activity coefficients = 1), calculate the exact equilibrium concentration of $\ce{H+}$ ions in the solution.

A half-cell is constructed at $298 \text{ K}$ by immersing a platinum electrode in a $1.0 \text{ L}$ aqueous solution containing $0.01 \text{ M}$ \ce{K2Cr2O7}, $0.1 \text{ M}$ \ce{CrCl3}, and $0.1 \text{ M}$ of a weak monobasic acid \ce{HA} ($K_a = 10^{-5}$). This half-cell is connected to a Standard Hydrogen Electrode (SHE) to form a galvanic cell.

Assuming the degree of dissociation of \ce{HA} is small ($\alpha \ll 1$) and neglecting any volume changes or complexation, what is the initial electromotive force (EMF) of the cell?

(Given: $E^\circ_{\ce{Cr2O7^{2-}/Cr^{3+}}} = 1.33 \text{ V}$, $\frac{2.303 RT}{F} = 0.06 \text{ V}$ at $298 \text{ K}$)

An aqueous solution of a weak monoprotic acid \ce{HA} (molar mass = $350 \text{ g mol}^{-1}$) has a molality of $2.0 \text{ m}$ and a density of $1.02 \text{ g mL}^{-1}$ at $298 \text{ K}$. The acid dissociation constant $K_a$ of \ce{HA (aq)} at this temperature is $0.1$. Given that the cryoscopic constant of water is $K_f = 2.0 \text{ K kg mol}^{-1}$, what is the freezing point depression ($\Delta T_f$) of the solution? (Assume the standard definition of $K_a$ based on molarity, and that activity coefficients are unity.)

A chemist investigates the kinetics of Schiff base formation between aniline ($\ce{PhNH2}$) and benzaldehyde ($\ce{PhCHO}$) at $298 \text{ K}$. To maintain a constant acidic environment, the reaction is performed in an aqueous solution prepared by dissolving $0.10 \text{ mol}$ of anilinium chloride ($\ce{PhNH3Cl}$) and $3.0 \times 10^{-3} \text{ mol}$ of $\ce{HCl}$ in $1.0 \text{ L}$ of water.

Given the acid dissociation constant of the anilinium ion is $K_a = 4.0 \times 10^{-5}$ at $298 \text{ K}$.

Calculate the true degree of dissociation ($\alpha$) of the anilinium ion (assume $1-\alpha \approx 1$ for the undissociated salt concentration), and identify the primary kinetic and electronic consequence of this specific pH environment on the aniline molecule.

Anilinium hydrogen sulfate, \ce{[C6H5NH3+][HSO4-]}, is dissolved in water at $363 \text{ K}$ to form a $0.10 \text{ M}$ solution. At this temperature, the ionic product of water is $K_w = 1.0 \times 10^{-12}$. The base dissociation constant of aniline is $K_b = 1.0 \times 10^{-10}$ and the acid dissociation constant of the hydrogen sulfate ion is $K_a = 1.0 \times 10^{-2}$. Assuming ideal behavior and neglecting the autoionization of water, what is the exact hydrogen ion concentration $[\ce{H+}]$ of the solution?

The synthesis of a Schiff base via the condensation of benzaldehyde and a primary amine $\ce{RNH2}$ ($K_b = 1.0 \times 10^{-9}$) is highly pH-dependent. The reaction is conducted at $298\text{ K}$ in an aqueous solution initially containing only a weak acid $\ce{HA}$ ($K_a = 1.0 \times 10^{-5}$) at a concentration of $1.0 \times 10^{-5}\text{ M}$. A trace amount of the amine and benzaldehyde is added such that their presence does not perturb the bulk $[\ce{H+}]$ established by $\ce{HA}$.

The concentration of the active nucleophile depends strictly on the fraction of the amine in its unprotonated state. Calculate the true percentage of the amine available for nucleophilic attack, and the percentage a student would calculate if they erroneously applied the standard weak electrolyte approximation ($1 - \alpha \approx 1$) to determine the pH of the $\ce{HA}$ solution. (Given: $\sqrt{5} = 2.236$, $K_w = 1.0 \times 10^{-14}$)

A localized corrosion cell forms on a structural iron beam exposed to an aerated aqueous solution of $0.1\text{ M } AlCl_3$ at $298\text{ K}$. The anodic process involves the oxidation of $Fe(s)$ to $Fe^{2+}(aq)$, while the cathodic process is the reduction of $O_2(g)$ to $OH^-(aq)$. Given the standard reduction potentials: $E^\circ_{Fe^{2+}/Fe} = -0.44\text{ V}$ and $E^\circ_{O_2/OH^-} = +0.40\text{ V}$. The first hydrolysis constant ($K_h$) of $[Al(H_2O)_6]^{3+}$ is $1.0 \times 10^{-5}$. Assume the partial pressure of $O_2(g)$ is $1\text{ bar}$, the local concentration of $Fe^{2+}$ is $1.0 \times 10^{-2}\text{ M}$, and $K_w = 1.0 \times 10^{-14}$. Calculate the operating cell potential ($E_{cell}$) of this corrosion cell. (Take $\frac{2.303 RT}{F} = 0.06\text{ V}$ at $298\text{ K}$).

Solid \ce{Zn(OH)2} is suspended in an aqueous solution maintained at a constant pH of 11.0 and a constant free \ce{NH3} concentration of $0.2 \text{ M}$ at $298 \text{ K}$. Given the solubility product constant $K_{sp}$ of \ce{Zn(OH)2} is $1.0 \times 10^{-16}$ and the overall formation constant $\beta_4$ for the complex ion \ce{[Zn(NH3)4]^{2+}} is $1.0 \times 10^9$, calculate the total molar solubility of \ce{Zn(OH)2} in this solution. (Assume no other complex ions, such as hydroxo complexes, are formed).

Azo coupling is highly sensitive to the pH of the reaction medium due to the competitive ionization of the coupling partners. Consider an equimolar aqueous mixture of a phenol derivative $\textbf{X}$ ($pK_a = 10.0$) and an aniline derivative $\textbf{Y}$ ($pK_b = 9.0$). A small amount of a diazonium salt, $\ce{ArN2+ Cl-}$, is added to the mixture at $298\text{ K}$.

The intrinsic second-order rate constant for the electrophilic attack of $\ce{ArN2+}$ on the phenoxide ion of $\textbf{X}$ is $k_X = 1.0 \times 10^9 \text{ M}^{-1}\text{s}^{-1}$. The intrinsic second-order rate constant for the attack on the neutral free amine form of $\textbf{Y}$ is $k_Y = 2.0 \times 10^4 \text{ M}^{-1}\text{s}^{-1}$.

Assuming that coupling with neutral $\textbf{X}$ and protonated $\textbf{Y}$ are negligible, at what pH will the initial rate of formation of the azo product from $\textbf{X}$ be exactly equal to that from $\textbf{Y}$?

(Given: $K_w = 10^{-14}$ at $298\text{ K}$, $\log 2 = 0.30$, $\log 5 = 0.70$)

An extremely dilute aqueous solution of \ce{HCl} is prepared at $298\text{ K}$ with a concentration of $1.0 \times 10^{-7}\text{ M}$. The limiting molar conductivities at $298\text{ K}$ are $\lambda^\circ_{\ce{H^+}} = 350\text{ S cm}^2\text{ mol}^{-1}$, $\lambda^\circ_{\ce{OH^-}} = 200\text{ S cm}^2\text{ mol}^{-1}$, and $\lambda^\circ_{\ce{Cl^-}} = 75\text{ S cm}^2\text{ mol}^{-1}$. The ionic product of water is $K_w = 1.0 \times 10^{-14}$. Calculate the true specific conductance ($\kappa_{sol}$) of this solution, and the apparent molar conductivity of \ce{HCl} ($\Lambda_{m, app}$) if it is calculated using the standard formula $\Lambda_{m, app} = \frac{1000 \times (\kappa_{sol} - \kappa_{\ce{H_2O}})}{C_{\ce{HCl}}}$, where $C_{\ce{HCl}}$ is the molarity of \ce{HCl} and $\kappa_{\ce{H_2O}}$ is the specific conductance of pure water. (Use $\sqrt{5} \approx 2.236$)

Aniline ($pK_a$ of $\ce{PhNH3+} = 4.6$) reacts with benzaldehyde to form a Schiff base. The reaction is carried out in an aqueous solution of $10^{-8}\text{ M}$ $\ce{HCl}$ at $298\text{ K}$. Calculate the true pH of the reaction medium and the ratio of unprotonated to protonated aniline ($\frac{[\ce{PhNH2}]}{[\ce{PhNH3+}]}$) present at equilibrium.

(Given: $K_w = 10^{-14}$, $\sqrt{401} \approx 20.02$, $\log_{10} 1.05 \approx 0.02$, $10^{0.38} \approx 2.4$, $10^{0.4} \approx 2.5$)

A $1.0\text{ L}$ aqueous solution of a weak acid salt $\ce{HACl}$ (initial concentration $= \sqrt{2} \times 10^{-7}\text{ M}$) is kept in a closed vessel of total volume $2.0\text{ L}$ at $300\text{ K}$. The cation $\ce{HA+}$ dissociates according to the following equilibrium: $\ce{HA+(aq) <=> H+(aq) + A(g)}$ At equilibrium, the pH of the solution is found to be $7 - \frac{1}{2} \log 2$. Assuming the gas $\ce{A}$ is completely insoluble in water and behaves ideally, calculate the thermodynamic equilibrium constant $K$ for the reaction. (Given: $K_w = 1.0 \times 10^{-14}$ at $300\text{ K}$, $R = 0.08\text{ L atm K}^{-1}\text{ mol}^{-1}$. Standard states: $1\text{ M}$ for aqueous species, $1\text{ atm}$ for gases.)

What is the minimum initial concentration of $\ce{NH_3}$ required to completely dissolve $0.05\text{ moles}$ of $\ce{AgCl(s)}$ in $1.0\text{ L}$ of a $0.15\text{ M } \ce{NaCl}$ aqueous solution at $298\text{ K}$?

Given: $K_{sp}(\ce{AgCl}) = 1.0 \times 10^{-10}$ Formation constant ($\beta_2$) for $\ce{[Ag(NH_3)_2]^+} = 2.5 \times 10^7$ Assume no change in volume upon the addition of solid $\ce{AgCl}$ and $\ce{NH_3}$.

Solid $\ce{CaHPO4}$ is equilibrated with a buffer solution at $298\text{ K}$ maintained at $\text{pH} = 5.0$. The solution initially contains $0.10\text{ M } \ce{CaCl2}$. Given the solubility product of $\ce{CaHPO4}$ is $K_{sp} = 1.0 \times 10^{-7}$ and the successive acid dissociation constants for $\ce{H3PO4}$ are $K_{a1} = 10^{-3}$, $K_{a2} = 10^{-7}$, and $K_{a3} = 10^{-12}$. Assuming infinite buffer capacity and ideal behavior, what is the molar solubility of $\ce{CaHPO4}$ in this solution?

The thermodynamic solubility product ($K_{sp}^{\circ}$) of \ce{Mg(OH)2 (s)} at $298\text{ K}$ is $1.25 \times 10^{-11}$. An electrochemical cell is constructed to measure the solubility of \ce{Mg(OH)2} in a $0.10\text{ M}$ \ce{NaOH (aq)} solution. During the measurement, a liquid junction potential ($E_{lj}$) of $+1.5\text{ mV}$ is observed at the salt bridge interface. If the mean ionic activity coefficient ($\gamma_{\pm}$) of \ce{Mg(OH)2} in this highly ionic medium is $0.50$, what is the true molar solubility of \ce{Mg(OH)2} in the solution? (Assume the contribution of dissolved \ce{Mg(OH)2} to the ionic strength and \ce{OH-} concentration is negligible).

A galvanic cell at 298 K is constructed with a Standard Hydrogen Electrode (SHE) as the anode and a silver electrode as the cathode. The cathode compartment contains a saturated aqueous solution of a sparingly soluble silver salt, $\ce{AgA}$ ($K_{sp} = 10^{-14}$), and is maintained at a constant $pH = 4.0$ using a buffer. The conjugate acid $\ce{HA}$ is a weak acid with $K_a = 10^{-8}$.

Solid $\ce{NaA}$ is then dissolved in the cathode compartment such that the total analytical concentration of the added $\ce{A}$ species is $0.1 \text{ M}$. Assuming no change in the volume of the solution and that the buffer perfectly maintains $pH = 4.0$, calculate the shift in the electromotive force ($\Delta E_{cell}$) of the cell.

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

To study redox precipitation constraints, solid silver hypochlorite, $\ce{AgClO(s)}$, is added to an aqueous solution. The hypochlorite ion undergoes disproportionation to chlorate and chloride ions, superseding simple solubility equilibrium. \ Given the following standard reduction potentials and solubility product constants at $298\text{ K}$: \ $E^\circ_{\ce{ClO^-/Cl^-}} = 0.89\text{ V}$ \ $E^\circ_{\ce{ClO_3^-/ClO^-}} = 0.62\text{ V}$ \ $K_{sp}(\ce{AgClO}) = 10^{-4}$ \ $K_{sp}(\ce{AgCl}) = 10^{-10}$ \ Assume $\frac{2.303RT}{F} = 0.06\text{ V}$. \ Determine the equilibrium constant ($K_{eq}$) for the following overall transformation: \ $3\ce{AgClO(s)} \rightleftharpoons 2\ce{AgCl(s)} + \ce{Ag+(aq)} + \ce{ClO_3^{-}(aq)}$

A $100\text{ mL}$ aqueous buffer solution contains $0.16\text{ M}$ of a weak acid $\ce{HA}$ ($pK_a = 3.0$) and $0.04\text{ M}$ of its sodium salt $\ce{NaA}$. To this solution, $4.0 \times 10^{-3}\text{ moles}$ of $\ce{KMnO4}$ and $5.0 \times 10^{-3}\text{ moles}$ of $\ce{H2O2}$ are added at $298\text{ K}$. The $\ce{H2O2}$ completely reacts to reduce a portion of the $\ce{MnO4-}$ to $\ce{Mn^{2+}}$. Assuming the volume of the solution remains constant at $100\text{ mL}$ and no other side reactions (e.g., formation of $\ce{MnO2}$) occur, what is the reduction potential of the $\ce{MnO4- / Mn^{2+}}$ half-cell at the final equilibrium state?

(Given: $E^\circ_{\ce{MnO4- / Mn^{2+}}} = 1.510\text{ V}$, $\frac{2.303 RT}{F} = 0.059\text{ V}$, $\log 2 = 0.30$)

An analytical chemist prepares a buffer solution at $50^\circ\text{C}$ by mixing a weak base $B$ ($pK_b = 5.0$ at $50^\circ\text{C}$) and its conjugate acid chloride $BHCl$ such that the solution exhibits maximum buffer capacity. Given that the ionic product of water at $50^\circ\text{C}$ is $K_w = 1.0 \times 10^{-13}$, determine the exact $pH$ of this buffer solution and its relative basicity, defined as $\Delta pH = pH_{\text{buffer}} - pH_{\text{neutral}}$, at this temperature.

A buffer solution is to be prepared using acetic acid (\ce{CH3COOH}) and sodium acetate (\ce{CH3COONa}) in a $60\%$ (v/v) ethanol-water mixed solvent at $298\text{ K}$. The aqueous $pK_a$ of acetic acid is $4.76$. The standard free energy of transfer ($\Delta G^\circ_{tr}$) for the dissociation reaction $\ce{CH3COOH \rightleftharpoons CH3COO- + H+}$ from pure water to the mixed solvent is $+5.70\text{ kJ mol}^{-1}$ due to the lower dielectric constant of the medium. (Given: $2.303 RT = 5.70\text{ kJ mol}^{-1}$ at $298\text{ K}$).

Which of the following correctly identifies (i) the pH at which this buffer exhibits maximum buffer capacity in the mixed solvent, and (ii) the required molar ratio of $\frac{[\ce{CH3COONa}]}{[\ce{CH3COOH}]}$ to prepare a buffer of $\text{pH} = 4.76$ in this specific medium?

A $1.0 \text{ L}$ aqueous solution at $298 \text{ K}$ contains $0.01 \text{ mol}$ of $\ce{FeSO4}$ and $0.1 \text{ mol}$ of $\ce{KCN}$. The solution is buffered at $\text{pH} = 2.2$. Given the acid dissociation constant of $\ce{HCN}$ is $K_a = 10^{-9.2}$ and the overall formation constant of $\ce{[Fe(CN)6]^{4-}}$ is $\beta_6 = 10^{35}$. Assuming no volume change and no precipitation of any solid species, what is the spin-only magnetic moment (in Bohr Magnetons) and the Crystal Field Stabilization Energy (CFSE) of the predominant iron complex in the solution? (Ignore pairing energy, $\Delta_o$ is the octahedral splitting energy for the predominant complex).

Consider a sparingly soluble salt \ce{MA} with a solubility product $K_{sp} = 1.0 \times 10^{-6}$ at $298 \text{ K}$. The anion \ce{A^-} is the conjugate base of a weak acid \ce{HA} with an ionization constant $K_a = 1.0 \times 10^{-9}$. Calculate the solubility of \ce{MA} in a $0.10 \text{ M}$ aqueous solution of \ce{MCl}.

(Assume \ce{MCl} is completely dissociated, no other complexation or precipitation reactions occur, and ignore activity coefficients.)

An electrochemical cell is constructed at $298 \text{ K}$:

$\ce{Pt(s) | H2(g, 1 bar) | 0.1 M Na2HPO4(aq) || 0.1 M NaH2PO4(aq) | H2(g, 1 bar) | Pt(s)}$

Given the acid dissociation constants for phosphoric acid ($\ce{H3PO4}$) at $298 \text{ K}$: $pK_{a1} = 2.2$, $pK_{a2} = 7.2$, $pK_{a3} = 12.4$.

Assume $\frac{2.303 RT}{F} = 0.06 \text{ V}$.

Calculate the cell potential ($E_{cell}$).

During the titration of a weak monoprotic acid $\ce{HA}$ with a strong base $\ce{MOH}$ at $298\text{ K}$, the equivalence point is reached. The standard textbook approximation for the pH at the equivalence point is given by $pH_{ideal} = \frac{1}{2}(pK_w + pK_a + \log C)$, where $C$ is the concentration of the salt $\ce{MA}$, and $pK_w$, $pK_a$ are thermodynamic constants.

However, in real solutions, the finite ionic strength ($I$) affects the activity coefficients ($\gamma$) of the species. Assuming the activity coefficient of the neutral un-ionized acid $\gamma_{HA} \approx 1$, and for the univalent ions $\gamma_{A^-} < 1$ and $\gamma_{OH^-} < 1$.

Which of the following expressions correctly relates the true thermodynamic pH ($pH_{true} = -\log a_{H^+}$) to $pH_{ideal}$, and what is the physical consequence on the equivalence point?

A $1.0\text{ L}$ aqueous buffer solution at $298\text{ K}$ contains $0.20\text{ mol}$ of a weak base $B$ and $0.12\text{ mol}$ of its conjugate acid $BH^+$ ($pK_a = 11.0$). To this, $0.50\text{ mol}$ of benzaldehyde ($\ce{PhCHO}$) is added. The base-catalyzed Cannizzaro reaction proceeds, consuming $\ce{OH^-}$ ions. Assume the reaction effectively ceases due to kinetic inhibition exactly when the weak base $B$ is completely exhausted. Given $pK_a(\ce{PhCOOH}) = 4.0$ and $K_w = 10^{-14}$, calculate the final pH of the solution at this point. (Assume no volume change and $\log 2 = 0.30$)

Consider the esterification of a weak organic acid \ce{RCOOH} with an alcohol \ce{R'OH} in an aqueous buffer solution at $373\text{ K}$: $\ce{RCOOH(aq) + R'OH(aq) <=> RCOOR'(aq) + H2O(l)}$ The true thermodynamic equilibrium constant for this reaction, expressed in terms of the un-ionized acid, is $K_{eq} = 10.0\text{ M}^{-1}$ at $373\text{ K}$. The acid dissociation constant of \ce{RCOOH} at $373\text{ K}$ is $K_a = 4.0 \times 10^{-6}$. The reaction is carried out in a heavily buffered aqueous solution containing a weak base $B$ and its conjugate acid $BH^+$ in a molar ratio of $[B]:[BH^+] = 1:10$. At $373\text{ K}$, the ionic product of water is $K_w = 1.0 \times 10^{-12}$ and the base dissociation constant of $B$ is $K_b = 1.0 \times 10^{-5}$. Initially, the total analytical concentration of the acid species ($[\ce{RCOOH}] + [\ce{RCOO^-}]$) is $1.0\text{ M}$ and the concentration of \ce{R'OH} is $1.0\text{ M}$. Assuming the buffer capacity is large enough to maintain a strictly constant pH, what is the equilibrium concentration of the ester \ce{RCOOR'}?

Excess $\ce{AgCl(s)}$ is added to $1.0 \text{ L}$ of an aqueous solution at $298 \text{ K}$. The solution contains a total analytical concentration of $0.20 \text{ M}$ glycine ($\ce{H2N-CH2-COOH}$) and is externally buffered to maintain a constant $\text{pH} = 9.6$.

Given:

• For glycine, $pK_{a1} = 2.3$ and $pK_{a2} = 9.6$.
• $K_{sp}(\ce{AgCl}) = 1.0 \times 10^{-10}$.
• The overall formation constant for the complex $[\ce{Ag(Gly)2}]^-$ is $\beta_2 = 1.0 \times 10^8$.

Assuming the external buffer has sufficient capacity to maintain the pH exactly at $9.6$, what is the exact molar solubility of $\ce{AgCl}$ in this solution?

A rigid container of total volume $3.0\text{ L}$ is divided into two compartments, A and B, by a freely movable, strictly solvent-permeable semi-permeable membrane. Compartment A initially has a volume of $2.0\text{ L}$ and contains an aqueous solution comprising $0.20\text{ mol}$ of $\ce{MNO3}$ and $0.20\text{ mol}$ of $\ce{NaX}$. Compartment B initially has a volume of $1.0\text{ L}$ and contains an aqueous solution of $1.20\text{ mol}$ of sucrose (a non-electrolyte). For the sparingly soluble salt $\ce{MX}$, the solubility product is $K_{sp} = 1.0 \times 10^{-2}$ at $298\text{ K}$. Assume ideal behavior for all solutions, complete dissociation of strong electrolytes, and that the volume of the solid precipitate is negligible. When the system reaches thermodynamic equilibrium at $298\text{ K}$, what is the mass of $\ce{MX}$ precipitated? (Given: Molar mass of $\ce{MX} = 150\text{ g mol}^{-1}$)

An aqueous solution contains $1.0 \times 10^{-3}\text{ M } AgNO_3$ at $298\text{ K}$. To this, an equal volume of a buffer solution containing $0.2\text{ M}$ of an aromatic amine ($L$) and $0.2\text{ M}$ of its corresponding hydrochloride salt ($L \cdot HCl$) is added. This experiment is performed separately with three different amines: Ligand I: 4-Methylaniline Ligand II: Aniline Ligand III: 2-Methylaniline