Thermodynamics - Class 11 Part 1

Consider a galvanic cell operating at $300\text{ K}$ with the following balanced cell reaction:

$\ce{M(s) + X^{2+}(aq) <=> M^{2+}(aq) + X(s)}$

The standard cell potential at $300\text{ K}$ is $E^{\circ} = 0.10\text{ V}$. The temperature coefficient of the standard cell potential at constant pressure is $\left(\frac{\partial E^{\circ}}{\partial T}\right)_P = -5.0 \times 10^{-4}\text{ V K}^{-1}$.

Assuming that the standard enthalpy ($\Delta H^{\circ}$) and standard entropy ($\Delta S^{\circ}$) of the reaction are independent of temperature, determine the standard enthalpy of the reaction at $300\text{ K}$ and the temperature ($T_{eq}$) at which the cell reaction reverses its direction of spontaneity under standard conditions.

(Given: Faraday's constant $F = 96500\text{ C mol}^{-1}$)

A $0.1\text{ M}$ aqueous solution of a weak monoprotic acid, \ce{HA}, exhibits an osmotic pressure of $2.75\text{ atm}$ at $300\text{ K}$ and $3.60\text{ atm}$ at $360\text{ K}$. Assuming the volume of the solution is independent of temperature, calculate the standard enthalpy of dissociation ($\Delta H^\circ$) of the acid in $\text{kJ/mol}$.

[Given: $R = \frac{1}{12}\text{ L atm K}^{-1}\text{ mol}^{-1}$ for osmotic pressure, $R = 8.3\text{ J K}^{-1}\text{ mol}^{-1}$ for thermodynamic calculations. $\ln(2) = 0.70$, $\ln(3) = 1.10$, $\ln(4.5) = 1.50$]

Consider the gas-phase synthesis reaction at $300\text{ K}$:

$\ce{A(g) + 2B(g) -> C(g)}$

The standard thermodynamic parameters at $300\text{ K}$ are provided below:

$\begin{array}{|c|c|c|c|} \hline \text{Species} & \Delta H^\circ_f \text{ (kJ mol}^{-1}\text{)} & S^\circ_m \text{ (J K}^{-1}\text{mol}^{-1}\text{)} & C^\circ_{p,m} \text{ (J K}^{-1}\text{mol}^{-1}\text{)} \\ \hline \ce{A(g)} & 20 & 250 & 50 \\ \ce{B(g)} & 10 & 150 & 40 \\ \ce{C(g)} & 0 & 350 & 30 \\ \hline \end{array}$

Assuming the molar heat capacities at constant pressure ($C^\circ_{p,m}$) are independent of temperature, determine the standard Gibbs free energy change ($\Delta G^\circ$) for the reaction at $500\text{ K}$.

(Given: $\ln(5/3) = 0.51$)

The standard enthalpy of ionization of a weak monoprotic acid \ce{HA} is $\Delta H^\circ_{\text{ion}} = 54.45 \text{ kJ mol}^{-1}$ and is assumed to be independent of temperature. At $298 \text{ K}$, the acid dissociation constant is $K_a = 1.0 \times 10^{-5}$. The ionic product of water at $318 \text{ K}$ is $K_w = 2.5 \times 10^{-14}$. Calculate the pH of a $0.1 \text{ M}$ aqueous solution of the sodium salt \ce{NaA} at $318 \text{ K}$.

(Given: $R = 8.314 \text{ J K}^{-1} \text{ mol}^{-1}$, $2.303 \times R = 19.15 \text{ J K}^{-1} \text{ mol}^{-1}$, $\log 2.5 = 0.4$)

Given the standard reduction potentials at $298\text{ K}$: $E^\circ_{\ce{Cu^+/Cu}} = +0.52\text{ V}$ $E^\circ_{\ce{Cu^{2+}/Cu^+}} = +0.16\text{ V}$ And the overall formation constants ($\beta$) for copper-ammonia complexes: $\beta_2 \text{ for } \ce{[Cu(NH_3)_2]^+} = 10^{11}$ $\beta_4 \text{ for } \ce{[Cu(NH_3)_4]^{2+}} = 10^{13}$ Consider the comproportionation reaction in an aqueous medium where the equilibrium concentration of free $\ce{NH_3}$ is maintained at $1.0\text{ M}$: $\ce{Cu(s) + [Cu(NH_3)_4]^{2+}(aq) \rightleftharpoons 2[Cu(NH_3)_2]^+(aq)}$ Which of the following correctly identifies the standard potential of this reaction ($E^\circ_{\text{rxn}}$) under these conditions and the thermodynamic stability of the $\ce{Cu(I)}$ state? (Take $\frac{2.303 RT}{F} = 0.06\text{ V}$)

The standard Gibbs free energy of dissociation ($\Delta G^\circ_{diss}$) for acetic acid ($\ce{CH3COOH}$) and pivalic acid ($\ce{(CH3)3CCOOH}$) are evaluated at $298\text{ K}$ in two different media: Water ($\epsilon = 78.4$) and the Gas Phase ($\epsilon = 1$). Let $\Delta \Delta G^\circ_{aq} = \Delta G^\circ_{diss, aq}(\text{pivalic}) - \Delta G^\circ_{diss, aq}(\text{acetic})$ and $\Delta \Delta G^\circ_{gas} = \Delta G^\circ_{diss, gas}(\text{pivalic}) - \Delta G^\circ_{diss, gas}(\text{acetic})$. Which of the following correctly identifies the signs of $\Delta \Delta G^\circ_{aq}$ and $\Delta \Delta G^\circ_{gas}$, and the dominant physical phenomenon dictating the gas-phase thermodynamic trend?

The specific acid-catalyzed hydrolysis of ethyl acetate in a dilute aqueous $\ce{HCl}$ solution follows the rate law: $\text{Rate} = k[\text{Ester}][\ce{H+}][\ce{H2O}]$, where $k$ is the true third-order rate constant. The rate constant $k$ follows the Arrhenius equation with an activation energy parameter $\frac{E_a}{2.303 R} = 3000 \text{ K}$.

The reaction is studied at two different temperatures, $300 \text{ K}$ and $375 \text{ K}$, in a variable-volume reactor maintained at a pressure sufficient to keep the mixture entirely in the liquid state. The density of the reaction mixture is $1.00 \text{ g mL}^{-1}$ at $300 \text{ K}$ and drops to a hypothetical value of $0.80 \text{ g mL}^{-1}$ at $375 \text{ K}$.

If $k_{obs}$ is the observed pseudo-first order rate constant (defined by $\text{Rate} = k_{obs}[\text{Ester}]$), determine the ratio $\frac{k_{obs}(375 \text{ K})}{k_{obs}(300 \text{ K})}$.

(Assume the total mass of the mixture and the moles of the acid catalyst remain constant, and the ester is present in trace amounts.)

Consider the following ligand exchange reaction for a \ce{Ni^{2+}} ($d^8$) complex in aqueous solution at $300 \text{ K}$:

$\ce{[Ni(A)6]^{2+}(aq) + 3B(aq) <=> [Ni(B)3]^{2+}(aq) + 6A(aq)}$

where A is a neutral monodentate ligand and B is a neutral bidentate ligand.

The crystal field splitting energy ($\Delta_o$) for \ce{[Ni(A)6]^{2+}} is $150 \text{ kJ mol}^{-1}$ and for \ce{[Ni(B)3]^{2+}} is $120 \text{ kJ mol}^{-1}$. Assume that the standard enthalpy change ($\Delta H^\circ$) of the exchange reaction is determined entirely by the difference in Crystal Field Stabilization Energy (CFSE) between the two complexes.

If the standard entropy change ($\Delta S^\circ$) for the exchange reaction is $+140 \text{ J K}^{-1} \text{ mol}^{-1}$, calculate the standard Gibbs free energy change ($\Delta G^\circ$) and identify the thermodynamically more stable complex at $300 \text{ K}$.

A $0.10 \text{ mol kg}^{-1}$ aqueous solution of a weak monoprotic acid \ce{HA (aq)} freezes at $-0.2232^\circ\text{C}$ at $1 \text{ atm}$. The cryoscopic constant of water is $K_f = 1.86 \text{ K kg mol}^{-1}$. Assume the solution behaves ideally with respect to the solvent (osmotic coefficient $\phi = 1$), such that the freezing point depression is strictly proportional to the total molality of all solute particles. However, the acid dissociation equilibrium is non-ideal. If the mean ionic activity coefficient of the ions \ce{H+ (aq)} and \ce{A- (aq)} in this solution is $\gamma_{\pm} = 0.80$ and the activity coefficient of the unionized acid is $\gamma_{HA} = 1.0$, what is the true thermodynamic acid dissociation constant ($K_a^\circ$) of \ce{HA} at the freezing point?

$n$ moles of a real gas obeying the van der Waals equation of state undergoes a reversible isothermal expansion from an initial volume $V_1$ to a final volume $V_2$ at a constant temperature $T$. Which of the following expressions correctly represents the change in internal energy ($\Delta U$) and the heat absorbed ($q$) by the gas? (Given: $a$ and $b$ are van der Waals constants)

Consider the elementary reversible gas-phase reaction $\ce{A(g) <=> B(g)}$ at $300\text{ K}$. The standard Gibbs free energy change is $\Delta G^\circ = -300 R\text{ J mol}^{-1}$ and the standard entropy change is $\Delta S^\circ = -30 R\text{ J K}^{-1}\text{ mol}^{-1}$. The temperature coefficient ($Q_{10}$) for the forward reaction between $300\text{ K}$ and $310\text{ K}$ is given by $\ln(Q_{10, f}) = 1.5$. Assuming $\Delta H^\circ$ and $\Delta S^\circ$ are independent of temperature, what is the value of $\ln(Q_{10, b})$ for the backward reaction in the same temperature range? ($R$ is the universal gas constant)

An aqueous solution of D-ribose (\ce{C5H10O5}, molar mass $= 150 \text{ g mol}^{-1}$) is prepared at $298 \text{ K}$. At this temperature, the molarity of the solution is $2.0 \text{ M}$ and its density is $1.20 \text{ g mL}^{-1}$. The solution is subsequently heated to $353 \text{ K}$ in a sealed vessel (assume negligible solvent vaporization). At $353 \text{ K}$, the density of the solution decreases to $1.15 \text{ g mL}^{-1}$ due to thermal expansion. Given that the vapor pressure of pure water at $353 \text{ K}$ is $260 \text{ Torr}$, determine the vapor pressure of the solution at $353 \text{ K}$. Assume ideal behavior for the vapor phase and that Raoult's Law is applicable.

Consider the following electrochemical cell at 298 K:

$\ce{Pt(s) | H2(g, 1 bar) | HA(aq, C) || Cu^{2+}(aq, 1 M) | Cu(s)}$

where $\ce{HA}$ is a weak monoprotic acid with a dissociation constant $K_a = 1.0 \times 10^{-5}$. When the concentration of $\ce{HA}$ is $C_1 = 0.1 \text{ M}$, the measured cell potential is $E_1$. When the $\ce{HA}$ solution in the anodic compartment is diluted to a concentration $C_2 = 7.5 \times 10^{-6} \text{ M}$, the cell potential becomes $E_2$. Calculate the magnitude of the change in cell potential, $|E_2 - E_1|$ in mV.

(Given: $\frac{2.303 RT}{F} = 0.06 \text{ V}$, $\log 2 = 0.30$, $\log 3 = 0.48$. Assume activity coefficients are unity.)

Consider $1\text{ mol}$ of a real gas obeying the Van der Waals equation of state, $\left(P + \frac{a}{V^2}\right)(V - b) = RT$, undergoing a reversible adiabatic expansion. Recognizing that the internal energy $U$ of a real gas is a function of both temperature $T$ and volume $V$, which of the following expressions correctly represents the slope of the adiabatic curve, $\left(\frac{\partial P}{\partial V}\right)_S$, on a $P-V$ indicator diagram? (Assume the molar heat capacity at constant volume, $C_V$, is independent of temperature).

Consider the following gas-phase equilibria at $298\text{ K}$:

(1) $\ce{X(g) + 2Y(g) \rightleftharpoons 2Z(g)} \quad K_{p1} = 2 \times 10^8$ (2) $\ce{Z(g) \rightleftharpoons W(g) + Y(g)} \quad K_{p2} = 5 \times 10^{-4}$

A rigid reaction vessel at $298\text{ K}$ contains the gases with the following instantaneous partial pressures: $P_X = 0.5\text{ bar}$, $P_Y = 2.0\text{ bar}$, $P_Z = 1.0\text{ bar}$, and $P_W = 5.0\text{ bar}$.

For the overall transformation $\ce{X(g) \rightleftharpoons 2W(g)}$, determine the standard Gibbs free energy change ($\Delta G^\circ$) and the actual Gibbs free energy change ($\Delta G$) at this instant.

Calculate the standard enthalpy of formation ($\Delta_f H^\circ$) of the methyl radical, $\ce{CH3^\bullet_{(g)}}$, at $298\text{ K}$.

Given the following thermodynamic data at $298\text{ K}$: Standard enthalpy of formation of $\ce{CH4_{(g)}} = -75\text{ kJ mol}^{-1}$ Enthalpy of sublimation of $\ce{C_{(s, graphite)}} = 717\text{ kJ mol}^{-1}$ Bond dissociation enthalpy of $\ce{H2_{(g)}} = 436\text{ kJ mol}^{-1}$ First bond dissociation enthalpy of $\ce{CH4_{(g)}}$ ($\ce{CH4_{(g)} -> CH3^\bullet_{(g)} + H^\bullet_{(g)}}$) $= 439\text{ kJ mol}^{-1}$

An electrochemical cell is constructed at 298 K:

\ce{Pt(s) | I^- (0.1 M) | I_2(s) || Fe^{2+} (0.1 M), Fe(OH)_3(s) | Pt(s)}

The right half-cell is buffered at $\text{pH} = 3.0$.

Given: $E^\circ_{\ce{Fe^{3+}/Fe^{2+}}} = 0.77\text{ V}$ $E^\circ_{\ce{I_2/I^-}} = 0.53\text{ V}$ $K_{sp}(\ce{Fe(OH)_3}) = 10^{-38}$ $\frac{2.303 RT}{F} = 0.06\text{ V}$ $1\text{ F} = 96500\text{ C/mol}$

Calculate the cell potential ($E_{cell}$) and the Gibbs free energy change ($\Delta G$) for the cell reaction as written: \ce{2 Fe^{3+}(aq) + 2 I^-(aq) -> 2 Fe^{2+}(aq) + I_2(s)}

Consider the oxidation of the triiodide ion (\ce{I_3^-}) by permanganate (\ce{MnO_4^-}) in a basic medium at 298 K, producing iodate (\ce{IO_3^-}) and solid manganese dioxide (\ce{MnO_2(s)}).

The standard reduction potentials at 298 K are: $E^\circ_{\ce{MnO_4^-}/\ce{MnO_2}, \ce{OH^-}} = +0.600 \text{ V}$ $E^\circ_{\ce{IO_3^-}/\ce{I_3^-}, \ce{OH^-}} = +0.615 \text{ V}$

An electrochemical cell is constructed using these half-reactions. All soluble species except \ce{OH^-} are maintained at a concentration of 1.0 M.

Calculate the Gibbs Free Energy change ($\Delta G$) for the cell reaction, scaled to exactly one mole of \ce{I_3^-}, when the pH of the solution is maintained at 10.0.

(Given: $\frac{2.303 RT}{F} = 0.060 \text{ V}$ at 298 K, $F = 96500 \text{ C mol}^{-1}$)

One mole of a real gas, obeying the van der Waals equation of state, undergoes an isothermal reversible expansion from an initial volume $V_1$ to a final volume $V_2$ at a constant temperature $T$. Which of the following expressions correctly represents the heat absorbed ($q$) by the gas during this process? (Given: $a$ and $b$ are van der Waals constants, and $R$ is the universal gas constant).

Consider the following electrochemical cell operating reversibly at $300\text{ K}$ and $1\text{ bar}$:

\ce{Zn(s) | Zn^{2+}(aq, 1 M) || H+(aq, 1 M) | H2(g, 1 bar) | Pt(s)}

The standard EMF of the cell at $300\text{ K}$ is $0.76\text{ V}$, and its temperature coefficient is $\left(\frac{\partial E^\circ}{\partial T}\right)_P = -5.0 \times 10^{-4}\text{ V K}^{-1}$. Assuming the gases behave ideally, calculate the total reversible work ($w_{rev}$) and the standard internal energy change ($\Delta U^\circ$) for the consumption of $1\text{ mole}$ of \ce{Zn(s)}.

(Given: $F = 96500\text{ C mol}^{-1}$, $R = 8.3\text{ J K}^{-1}\text{ mol}^{-1}$)

A gaseous reaction \ce{A(g) -> 2B(g)} follows zero-order kinetics. When the reaction is carried out in a rigid $1.0 \text{ L}$ flask at $300 \text{ K}$, the total pressure increases at a constant rate of $0.2463 \text{ atm min}^{-1}$.

In a separate experiment, $1.0 \text{ mole}$ of pure \ce{A(g)} is placed in a cylinder fitted with a frictionless piston at $300 \text{ K}$. The initial volume is $10.0 \text{ L}$. The gas undergoes a reversible isothermal expansion at $300 \text{ K}$, during which the volume of the cylinder increases at a constant rate of $0.5 \text{ L min}^{-1}$.

Assuming ideal gas behavior, calculate the magnitude of the work done by the gas (in Joules) during the first $20 \text{ minutes}$ of the expansion.

(Given: $R = 0.0821 \text{ L atm K}^{-1} \text{ mol}^{-1} = 8.314 \text{ J K}^{-1} \text{ mol}^{-1}$, $\ln 2 = 0.693$)

Initially, $1.0 \text{ mol}$ of $\ce{N2O4(g)}$ is placed in a cylinder fitted with a frictionless piston maintained at a constant pressure of $1.0 \text{ atm}$ and a constant temperature $T$. At equilibrium, the degree of dissociation of $\ce{N2O4(g)}$ into $\ce{NO2(g)}$ is $0.5$.

Subsequently, $7.8 \text{ mol}$ of an ideal inert gas (He) is reversibly and isothermally added to the cylinder, maintaining the constant pressure of $1.0 \text{ atm}$.

What is the magnitude of the expansion work done by the system during the addition of the inert gas? (Assume ideal gas behavior for all gases)

Consider a hydrogen-oxygen fuel cell operating reversibly and isothermally at $298 \text{ K}$ and $1 \text{ bar}$. The overall cell reaction is: $\ce{H2(g) + 1/2O2(g) -> H2O(l)}$ Given the standard thermodynamic data at $298 \text{ K}$: $\Delta_f H^\circ(\ce{H2O, l}) = -285.8 \text{ kJ mol}^{-1}$ $\Delta_f S^\circ(\ce{H2O, l}) = -163.3 \text{ J K}^{-1} \text{ mol}^{-1}$ $R = 8.314 \text{ J K}^{-1} \text{ mol}^{-1}$ Assume ideal gas behavior and negligible volume for liquid water. Which of the following represents the correct values of heat exchanged ($q$) and the change in internal energy ($\Delta U$) for the consumption of $1 \text{ mole}$ of $\ce{H2(g)}$?

For a transition metal $M$ which forms a $3d^8$ $M^{2+}$ ion, the following thermodynamic parameters are given at $300 \text{ K}$:

$\bullet$ Enthalpy of atomization, $\Delta H_{at} = 430 \text{ kJ mol}^{-1}$ $\bullet$ Sum of first and second ionization enthalpies, $IE_1 + IE_2 = 2490 \text{ kJ mol}^{-1}$ $\bullet$ Isotropic (spherical) hydration enthalpy of $M^{2+}(g)$, $\Delta H_{hyd, iso} = -1980 \text{ kJ mol}^{-1}$ $\bullet$ Crystal field splitting energy ($\Delta_o$) for the octahedral aquo complex $[M(H_2O)_6]^{2+}$ is $100 \text{ kJ mol}^{-1}$ $\bullet$ Standard entropy change for the half-reaction $M(s) \rightarrow M^{2+}(aq) + 2e^-$, $\Delta S^\circ = +100 \text{ J K}^{-1} \text{ mol}^{-1}$ $\bullet$ Absolute standard Gibbs free energy of oxidation for $H_2(g) \rightarrow 2H^+(aq) + 2e^-$ is $+838.25 \text{ kJ mol}^{-1}$

Calculate the standard reduction potential $E^\circ_{M^{2+}/M}$ at $300 \text{ K}$. (Given: $F = 96500 \text{ C mol}^{-1}$)

Consider the acid-catalyzed esterification of octadecanoic acid with octan-1-ol to form octyl octadecanoate and water. At $298\text{ K}$, the standard enthalpy and entropy of the reaction producing liquid water are $\Delta H^\circ_{rxn} = -12.0\text{ kJ mol}^{-1}$ and $\Delta S^\circ_{rxn} = -10.0\text{ J K}^{-1}\text{ mol}^{-1}$, respectively.

The standard enthalpy of vaporization of water is $\Delta H^\circ_{vap} = 42.0\text{ kJ mol}^{-1}$ and its standard entropy of vaporization is $\Delta S^\circ_{vap} = 110.0\text{ J K}^{-1}\text{ mol}^{-1}$.

The reaction is carried out at $400\text{ K}$ in a continuous flow reactor. Due to the high boiling points of the organic species, they remain entirely in the liquid phase, but water is formed exclusively in the vapor phase. Assuming all standard enthalpies and entropies are independent of temperature, what is the value of $\ln K$ for this specific equilibrium at $400\text{ K}$?

($R$ is the universal gas constant in $\text{J K}^{-1}\text{ mol}^{-1}$)

Consider the homolytic cleavage of the $C_3-H$ bond in 1,4-pentadiene, \ce{C5H8(g)}, to form the pentadienyl radical, \ce{C5H7^\bullet(g)}, and a hydrogen atom, \ce{H^\bullet(g)}, at $298 \text{ K}$. You are given the following thermodynamic data at $298 \text{ K}$: Standard enthalpy of formation of 1,4-pentadiene, $\Delta H_f^\circ(\ce{C5H8}, g) = 106 \text{ kJ mol}^{-1}$ Standard enthalpy of formation of a hydrogen atom, $\Delta H_f^\circ(\ce{H^\bullet}, g) = 218 \text{ kJ mol}^{-1}$ Standard localized $C(sp^3)-H$ bond dissociation enthalpy $= 415 \text{ kJ mol}^{-1}$ Resonance energy of the pentadienyl radical $= 82 \text{ kJ mol}^{-1}$ (Assume the resonance energy of 1,4-pentadiene is negligible). Calculate the standard enthalpy of formation of the pentadienyl radical, $\Delta H_f^\circ(\ce{C5H7^\bullet}, g)$.

Consider the $S_N1$ solvolysis of the optically active alkyl halide (2S,3S)-2-chloro-3-methylpentane in \ce{H2O} at 298 K. The reaction proceeds via an initial carbocation $C_1^+$ which undergoes a rapid 1,2-shift to form a more thermodynamically stable carbocation $C_2^+$, ultimately yielding the major product $P$. The global enthalpy path shifts due to the superior hyperconjugative stabilization of $C_2^+$.

Let $n_1$ and $n_2$ be the number of hyperconjugable $\alpha$-hydrogens in $C_1^+$ and $C_2^+$, respectively. Identify the major product $P$, its optical activity, and the difference in hyperconjugable hydrogens ($\Delta n = n_2 - n_1$) that provides the thermodynamic driving force ($\Delta H_{\text{rearr}} < 0$) for the shift.

To investigate the thermodynamic stability of high oxidation state fluorides, a student models the formation of gaseous sulfur hexafluoride, $\ce{SF6(g)}$, using a hypothetical gas-phase ionic Born-Haber cycle. The student assumes the formation of isolated $\ce{S^{6+}(g)}$ and $\ce{F^{-}(g)}$ ions, which then condense to form the covalent $\ce{SF6(g)}$ molecule. Given the following thermodynamic data at 298 K: Standard enthalpy of formation of $\ce{SF6(g)}$, $\Delta_f H^\circ = -1209 \text{ kJ/mol}$ Enthalpy of atomization of $\ce{S(s)}$, $\Delta_{atom} H^\circ = 277 \text{ kJ/mol}$ Bond dissociation enthalpy of $\ce{F2(g)}$, $\Delta_{diss} H^\circ = 159 \text{ kJ/mol}$ Sum of the first six ionization enthalpies of $\ce{S(g)}$, $\sum_{i=1}^6 IE_i = 27700 \text{ kJ/mol}$ Electron gain enthalpy of $\ce{F(g)}$, $\Delta_{eg} H^\circ = -328 \text{ kJ/mol}$ Calculate the hypothetical 'gas-phase condensation enthalpy' ($\Delta_{cond} H^\circ$) in kJ/mol—the energy released when one mole of $\ce{S^{6+}(g)}$ and six moles of $\ce{F^{-}(g)}$ combine to form one mole of $\ce{SF6(g)}$.

Calculate the boiling point elevation ($\Delta T_b$) of a $2.00\text{ molal}$ aqueous solution of a non-volatile, non-electrolyte solute.

Given data: Standard enthalpy of vaporization of water at $298\text{ K}$, $\Delta H_{vap}^\circ(298\text{ K}) = 44.0\text{ kJ mol}^{-1}$ Molar heat capacities at constant pressure: $C_{p,m}(\ce{H2O}, l) = 75.0\text{ J K}^{-1}\text{mol}^{-1}$, $C_{p,m}(\ce{H2O}, g) = 35.0\text{ J K}^{-1}\text{mol}^{-1}$ Boiling point of pure water, $T_b^\circ = 373\text{ K}$ Universal gas constant, $R = 8.314\text{ J K}^{-1}\text{mol}^{-1}$ Molar mass of water, $M = 18.0\text{ g mol}^{-1}$

Assume $C_{p,m}$ values are independent of temperature.

One mole of liquid \ce{X2} at $500\text{ K}$ and $1\text{ atm}$ is placed in an insulated, constant-pressure chamber. A catalyst is introduced, triggering the endothermic decomposition and phase change: $\ce{X2(l) -> 2X(g)}$ The reaction proceeds via a highly reactive steady-state intermediate \ce{X^*(g)}, ensuring the process is strictly irreversible. The reaction goes to completion. Given the following thermodynamic parameters at $500\text{ K}$: $\Delta H^\circ_{\text{rxn}} = 10.0\text{ kJ mol}^{-1}$ $S^\circ_{m}(\ce{X2}, l) = 100\text{ J K}^{-1}\text{mol}^{-1}$ $S^\circ_{m}(\ce{X}, g) = 100\text{ J K}^{-1}\text{mol}^{-1}$ $C_{p,m}(\ce{X}, g) = 25\text{ J K}^{-1}\text{mol}^{-1}$ (assumed independent of temperature) Assume the heat capacity of the chamber is negligible. What is the total entropy change of the universe ($\Delta S_{\text{univ}}$) for this process? (Take $\ln(0.6) = -0.51$)

Consider two independent systems at $298 \text{ K}$:

\textbf{System I:} $x$ moles of an ideal monoatomic gas undergoes adiabatic free expansion from a volume $V$ to $10V$. The entropy change of the universe for this process is $\Delta S_I$.

\textbf{System II:} A galvanic cell is constructed as follows: $\ce{Pt | H2 (g, 1 bar) | HA (aq, 0.1 M) || HCl (aq, 0.1 M) | H2 (g, 1 bar) | Pt}$ where $\ce{HA}$ is a weak monoprotic acid with $K_a = 10^{-5}$ at $298 \text{ K}$. The standard enthalpy of dissociation of $\ce{HA}$ is $\Delta H^\circ_{diss} = 17.115 \text{ kJ mol}^{-1}$. The entropy change of the cell reaction ($\Delta S_{rxn} = nF \left(\frac{\partial E}{\partial T}\right)_P$) per mole of $\ce{H2}$ oxidized is $\Delta S_{II}$.

Given that $\Delta S_I = \Delta S_{II}$, and assuming ideal behavior for all solutions and gases, what is the value of $x$ and the initial cell potential $E_{cell}$ at $298 \text{ K}$?

(Given: $\frac{2.303 RT}{F} = 0.059 \text{ V}$ and $2.303 RT = 5.705 \text{ kJ mol}^{-1}$ at $298 \text{ K}$)

An equilibrium mixture of $\ce{F2(g) + Cl2(g) <=> 2ClF(g)}$ ($\Delta H^\circ = -108 \text{ kJ/mol}$) is contained in a thermally insulated cylinder at temperature $T_1$, fitted with a locked, frictionless piston. $2.0 \text{ moles}$ of ideal Argon gas ($\ce{Ar}$) are injected into the cylinder. Immediately after, the piston is unlocked, allowing the gas mixture to undergo adiabatic expansion against a constant external pressure until mechanical equilibrium is reached at a new temperature $T_2$. Let $\Delta T = T_2 - T_1$ and $\Delta n_{\ce{ClF}}$ be the net change in the number of moles of $\ce{ClF}$ from the initial to the final equilibrium state. Which of the following correctly represents $\Delta T$ and $\Delta n_{\ce{ClF}}$?

A non-linear polyatomic ideal gas \ce{X} undergoes a unimolecular isomerization \ce{X(g) -> Y(g)} following the Lindemann-Hinshelwood mechanism. At $300\text{ K}$, the reaction operates strictly at the low-pressure limit. The gas is initially confined in a vessel of volume $V_0$ and is essentially pure \ce{X}. It then undergoes an adiabatic free expansion against zero external pressure until its volume becomes $8V_0$.

Let $k_{eff}$ be the effective first-order rate constant defined by $\text{Rate} = k_{eff}[\ce{X}]$. Given that the activation energy for the bimolecular energization step is $E_a = 300 R \ln(4)$ and assuming the Arrhenius pre-exponential factor is independent of temperature, what is the ratio of the effective first-order rate constant immediately after the expansion to that before the expansion, $\frac{k_{eff,2}}{k_{eff,1}}$?

An electrochemical cell is constructed as follows:

$\ce{M(s) | M^{2+}(aq, 1 M) || X^{-}(aq, 1 M) | X_2(g, V, T) | Pt(s)}$

The cathodic compartment initially contains $10 \text{ moles}$ of a real gas $\ce{X_2}$ in a volume of $2.0 \text{ L}$ at $400 \text{ K}$. The gas undergoes adiabatic free expansion against a vacuum until its volume becomes $10.0 \text{ L}$.

Given for $\ce{X_2(g)}$: van der Waals constants: $a = 6.0 \text{ L}^2 \text{ atm mol}^{-2}$, $b = 0$. Molar heat capacity at constant volume, $C_{v,m} = 3 R$.

Standard electrode potentials (assumed independent of temperature): $E^\circ_{\ce{M^{2+}/M}} = -0.76 \text{ V}$ $E^\circ_{\ce{X_2/X^-}} = +1.36 \text{ V}$

Assume the activity of $\ce{X_2(g)}$ in the Nernst equation is numerically equal to its pressure in atm. Use $R = 0.08 \text{ L atm K}^{-1} \text{ mol}^{-1}$, $\frac{2.303 R \times 300}{F} = 0.06 \text{ V}$. $\log_{10} 2 = 0.30$, $\log_{10} 3 = 0.48$.

Calculate the EMF of the cell after the expansion is complete.

The standard enthalpy of formation of gaseous Molybdenum hexachloride, $\ce{MoCl6}(g)$, is $-840 \text{ kJ mol}^{-1}$ and that of Molybdenum tetrachloride, $\ce{MoCl4}(g)$, is $-380 \text{ kJ mol}^{-1}$ at $298 \text{ K}$. The standard enthalpy of atomization of solid Molybdenum, $\Delta_{atom}H^\circ(\ce{Mo}, s)$, is $660 \text{ kJ mol}^{-1}$ and the bond dissociation enthalpy of $\ce{Cl2}(g)$ is $240 \text{ kJ mol}^{-1}$.

A student calculates the enthalpy of the decomposition reaction: $\ce{MoCl6}(g) \rightarrow \ce{MoCl4}(g) + \ce{Cl2}(g)$ by erroneously assuming that the average $\ce{Mo-Cl}$ bond enthalpy remains constant across different oxidation states and is equal to its value in $\ce{MoCl6}(g)$.

What is the true standard enthalpy of the decomposition reaction, and what is the value calculated by the student, respectively?

Compound A, 2-($\beta$-D-glucopyranosyloxy)pyridine, undergoes enzymatic hydrolysis in an aqueous medium at $300\text{ K}$ to yield D-glucose and 2-hydroxypyridine. The released 2-hydroxypyridine subsequently undergoes complete tautomerization to its more stable keto form, 2-pyridone.\n\nGiven the following thermodynamic data at $300\text{ K}$:\n\nStandard enthalpy of combustion ($\Delta H^\circ_c$) in $\text{kJ/mol}$:\nCompound A (s) = $-4500$\nD-Glucose (s) = $-2800$\n2-Hydroxypyridine (s, enol form) = $-1715$\n\nStandard enthalpy of solution ($\Delta H^\circ_{sol}$) in $\text{kJ/mol}$:\nCompound A (s) = $+30$\nD-Glucose (s) = $+12$\n2-Hydroxypyridine (s, enol form) = $+8$\n\nThermodynamic parameters for the tautomerization $\ce{2-Hydroxypyridine (aq) <=> 2-Pyridone (aq)}$:\n$\Delta G^\circ_{taut} = -25.0\text{ kJ/mol}$\n$\Delta S^\circ_{taut} = -30.0\text{ J K}^{-1}\text{mol}^{-1}$\n\nCalculate the standard observed enthalpy of the overall reaction ($\Delta H^\circ_{obs}$) for the hydrolysis of Compound A (aq) yielding D-glucose (aq) and 2-pyridone (aq) at $300\text{ K}$.

Consider the standard Gibbs free energy changes ($\Delta G^\circ$) for the following processes at 298 K:

For a transition metal $M$: Atomization: $M(s) \rightarrow M(g)$, $\Delta G^\circ = 330 \text{ kJ mol}^{-1}$ Ionization: $M(g) \rightarrow M^{2+}(g) + 2e^-$, $\Delta G^\circ = 2700 \text{ kJ mol}^{-1}$ Hydration: $M^{2+}(g) \rightarrow M^{2+}(aq)$, $\Delta G^\circ = -1998.8 \text{ kJ mol}^{-1}$

For a transition metal $N$: Atomization: $N(s) \rightarrow N(g)$, $\Delta G^\circ = 130 \text{ kJ mol}^{-1}$ Ionization: $N(g) \rightarrow N^{2+}(g) + 2e^-$, $\Delta G^\circ = 2640 \text{ kJ mol}^{-1}$ Hydration: $N^{2+}(g) \rightarrow N^{2+}(aq)$, $\Delta G^\circ = -1950 \text{ kJ mol}^{-1}$

The absolute standard Gibbs free energy change for the oxidation half-reaction $H_2(g) \rightarrow 2H^+(aq) + 2e^-$ is $965.6 \text{ kJ mol}^{-1}$.

An aqueous solution containing $1.0 \text{ M}$ of $M^{2+}$ and $1.0 \text{ M}$ of $N^{2+}$ at pH = 0 is electrolyzed using inert electrodes at 298 K and 1 bar pressure. The overpotential for $H_2$ evolution at the cathode is $0.40 \text{ V}$. Given $F = 96500 \text{ C mol}^{-1}$.

What are the standard reduction potentials $E^\circ_{M^{2+}/M}$ and $E^\circ_{N^{2+}/N}$ respectively, and which species is discharged first at the cathode?

A closed reaction vessel at $298 \text{ K}$ contains a standard acidic aqueous solution ($[\ce{H^+}] = 1.0 \text{ M}$). To this, $1.0 \text{ mole}$ of $\ce{KMnO4}$ and $4.0 \text{ moles}$ of $\ce{MnSO4}$ are added. The system is allowed to reach complete thermodynamic equilibrium.

Given the standard reduction potentials at $298 \text{ K}$: $E^\circ(\ce{MnO4^- / Mn^{2+}}) = +1.51 \text{ V}$ $E^\circ(\ce{MnO2 / Mn^{2+}}) = +1.23 \text{ V}$ $E^\circ(\ce{Mn^{3+} / Mn^{2+}}) = +1.51 \text{ V}$

Calculate the total standard Gibbs free energy change ($\Delta G^\circ_{total}$) for the complete conversion of the given amounts of reactants to their most thermodynamically stable products. (Assume ideal behavior, standard states for all species except the initial stoichiometric amounts provided, and $F = 96500 \text{ C mol}^{-1}$)

Consider the gas-phase heterolytic cleavage of 2-chloro-3-methylbutane ($\ce{C5H11Cl}$) to form the thermodynamically most stable rearranged carbocation intermediate ($R'^+$) and a chloride ion ($\ce{Cl^-}$) at 298 K.

Given the following thermodynamic parameters: $\bullet$ Standard enthalpy of formation of $\ce{C5H11Cl} (g)$, $\Delta H_f^\circ = -140 \text{ kJ/mol}$ $\bullet$ Standard enthalpy of formation of $\ce{Cl^-} (g)$, $\Delta H_f^\circ = -230 \text{ kJ/mol}$ $\bullet$ The theoretical standard enthalpy of formation of the localized (un-hyperconjugated) $3^\circ$ carbocation $R'^+$ is $+880 \text{ kJ/mol}$. $\bullet$ The hyperconjugative stabilization energy is $12 \text{ kJ/mol}$ per $\alpha$-C-H bond.

Calculate the actual enthalpy of reaction ($\Delta H_{rxn}^\circ$) for this heterolytic cleavage, accounting for the hyperconjugative stabilization of the final rearranged carbocation.

Calculate the standard enthalpy of formation ($\Delta_f H^\circ$) of liquid tetrachloromethane, $\ce{CCl4(l)}$, at $298\text{ K}$ and $1\text{ bar}$, given the following thermodynamic data: