Show that for nonstandard conditions the temperature variation of a cell potential is $$E\left(T_{1}\right)-E\left(T_{2}\right)=\left(T_{1}-T_{2}\right) \frac{\left(\Delta S^{\circ}-R \ln Q\right)}{z F}$$ where \(E\left(T_{1}\right)\) and \(E\left(T_{2}\right)\) are the cell potentials at \(T_{1}\) and \(T_{2},\) respectively. We have assumed that the value of \(Q\) is maintained at a constant value. For the nonstandard cell below, the potential drops from \(0.394 \mathrm{V}\) at \(50.0^{\circ} \mathrm{C}\) to \(0.370 \mathrm{V}\) at \(25.0^{\circ} \mathrm{C} .\) Calculate \(Q\) \(\Delta H^{\circ},\) and \(\Delta S^{\circ}\) for the reaction, and calculate \(K\) for the two temperatures. $$\mathrm{Cu}(\mathrm{s})\left|\mathrm{Cu}^{2+}(\mathrm{aq}) \| \mathrm{Fe}^{3+}(\mathrm{aq}), \mathrm{Fe}^{2+}(\mathrm{aq})\right| \mathrm{Pt}(\mathrm{s})$$ Choose concentrations of the species involved in the cell reaction that give the value of \(Q\) that you have calculated, and then determine the equilibrium concentrations of the species at \(50.0^{\circ} \mathrm{C}\)

Step by step solution

01

Assemble Knowns and Calculate Q

Start by assembling the known values, as well as calculating Q from the cell potential using the Gibbs free energy formula. Since \(E(T_1)\) = 0.394V, \(E(T_2)\) = 0.370V, \(T_1\) = 50 degrees Celsius or 323.15K, \(T_2\) = 25 degrees Celsius or 298.15K, n = 2 (number of electrons involved in the reaction), F = 96485 C/mol (Faraday’s constant), and R = 8.314 J/mol K (Universal gas constant): \[Q = e^{[(E(T_2) - E(T_1)) \cdot n \cdot F / R(T_1 - T_2)]}\]

02

Compute \(\Delta H^\circ\) and \(\Delta S^\circ\)

\(\Delta H^\circ\) and \(\Delta S^\circ\) can be found via the Gibbs-Helmholtz equation by rearranging for \(\Delta S^\circ\) and hence \(\Delta H^\circ\). That gives us: \[\Delta S^\circ = (\Delta E \cdot n \cdot F / (\Delta T)) + R \cdot ln(Q)\] and \[\Delta H^\circ = \Delta E \cdot n \cdot F + T \cdot \Delta S^\circ\]

03

Determine the Equilibrium Constant (K)

Apply the Van't Hoff equation to determine K at T1 and T2: \[K(T_{1}) = e^{-\Delta H^\circ/R \cdot T_{1} + \Delta S^\circ/ R}\] and \[K(T_{2}) = e^{-\Delta H^\circ / R \cdot T_{2}+ \Delta S^\circ/R}\]

04

Establish Equilibrium Concentrations

Write the expression for the equilibrium constant (K) for the cell reaction, and from that expression find the concentrations of the species in equilibrium at 50 degrees Celsius.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!