Inorganic chemistry Assistance Lecturer Amjad Ahmed Jumaa  Predicting whether a (redox) reaction is spontaneous.  Calculating (ΔG°) and (K) from (E°).  The Nernst equation. 1

Spontaneity of (Redox) reactions: Predicting whether a (redox) reaction is spontaneous: There is a relationship between free energy change and cell emf. ΔG = - nFE cell ……………… (1) n is the number of moles of electrons transferred during the (redox) reaction. F is the Faraday constant, which is the electrical charge contained in (1 mol) of electrons. 1F = 96,500 C/mol. = 96,500 J/V.mol. Both (n) and (F) are positive quantities. (ΔG) is negative for a spontaneous process. Therefore, (E cell ) is positive for a spontaneous process.

ΔG° = - nFE ° cell ……………… (2) Example: Predict whether a spontaneous reaction will occur when the following reactants and products are in their standard states. Fe 2+ + Cr 2 O 7 2- → Fe 3+ + Cr 3+ Solution: Let's calculate the (standard cell emf ) for this reaction. From the sign of (E ° cell ), we can determine if the reaction is spontaneous. Separate the reaction into half-reactions to calculate the standard cell emf. Fe 2+ (aq) Fe 3+ (aq) + e - E ° anode = +0.77V

At this point, we could balance the reduction half-reaction and then come up with the overall balanced equation. And finding a reaction that contains both (Cr 2 O 7 2- ) and (Cr 3+ ). Cr 2 O 7 2- (aq) + 14H + (aq) + 6e - 2Cr 3+ (aq)+7 H 2 O(l) E ° cathode = +1.33V We have not come up with a balanced equation, but we do not need a balanced equation to calculate (E° cell ). E° cell = E° cathode - E° anode E° cell = 1.33V – 0.77V = V Since the (E° cell ). Is positive the reaction is spontaneous.

Calculating (ΔG°) and (K) from (E°): Equation (2) shows that the relationship between standard free energy change and (standard emf) is: ΔG° = - nFE ° cell ……………… (2) We knew that the free energy change for a reaction is related to its equilibrium constant (K) by the following equation: ΔG° = - RT ln K Therefore, from the two equations we obtain: - nFE ° cell = - RT ln K Solving for (E ° cell ), we obtain:

E ° cell = ……………… (3) At, 298 K, we can simplify equation (3) by substitution for ( R ) and ( F). E ° cell = E ° cell = ……………………….. (4)

Example: Calculate (ΔG°) and the equilibrium constant (K) at (25°) for the reaction. 2Br - (aq) + I 2 (s) → Br 2 (l) + 2I - (aq) Solution: First, we can calculate the (standard cell emf) from half-reaction potentials. Then, the standard Gibbs free energy change is: ΔG° = - nFE ° cell ……………… (2) The equilibrium constant at (25°) is related to (E ° cell ) by: E ° cell = ………………. (4)

Step(1): separate the reaction into half-reactions to calculate the standard cell emf. Step(2): substitute (E ° cell ) into equation(2) to calculate the standard free energy change. In the balanced equation above, (2 moles) of electrons are transferred. Therefore, (n = 2). I 2 (s) + 2e - 2I - (aq) E ° cathode = V 2Br - (aq) Br 2 (l) + 2e - E ° anode = +1.07V 2Br - (aq) + I 2 (s) → Br 2 (l) + 2I - (aq) E ° cell = E ° cathode - E ° anoode = -0.54V

ΔG° = - nFE ° cell ΔG° = - (2) (96,500 J/V.mol) ( V) ΔG° = 1.04 x 10 5 J/mol. = 104 kJ /mol. positive value of (ΔG°) and the negative value for (E ° cell ) indicate that the reaction is non spontaneous under standard-state conditions. Step(3): Rearrange equation(4) to solve for the equilibrium constant, (K). ln K =

ln K = = K = e = 6 x The Nernst equation: There are a relationship between free energy change (ΔG) and the standard free energy change (ΔG°). ΔG = ΔG° + RT lnQ (Q) is the reaction quotient. We also know that: ΔG = - nFE cell and, ΔG° = - nFE° cell

Substituting for (ΔG) and (ΔG°) in the first equation, we find: - nFE cell = - nFE° cell + RT lnQ Dividing both sides of the equation by (-nF) gives: E = E° - ………… (5) Equation (5), is known as the Nernst equation. At (298 K), equation (5) can be simplified by substituting (R,T) and (F) into the equation. E = E° - ………… (6) At equilibrium

E = 0 ; and K = Q ; where (K) is equilibrium constant of the (redox) reaction. Substituting into equation (6) gives: E° = Using the Nernst equation to predict the spontaneity of a (redox) reaction: can use the Nernst equation to calculate the (cell emf) (E). If the ( cell emf) is positive, the (redox) reaction is spontaneous. if the ( cell emf) is negative, the (redox) reaction is non spontaneous in the direction written.