Connection between Cell Potential, ∆G, and K | Chemistry Optional Notes for UPSC PDF Download

∆G: Gibbs Energy

• ∆G is the change of Gibbs (free) energy for a system and ∆G° is the Gibbs energy change for a system under standard conditions (1 atm, 298K). On an energy diagram, ∆G can be represented as:
  
• Where ∆G is the difference in the energy between reactants and products. In addition ∆G is unaffected by external factors that change the kinetics of the reaction. For example if Ea(activation energy) were to decrease in the presence of a catalyst or the kinetic energy of molecules increases due to a rise in temperature, the ∆G value would remain the same.

E°cell: Standard Cell Potential

• E°_cell is the electromotive force (also called cell voltage or cell potential) between two half-cells. The greater the E°cell of a reaction the greater the driving force of electrons through the system, the more likely the reaction will proceed (more spontaneous). E°cell is measured in volts (V). The overall voltage of the cell = the half-cell potential of the reduction reaction + the half-cell potential of the oxidation reaction. To simplify,
  E^o_cell =E^o_reduction − E^o_oxidation (1)
  or
  E^o_cell = Eo_cathode − E^o_anode (2)
• The potential of an oxidation reduction (loss of electron) is the negative of the potential for a reduction potential (gain of electron). Most tables only record the standard reduction half-reactions as standard reduction potentials. To find the standard oxidation potential, simply reverse the sign of the standard reduction potential.

Note

The more positive reduction potential of reduction reactions are more spontaneous. When viewing a cell reduction potential table, the higher the cell is on the table, the higher potential it has as an oxidizing agent.

Difference between Ecell and Eº_cell

Eº_cell is the standard state cell potential, which means that the value was determined under standard states. The standard states include a concentration of 1 Molar (mole per liter) and an atmospheric pressure of 1. Similar to the standard state cell potential, Eº_cell, the Ecell is the non-standard state cell potential, which means that it is not determined under a concentration of 1 M and pressure of 1 atm. The two are closely related in the sense that the standard cell potential is used to calculate for the cell potential in many cases.
 (3)
Other simplified forms of the equation that we typically see:
 (4)
or in terms of log₁₀ (base 10) instead of the natural logarithm (base e)
 (5)
Both equations applies when the temperature is 25ºC. Deviations from 25ºC requires the use of the original equation. Essentially, Eº is E at standard conditions

Solved Examples

Example 1: What is the value of  E_cell  for the voltaic cell below:
Pt(s) | Fe²⁺ (0.1M), Fe³⁺ (0.2M) || Ag⁺(0.1M) | Ag(s)
Ans: To use the Nernst equation, we need to establish  E^o_cell and the reaction to which the cell diagram corresponds so that the form of the reaction quotient (Q) can be revealed. Once we have determined the form of the Nernst equation, we can insert the concentration of the species.

Now to determine  E_cell  for the reaction
Fe²⁺ (0.1M) + Ag⁺ (1.0M) → Fe³⁺(0.20M) + Ag(s)
Use the Nernst equation

Note that this reaction is spontaneous (positive  E_cell) as written under standard conditions, but is non-spontaneous (negative  E_cell) under the non-standard conditions of question.

K: The Equilibrium Constant

K  is the equilibrium constant of a general reaction
aA + bB ⇋ cC + dD (6)
and is expressed by the reaction quotient:
 (7)

Example 2: Given  K = 2.81×10⁻¹⁶  for the following reaction
Cu²⁺ (aq) + Ag(s) ⇌ Cu(s) + 2Ag⁺
Find ∆G.
Ans: Use the following formula:

The Relationship Between the Three

The relationship between ΔG, K, and E^∘_cell can be represented by the following diagram.

where

• R = 8.314 J mol C^-1
• T = Temp in K
• n = moles of e- from balanced redox reaction
• F = Faraday's constant = 96,485 C/mol

E^o_cell  can be calculated using the following formula:
 (8)

Example 3: Find the  E^o_cell  for the following coupled half-reactions.
Ans: 1. Determine the cathode and anode in the reaction
Zn(s) ⇌ Zn²⁺(aq) + 2e⁻
Anode, Oxidation half-reaction (since  Zn(s) increase oxidation state from 0 to +2)
Cu²⁺(aq) + 2e⁻ ⇌ Cu(s)
Cathode, Reduction half-rection (since  Cu²⁺(aq) decreases oxidation state from +2 to 0)
2. Determine the  E^o_cell  values using the standard reduction potential table (Table P1)

3. Use
E^∘_cell = E^∘_SRP(cathode) − E^o_SRP(anode)
= 0.340V − (−0.763V)
= 1.103V