Electrochemistry: Derive the Nernst Equation for a galvanic cell under non-standard conditions

Chapter: Electrochemistry – Nernst Equation

📘 What is the Nernst Equation?

The Nernst equation gives the electrode potential or cell potential of an electrochemical cell under non-standard conditions.

For a galvanic cell, the cell potential depends on the concentration or activity of the reacting species, in addition to the standard cell potential.

E_cell = E°_cell − (RT/nF) ln Q

At 298 K:

E_cell = E°_cell − (0.0591/n) log Q

🔬 Step 1: Consider a General Galvanic Cell

Consider the cell reaction:

aA + bB → cC + dD

For this reaction, the reaction quotient is:

Q = [C]^c[D]^d / [A]^a[B]^b

Pure solids and pure liquids are not included in Q because their activities are taken as unity.

🔬 Step 2: Gibbs Energy and Cell Potential

The Gibbs energy change for an electrochemical reaction is related to the cell potential by:

ΔG = −nFE

Under standard conditions:

ΔG° = −nFE°

Therefore:

ΔG = ΔG° + RT ln Q

🔬 Step 3: Substitute the Electrochemical Relations

We know:

ΔG = −nFE

and

ΔG° = −nFE°

Substituting these into:

ΔG = ΔG° + RT ln Q

we get:

−nFE = −nFE° + RT ln Q

🔬 Step 4: Rearrangement

Divide the entire equation by −nF:

E = E° − (RT/nF) ln Q

This is the Nernst Equation.

🌡️ Step 5: Nernst Equation at 298 K

At 298 K:

R = 8.314 J mol⁻¹ K⁻¹
F = 96485 C mol⁻¹

Changing natural logarithm to common logarithm gives:

E = E° − (0.0591/n) log Q

where n is the number of electrons transferred in the balanced cell reaction.

🔋 Application to a Zn–Cu Galvanic Cell

Consider the Daniell cell:

Zn(s) | Zn²⁺(aq) || Cu²⁺(aq) | Cu(s)

Overall reaction:

Zn + Cu²⁺ → Zn²⁺ + Cu

Here, two electrons are transferred:

n = 2

Therefore:

Q = [Zn²⁺]/[Cu²⁺]

Hence, at 298 K:

E_cell = E°_cell − (0.0591/2) log ([Zn²⁺]/[Cu²⁺])

✅ Final Derivation

Starting from:

ΔG = ΔG° + RT ln Q

Using:

ΔG = −nFE      and      ΔG° = −nFE°

we obtain:

E_cell = E°_cell − (RT/nF) ln Q

At 298 K:

E_cell = E°_cell − (0.0591/n) log Q

E = Cell potential under non-standard conditions

E° = Standard cell potential

R = Gas constant = 8.314 J mol⁻¹ K⁻¹

T = Absolute temperature in kelvin

n = Number of electrons transferred

F = Faraday constant ≈ 96485 C mol⁻¹

Q = Reaction quotient

Q. Write the Nernst equation for a galvanic cell.

Answer:

E_cell = E°_cell − (RT/nF) ln Q

At 298 K:

E_cell = E°_cell − (0.0591/n) log Q

Q. Derive the Nernst equation briefly.

1. For an electrochemical reaction: ΔG = ΔG° + RT ln Q.
2. Using ΔG = −nFE and ΔG° = −nFE°.
3. Substitution and rearrangement give: E = E° − (RT/nF) ln Q.

Q. Derive the Nernst equation at 298 K.

ΔG = ΔG° + RT ln Q

−nFE = −nFE° + RT ln Q

E = E° − (RT/nF) ln Q

At 298 K, converting ln to log:

E = E° − (0.0591/n) log Q

Q. Derive the Nernst equation for a galvanic cell and explain the terms involved.

For a general cell reaction:

aA + bB → cC + dD

The reaction quotient is:

Q = [C]^c[D]^d / [A]^a[B]^b

Thermodynamically:

ΔG = ΔG° + RT ln Q

For an electrochemical cell:

ΔG = −nFE
ΔG° = −nFE°

Therefore:

E = E° − (RT/nF) ln Q

At 298 K:

E = E° − (0.0591/n) log Q

Q. Derive the Nernst equation for a galvanic cell under non-standard conditions. Explain its significance.

Step 1: Gibbs Energy Relation

ΔG = ΔG° + RT ln Q

Step 2: Relation with Cell Potential

For a galvanic cell:

ΔG = −nFE

Under standard conditions:

ΔG° = −nFE°

Step 3: Substitute

−nFE = −nFE° + RT ln Q

Step 4: Rearrange

E = E° − (RT/nF) ln Q

Step 5: At 298 K

E = E° − (0.0591/n) log Q

Significance

• It calculates cell potential under non-standard conditions.
• It shows how ion concentration affects cell potential.
• It can be used to determine equilibrium conditions.
• It is useful for calculating the potential of individual electrodes.

Conclusion:

The Nernst equation connects the standard cell potential with the actual cell potential at a specified temperature and composition.

1. The Nernst equation is used to calculate:

A) Cell potential under non-standard conditions

B) Atomic mass

C) Boiling point

D) Density

✅ Answer

A

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2. Which equation is correct?

A) E = E° + RT ln Q/nF

B) E = E° − RT ln Q/nF

C) E = E°Q

D) E = E° + Q

✅ Answer

B

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3. At 298 K, the Nernst equation contains the factor:

A) 0.0591/n

B) 0.591n

C) 59.1n

D) 9.81/n

✅ Answer

A

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4. In the Nernst equation, n represents:

A) Number of ions

B) Number of electrons transferred

C) Number of atoms

D) Number of solutions

✅ Answer

B

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5. F in the Nernst equation represents:

A) Force constant

B) Faraday constant

C) Free energy

D) Frequency

✅ Answer

B

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6. The approximate value of Faraday constant is:

A) 8.314 C mol⁻¹

B) 96485 C mol⁻¹

C) 22.4 C mol⁻¹

D) 6.022 × 10²³ C mol⁻¹

✅ Answer

B

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7. Q in the Nernst equation represents:

A) Heat

B) Reaction quotient

C) Charge only

D) Volume

✅ Answer

B

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8. For a reaction aA + bB → cC + dD, Q is:

A) [A][B]/[C][D]

B) [C]ᶜ[D]ᵈ/[A]ᵃ[B]ᵇ

C) [A]ᵃ[B]ᵇ

D) [C][D]

✅ Answer

B

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9. Pure solids are generally:

A) Included in Q

B) Not included in Q

C) Always numerator

D) Always denominator

✅ Answer

B

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10. The relation between Gibbs energy and cell potential is:

A) ΔG = nFE

B) ΔG = −nFE

C) ΔG = RT

D) ΔG = E/nF

✅ Answer

B

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11. Under standard conditions:

A) ΔG° = −nFE°

B) ΔG° = nFE°

C) ΔG° = RT

D) ΔG° = Q

✅ Answer

A

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12. A galvanic cell converts:

A) Electrical energy into chemical energy spontaneously

B) Chemical energy into electrical energy

C) Heat into mass

D) Light into chemical energy only

✅ Answer

B

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13. If Q = 1, the Nernst equation gives:

A) E = E°

B) E = 0

C) E = 2E°

D) E = Q

✅ Answer

A

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14. At equilibrium, the cell potential of a reversible galvanic cell is:

A) Maximum

B) Zero

C) Negative infinity

D) Always 1 V

✅ Answer

B

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15. At equilibrium, the reaction quotient is equal to:

A) 0

B) 1

C) K

D) nF

✅ Answer

C) K

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16. For the Daniell cell Zn + Cu²⁺ → Zn²⁺ + Cu, n equals:

A) 1

B) 2

C) 3

D) 4

✅ Answer

B) 2

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17. For the Daniell cell, Q is:

A) [Cu²⁺]/[Zn²⁺]

B) [Zn²⁺]/[Cu²⁺]

C) [Zn][Cu]

D) [Zn²⁺][Cu²⁺]

✅ Answer

B

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18. Increasing the concentration of products generally makes Q:

A) Increase

B) Decrease

C) Zero

D) Always equal to 1

✅ Answer

A

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19. The Nernst equation is derived using the relation:

A) ΔG = ΔG° + RT ln Q

B) PV = nRT only

C) E = mc²

D) q = mcΔT

✅ Answer

A

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20. The Nernst equation is important because it relates cell potential to:

A) Concentration/activity of reacting species

B) Colour of the solution

C) Atomic radius

D) Melting point only

✅ Answer

A

🎯 Quick Revision

General equation: E = E° − (RT/nF) ln Q
At 298 K: E = E° − (0.0591/n) log Q
ΔG: ΔG = −nFE
Standard ΔG: ΔG° = −nFE°
Q: Reaction quotient
n: Number of electrons transferred
F: Faraday constant