Nernst Equation

Electrochemistry is the study of the relationship between electrical energy and chemical reactions. It deals with the transfer of electrons between substances and their chemical reactions. This subject provides the basis for understanding the behaviour of batteries, fuel cells, and other types of energy storage and energy-conversion systems.

Nernst equation is a fundamental equation in electrochemistry that predicts the voltage at which an electrochemical reaction takes place. It is used to calculate the standard electrode potential of an electrochemical cell and the voltage of a galvanic cell under non-standard conditions. Nernst equation is an important tool for understanding the behaviour of batteries and other electrochemical devices.

Nernst Equation

Nernst Equation is given by,

E = E° – (RT/nF) × ln(Q)

where
E is observed electrode potential
E° is the standard electrode potential
R is the gas constant
T is the temperature in Kelvin
n is the number of electrons transferred in the reaction
F is Faraday’s constant
ln(Q) is the natural logarithm of the reaction quotient

Derivation of Nernst Equation

The Nernst equation relates the standard electrode potential to the observed electrode potential and reaction quotient in an electrochemical cell. The equation is E = E° – (RT/nF) × ln(Q), where E is the observed electrode potential, E° is the standard electrode potential, R is the gas constant, T is the temperature in Kelvin, n is the number of electrons transferred in the reaction, F is Faraday’s constant, and ln(Q) is the natural logarithm of the reaction quotient. 

The Nernst equation is used to calculate the voltage produced by a galvanic cell, the standard electrode potential of an electrochemical cell, the reaction quotient, and the temperature required to increase the observed electrode potential to a certain value.

Gibbs free energy of the cell reaction can be expressed as,

ΔG = ΔG° + RT × ln(Q)

Standard Gibbs free energy change (ΔG°) can be related to the standard electrode potential (E°) as follows,

ΔG° = -nFE°

Substituting ΔG° in the expression for ΔG,

ΔG = -nFE° + RT × ln(Q)

Solving for the observed electrode potential (E),

E = ΔG / (-nF)

E = -nFE° / nF + RT × ln(Q) / (-nF)

E = E° – (RT / nF) × ln(Q)

Equilibrium Constant with Nernst Equation

The equilibrium constant (K) can be determined using the Nernst equation, which relates the standard electrode potential (E°) to the actual potential (E) and the concentration of the reactants and products in a cell. The equation can be written as follows:

E = E° – (RT/nF) × lnQ

where 

R is the gas constant, 
T is the temperature in kelvin, 
n is the number of electrons involved in the reaction, 
F is the Faraday constant,
Q is the reaction quotient. 

By rearranging the equation, we can calculate the equilibrium constant:

Q = e^{((E° – E)/(RT/nF))}

Applications of Nernst Equation

Various application of the Nernst Equation is,

• Electrochemistry: The Nernst equation is commonly used in electrochemistry to calculate the potential of an electrode in a cell and to determine the concentration of reactants and products in a reaction.
• Biochemistry: The Nernst equation can be used in biochemistry to calculate the potential of ion channels in biological membranes and to understand the transport of ions across membranes.
• Analytical Chemistry: The Nernst equation can be used in analytical chemistry to calculate the concentration of ions in a solution, such as the concentration of hydrogen ions in an acid-base titration.

Limitations of Nernst Equation

The Limitations of Nernst Equation are discussed below,

• Nernst equation is based on the assumption of ideal conditions, such as constant temperature and pressure, which may not always be the case in real-world applications.
• Nernst equation assumes that the electrode potential remains constant, which is not always the case in practical applications due to the effects of electrode polarization.
• Nernst equation is only applicable to reactions involving a limited number of ions and is not applicable to reactions involving complex species or to reactions involving multiple electron transfer processes.

Solved Example on Nernst Equation

Example 1: A hydrogen electrode is immersed in a hydrogen sulfate solution of 2 M at 50°C. Calculate the potential of the hydrogen electrode, given the standard electrode potential of H+/H₂, is 0.000 V.

Solution:

E = E° – (RT/nF) × ln(Q)

Using RT/F = 0.059 V and n = 2, the equation becomes:
E = 0.000 V – (0.059 V × (323 K) / 2) × ln([H₂SO₄]/2)
E = 0.000 V – (0.0295 V) × ln(2) = 0.000 V – 0.0192 V

So, the potential of the hydrogen electrode in a hydrogen sulfate solution of 2 M at 50°C is -0.0192 V.

Example 2: An electrochemical cell consists of a zinc electrode in a zinc sulfate solution and a silver electrode in a silver nitrate solution. The standard electrode potentials are -0.76 V and +0.80 V, respectively. Calculate the potential of the cell at 40°C.

Solution:

E = E° (Ag+/Ag) – E° (Zn²⁺/Zn) – (RT/nF) × ln(Q)

Using RT/F = 0.059 V and n = 2, the equation becomes
E = +0.80 V – (-0.76 V) – (0.059 V × (313 K) / 2) × ln(Q)
E = 1.56 V – (0.0295 V) × ln(Q)

The value of Q can be calculated from the balanced half-cell reactions
Zn(s) → Zn²⁺(aq) + 2e^– (at the zinc electrode)
Ag⁺(aq) + 1e^– → Ag(s) (at the silver electrode)

For a balanced cell, the reaction quotient should equal 1, so
1 = [Ag⁺] [Zn²⁺]/ [Ag]

Rearranging and substituting the value of Q into the Nernst equation, we get

E = 1.56 V – (0.0295 V) × ln(1) 
   = 1.56 V

So, the potential of the electrochemical cell at 40°C is 1.56 V.

Example 3: A cadmium electrode is immersed in a cadmium nitrate solution of 0.1 M at 25°C. Calculate the potential of the cadmium electrode, given the standard electrode potential of Cd²⁺/Cd, is -0.403 V.

Solution:

E = E° – (RT/nF) × ln(Q)

Using RT/F = 0.059 V and n = 2, the equation becomes
E = -0.403 V – (0.059 V × (298 K) / 2) × ln([Cd(NO₃)₂]/0.1)
E = -0.403 V – (0.0295 V) × ln(0.1) 
   = -0.403 V – (-0.2302 V)

So, the potential of the cadmium electrode in a cadmium nitrate solution of 0.1 M at 25°C is -0.17 V.

Example 4: An electrochemical cell consists of a copper electrode in a copper nitrate solution and a gold electrode in a gold chloride solution. The standard electrode potentials are +0.34 V and +1.69 V, respectively. Calculate the potential of the cell at 15°C.

Solution:

E = E° (Au⁺/Au) – E° (Cu²⁺/Cu) – (RT/nF) × ln(Q)

Using RT/F = 0.059 V and n = 2, the equation becomes:
E = +1.69 V – (+0.34 V) – (0.059 V × (288 K) / 2) × ln(Q)
E = 1.35 V – (0.0295 V) × ln(Q)

The value of Q can be calculated from the balanced half-cell reactions:
Cu²⁺(aq) + 2e^– → Cu(s) (at the copper electrode)
Au³⁺(aq) + 3e^– → Au(s) (at the gold electrode)

For a balanced cell, the reaction quotient should equal 1, so:
1 = [Au³⁺]³/ [Cu²⁺]² [Au]

Rearranging and substituting the value of Q into the Nernst equation, we get:
E = 1.35 V – (0.0295 V) × ln(1) 
   = 1.35 V

So, the potential of the electrochemical cell at 15°C is 1.35 V.

Example 5. A magnesium electrode is immersed in a magnesium sulfate solution of 0.05 M at 10°C. Calculate the potential of the magnesium electrode, given the standard electrode potential of Mg²⁺/Mg, is -2.37 V.

Solution: 

E = E° – (RT/nF) × ln(Q)

Using RT/F = 0.059 V and n = 2, the equation becomes,
E = -2.37 V – (0.059 V × (263 K) / 2) × ln([MgSO₄]/0.05)
E = -2.37 V – (0.0295 V) × ln(0.05) 
   = -2.37 V – (-1.055 V)

So, the potential of the magnesium electrode in a magnesium sulfate solution of 0.05 M at 10°C is -1.315 V.

FAQs on Nernst Equation

Question 1: What is the Nernst equation used for?

Answer:

The Nernst equation is used to calculate the potential of an electrode in a cell and to determine the concentration of reactants and products in a reaction.

Question 2: What are the assumptions of the Nernst equation?

Answer:

The Nernst equation is based on the assumptions of constant temperature and pressure and the assumption that the electrode potential remains constant.

Question 3: How does the Nernst equation relate to the equilibrium constant?

Answer:

The Nernst equation can be used to calculate the equilibrium constant by rearranging the equation and calculating the reaction quotient.

Question 4: Is the Nernst equation only applicable to electrochemistry?

Answer:

No, the Nernst equation has applications in various fields, including biochemistry and analytical chemistry.

Question 5: What are the limitations of the Nernst equation?

Answer:

The limitations of the Nernst equation include the assumptions of ideal conditions and the constant electrode potential and its limited applicability to reactions involving a limited number of ions.

Question 6: How does the Nernst equation calculate the concentration of ions in a solution?

Answer:

The Nernst equation can be used to calculate the concentration of ions in a solution by relating the electrode potential to the concentration of the reactants and products in a cell.

Question 7: Can the Nernst equation be used for reactions involving multiple electron transfer processes?

Answer:

No, the Nernst equation is not applicable to reactions involving multiple electron transfer processes.