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The Nernst Equation

Key Concepts

The Nernst Equation allows us to calculate the voltage produced by any electrochemical cell given E^o values for its electrodes and the concentrations of reactants and products.
The general Nernst Equation: E = E^o -(RT/nF)lnQ
E^o = standard electrochemical cell potential (voltage)
R = ideal gas constant
T = temperature
n = moles of electrons
F = Faraday constant = 96,485 C mol^-1
Q = mass-action expression (approximated by the equilibrium expression)
At 25^oC the Nernst Equation is simplified to: E = E^o -(0.0592/n)logQ
E^o = standard electrochemical cell potential (voltage)
n = moles of electrons
Q = mass-action expression (approximated by the equilibrium expression)
For a system at equilibrium at 25^oC, E = 0 and the Nernst Equation is simplified to:
E^o = (0.0592/n)logK
E^o = standard electrochemical cell potential (voltage)
n = moles of electrons
K = equilibrium expression

Example of E Calculation

Calculate the voltage produced by the cell Sn_(s)|Sn²⁺||Ag⁺|Ag_(s) at 25^oC given:
[Sn²⁺] = 0.15 M
[Ag⁺] = 1.7 M

Write the Nernst Equation for 25^oC:
E = E^o -0.0592/n x logQ

Calculate E^o for the cell:

anode:	Sn_(s)	→	Sn²⁺ + 2e	E^o = -0.14V
cathode:	2x [e^- + Ag⁺	→	Ag_(s)]	E^o = +0.80V

cell:	Sn_(s) + 2Ag⁺	→	Sn²⁺ + 2Ag_(s)	E^o = +0.66V

Wite the expression for Q:
Q = [Sn²⁺]/[Ag⁺]²
(concentrations of solids = 1)
Write the Nernst Equation for this example:
E = E^o -0.0592/n x log([Sn²⁺]/[Ag⁺]²)
Substitute the values:
E^o = +0.66V
n = 2 (2 moles of electrons transferred during the redox reaction)
[Sn²⁺] = 0.15 M
[Ag⁺] = 1.7 M
E = +0.66 -0.0592/2 x log([0.15]/[1.7]²)
Calculate Q:
E = +0.66 -0.0592/2 x log[0.0519]
Calculate logQ:
E = +0.66 -0.0592/2 x -1.285
Calculate E:
E = +0.66 -0.0592/2 x -1.285
E = +0.70V

E > E^o, and positive, so the cell reaction has a greater tendency to take place at these concentrations.

Example of K Calculation

Calculate the equilibrium constant, K, for the reaction Sn_(s)|Sn²⁺||Ag⁺|Ag_(s) at 25^oC.

Write the Nernst Equation: E = E^o -(0.0592/n)logQ
At equilibrium Q = K and E = 0
Write the Nernst Equation for the equilibrium expression: E^o = (0.0592/n)logK
Calculate E^o for the equation (as above) = +0.66
Calculate n (moles of electrons transferred) =2
Substitute E^o and n into the equation:
+0.66 = (0.0592/2)logK
Calculate logK:
+0.66 = 0.0296logK
logK = 22.3
Calculate K:
K = 10^22.3 = 1.98 x 10²²

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