Premed · Premed · General Chemistry 2
Lecture 15: The Nernst Equation
General Chemistry II
Learning Objectives
By the end of this lecture, students will be able to:
- State the Nernst equation and explain each variable
- Calculate cell potential under nonstandard conditions
- Determine the effect of concentration changes on cell potential
- Explain and calculate the EMF of concentration cells
- Relate the Nernst equation to the equilibrium condition (dead battery)
- Describe how electrochemical measurements can determine pH and ion concentrations
Lecture Content
I. The Nernst Equation
The standard cell potential E^0_cell applies only under standard conditions of 1 M concentration, 1 atm pressure, and 25 degrees C. Under any other conditions, the actual cell potential is given by the Nernst equation: E_cell = E^0_cell - (RT / nF) ln(Q). Here R is the gas constant (8.314 J/(molK)), T is the temperature in Kelvin, n is the number of moles of electrons transferred in the balanced equation, F is Faraday's constant (96,485 C/mol), and Q is the reaction quotient.
At 25 degrees C (298 K), the equation simplifies to the convenient form E_cell = E^0_cell - (0.0592 / n) log(Q), or equivalently E_cell = E^0_cell - (0.02569 / n) ln(Q). The Nernst equation is derived from the fundamental relationships Delta G = Delta G^0 + RT ln(Q) and Delta G = -nFE, combining them to express cell potential as a function of the reaction quotient.
II. Effect of Concentration on Cell Potential
The Nernst equation reveals how concentration changes affect cell potential through the reaction quotient Q = [products]^coefficients / [reactants]^coefficients. When Q < 1, meaning reactants are in excess, ln(Q) is negative and E_cell exceeds E^0_cell. When Q > 1, meaning products are in excess, ln(Q) is positive and E_cell falls below E^0_cell. At standard conditions where Q = 1, E_cell equals E^0_cell. At equilibrium where Q = K, E_cell equals zero, corresponding to a "dead battery."
This behavior connects directly to Le Chatelier's principle. Increasing reactant concentrations drives the reaction forward, increasing the cell potential. Increasing product concentrations opposes the reaction, decreasing the cell potential.
<image>A graph of E_cell vs. ln(Q) for a galvanic cell. The y-axis shows E_cell (volts) and the x-axis shows ln(Q). A straight line with negative slope (-RT/nF) starts at E^0_cell when ln(Q) = 0 and decreases linearly. Key points are labeled: at ln(Q) = 0, E_cell = E^0_cell (standard conditions); at ln(Q) = ln(K), E_cell = 0 (equilibrium, dead battery). The region where Q < 1 shows E_cell > E^0_cell; the region where Q > 1 shows E_cell < E^0_cell. The region where E_cell < 0 indicates the reverse reaction would be spontaneous.</image>
III. Nernst Equation Calculations
Consider the zinc-copper cell with nonstandard concentrations: Zn(s) + Cu^2+(aq) -> Zn^2+(aq) + Cu(s), where E^0_cell = 1.10 V and n = 2. If [Cu^2+] = 0.010 M and [Zn^2+] = 2.0 M, then Q = [Zn^2+] / [Cu^2+] = 2.0 / 0.010 = 200. Applying the Nernst equation: E_cell = 1.10 - (0.0592/2) log(200) = 1.10 - 0.0296 2.30 = 1.10 - 0.068 = 1.03 V. The cell potential has decreased from the standard value because Q > 1, reflecting the excess of products.
As the cell operates, Cu^2+ is consumed and Zn^2+ accumulates, so Q continually increases and E_cell progressively decreases. Eventually Q reaches K and E_cell drops to zero, at which point the cell has reached equilibrium and can no longer do work.
IV. Concentration Cells
A concentration cell is a galvanic cell in which both half-cells contain the same electrode material and the same electrolyte, but at different concentrations. Because the half-reactions are identical, E^0_cell = 0, and the cell operates solely because of the concentration difference. The Nernst equation becomes E_cell = -(0.0592/n) log(Q) = -(0.0592/n) log([dilute]/[concentrated]). Since the dilute concentration is smaller than the concentrated one, Q < 1, log(Q) is negative, and E_cell is positive.
The cell drives a spontaneous process that equalizes the concentrations: metal dissolves on the dilute side (the anode, increasing the concentration there) and metal ions deposit on the concentrated side (the cathode, decreasing the concentration there). The cell operates until both sides reach the same concentration, at which point Q = 1 and E = 0.
For example, a copper concentration cell with [Cu^2+] = 0.010 M on one side and 1.0 M on the other has the dilute side as the anode and the concentrated side as the cathode. The cell potential is E_cell = -(0.0592/2) * log(0.010/1.0) = -(0.0296)(-2) = 0.059 V.
<image>A diagram of a copper concentration cell. Two beakers each contain a copper electrode in CuSO4 solution, connected by a salt bridge and an external wire through a voltmeter. The left beaker has [Cu^2+] = 0.010 M (labeled "dilute, ANODE") and the right beaker has [Cu^2+] = 1.0 M (labeled "concentrated, CATHODE"). Arrows show: Cu dissolving at the anode (increasing [Cu^2+] on the left), Cu^2+ depositing as Cu at the cathode (decreasing [Cu^2+] on the right), and electrons flowing left to right. The voltmeter reads a small positive voltage. The net effect is equalization of concentrations.</image>
V. Finding K from the Nernst Equation
At equilibrium, E_cell = 0 and Q = K. Substituting these conditions into the Nernst equation gives 0 = E^0_cell - (0.0592/n) log(K), which rearranges to log(K) = n E^0_cell / 0.0592 and therefore K = 10^(nE^0_cell / 0.0592).
For the zinc-copper cell with E^0 = 1.10 V and n = 2, log(K) = 2(1.10) / 0.0592 = 37.2, giving K = 10^37.2 = 1.6 x 10^37. This astronomically large equilibrium constant confirms that the reaction is overwhelmingly product-favored.
VI. Electrochemical Measurement of pH
Galvanic cells can be used to measure the pH of a solution. If one half-cell involves H+ ions, the Nernst equation establishes a direct relationship between the cell potential and [H+], and since pH = -log[H+], measuring the voltage effectively measures pH.
The glass electrode pH meter exploits this principle. It contains a thin glass membrane that is selectively sensitive to H+ concentration. The voltage generated across this membrane is proportional to the pH of the solution and is converted electronically to a pH reading. The instrument is calibrated using buffer solutions of known pH. Other ion-selective electrodes operate on similar principles and can measure the concentrations of specific ions such as F-, Ca^2+, and Na+.
VII. Batteries as Galvanic Cells
Batteries are practical applications of galvanic cells. Primary batteries are non-rechargeable and include alkaline batteries (Zn/MnO2, approximately 1.5 V) and lithium batteries (Li/MnO2, approximately 3.0 V). Secondary batteries are rechargeable, with the lead-acid battery being a familiar example. Each cell contains Pb and PbO2 electrodes in H2SO4 and produces approximately 2.0 V; six cells in series give the 12 V car battery. The discharge reaction, Pb + PbO2 + 2H2SO4 -> 2PbSO4 + 2H2O, is reversed during charging by applying an external voltage. Lithium-ion batteries, with lithium intercalated in graphite and metal oxide electrodes, produce approximately 3.7 V and dominate modern portable electronics.
Fuel cells are continuous-feed galvanic cells in which the reactants are supplied continuously rather than being sealed inside the cell. The hydrogen fuel cell, with the reaction 2H2 + O2 -> 2H2O and E^0 approximately 1.23 V, produces electricity as long as hydrogen and oxygen are supplied. Water is the only byproduct.
<image>A cross-section diagram of a hydrogen fuel cell. The left side shows the anode where H2 gas is supplied and oxidized (H2 -> 2H+ + 2e-). The right side shows the cathode where O2 gas is supplied and reduced (O2 + 4H+ + 4e- -> 2H2O). A proton exchange membrane (PEM) in the center allows H+ ions to pass from anode to cathode. Electrons travel through an external circuit from anode to cathode, powering a load (light bulb). Water is produced at the cathode and exits as the only byproduct. The overall reaction 2H2 + O2 -> 2H2O is labeled, along with E^0_cell approximately 1.23 V.</image>
