# The Nernst Equation

To determine the active hydrogen ion concentration, a pH measurement is necessary.

## pH Measurement Methods

Three methods are commonly used to determine the pH value of an aqueous solution:

**The Visual Method:**

A color comparison with pH sensitive indicator paper (litmus) to a standard color scale.

**The Photometric Method:**

Using a spectrophotometer to measure the wave-length of the pH sensitive colored solution.

**The Potentiometric Method:**

An electrochemical measurement, measuring the voltage created by a chemical reaction, such as that which takes place between metals and dissolved salts.

The potentiometric method for determining the pH value of aqueous solutions is the most frequent and reliable design for process control applications therefore this article will deal exclusively with this method for further pH theory.

## Potentiometric (Electrochemical) Sensor Design

This pH measurement method is based on the Nernst equation which describes in a relatively simple form the relationship between the galvanic potential of a defined electrode assembly and the chemical activity of the ion concentration being measured.

A pH electrode assembly always consists of a measuring electrode which is sensitive to the ion activity to be measured and a reference electrode. The operation of an electrode assembly in its simplest form is demonstrated by the following example:

If two hydrogen electrodes (each a thin plate of polished platinum) are immersed in two solutions having different hydrogen ion concentrations, each electrode will generate a potential voltage which depends on the active hydrogen concentration of the solution in which the electrode is immersed. To measure this voltage potential, the two solutions are connected by a salt bridge (electrolyte bridge) and the two electrodes are connected to a high impedance voltmeter. Both solutions are saturated with pure hydrogen gas.

Since the current passing during such a measurement is negligible, the chemical composition of the sample solution is not altered. The bridge acts as a phase boundary between solution C_{1} and solution C_{2} and completes the electric circuit.

## The Nernst Equation

A voltage difference will be generated between the two platinum electrodes by the different active hydrogen ion concentrations in the solutions. The relationship is expressed by the Nernst equation shown below:

**pH Equation Variables**

E = potential difference (mV)

R = gas constant (8.31439 J x mol^{-1} x K^{-1})

F = Faraday constant (96495.7 °C x mol^{-1})

T = absolute temperature in Kelvin (K)

n = charge number of the measured ion (in this case n_{H} = 1)

C_{1} = active Hydrogen ion concentration in solution C_{1}

C_{2} = active Hydrogen ion concentration in solution C_{2}

If we select between C_{1} and C_{2} a concentration ratio of 10:1, the equation may be written as:

Variable E is known as the Nernst’s potential and given the symbol U_{N}. U_{N} corresponds to the change in potential with each ten-fold change in activity. The values of R and F are constant. The charge number n is specific for each type of ion and the temperature T can be calculated from the measured value in °C.

If we assume the temperature of the solutions to be 25°C, then:**T = 273.15 + 25 = 298.15 Kelvin**

This will give us the Nernst potential of:

Since the ion activity is temperature dependent, so is the Nernst potential (refer to the nernst equation). The following table illustrates the temperature dependency:

T °C |
U |

0 |
54.20 |

5 |
55.19 |

10 |
56.18 |

15 |
57.17 |

20 |
58.16 |

25 |
59.16 |

30 |
60.15 |

T °C |
U |

35 |
61.14 |

40 |
62.13 |

45 |
63.12 |

50 |
64.12 |

55 |
65.11 |

60 |
66.10 |

65 |
67.09 |

T °C |
U |

70 |
68.08 |

75 |
69.08 |

80 |
70.07 |

85 |
71.06 |

90 |
72.05 |

95 |
73.04 |

100 |
74.04 |

*Table: Temperature Dependency of the Nernst Potential.*

Due to the relationship of temperature and the Nernst potential it is common that modern combination pH sensor designs integrate a resistance temperature sensor within the sensor to compensate the Nernst potential.

Prior Article - The pH Scale

Next Article - The Ion and Dissociation

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