1.14.18: Electrochemical Units

Electric Current

The SI base electrical unit is the AMPERE which is that constant electric current which if maintained in two straight parallel conductors of infinite length and of negligible circular cross section and placed a metre apart in a vacuum would produce between these conductors a force equal to \(2 \times 10^{-7}\) newton per metre length. It is interesting to note that definition of the Ampere involves a derived SI unit, the newton. Except in certain specialised applications, electric currents of the order ‘amperes’ are rare. Starter motors in cars require for a short time a current of several amperes.

When a current of one ampere passes through a wire about \(6.2 \times 10^{18}\) electrons pass a given point in one second [1,2].

The coulomb (symbol \(\mathrm{C}\)) is the electric charge which passes through an electrical conductor when an electric current of one \(\mathrm{A}\) flows for one second. Thus

\[[\mathrm{C}]=[\mathrm{As}]\]

Electric Potential

In order to pass an electric current thorough an electrical conductor a difference in electric potential must exist across the electrical conductor. If the energy expended by a flow of one ampere for one second equals one Joule the electric potential difference across the electrical conductor is one volt [3].

Electrical Resistance and Conductance

If the electric potential difference across an electrical conductor is one volt when the electrical current is one ampere, the electrical resistance is one ohm, symbol \(\Omega\) [4]. The inverse of electrical resistance , the conductance, is measured using the unit siemens, symbol [S]. [4]

Ohm’s Law

This famous phenomenological law describes the ability of a system to conduct electrical charge. This law describes the relationship between three properties of an electrical conductor; e.g. a salt solution. The three properties are

1. electric current \(\mathrm{I}\) described using the SI unit , ampere, symbol \(\mathrm{A}\);
2. electric potential difference \(\mathrm{V}\) across the electrical conductor using the SI unit , volt, symbol \(\mathrm{V}\); and

Ohm’s Law is a phenomenological law in that it describes the phenomenon of electrical conductivity. Unlike the laws of thermodynamics, there are exceptions to Ohm’s law for very high voltages and high frequency alternating electric currents. The (electrical) conductance \(\mathrm{G}\) of a system is given by the inverse of its resistance \(\mathrm{R}\). The conductivity \(\kappa\) (\(\equiv \sigma\)) of a system is given by equation (c) [5].

\[\mathrm{j}=\kappa \, \mathrm{E}\]

Here \(\mathrm{j}\) is the electric current density and \(\mathrm{E}\) is the electric field strength. Chemists prefer to think in terms the charge carrying properties of a given system; i.e. the conductance \(\mathrm{G}\) (\(=1 / \mathrm{R}\)) using the unit siemens, symbol S [5].

An interesting contrast often emerges between chemists and physicists, the latter seem to emphasise the property of ‘resistance’ whereas chemists are more interested in how systems transport electrical charge. Certainly the classic subject in chemistry concerns the electrical conductivities of salt solutions where the charge carriers are ions. The subject is complicated by the fact that there are two types of charge carriers in a given solution, cations and anions. Moreover the subject is further complicated by the fact that these charge carriers move in opposite directions with different velocities. In a solution containing a single salt the fraction of electric current carried by the cations and anions are called transport numbers; i.e. \(\mathrm{t}_{+}\) and \(\mathrm{t}_{-}\) respectively for cations and anions where \(\mathrm{t}_{+}+\mathrm{t}_{-}=1\).

Ionic Mobility

A given aqueous salt solution, volume \(\mathrm{V}\), is prepared using \(\mathrm{n}_{1}\) moles of water and \(\mathrm{n}_{j}\) moles of a salt \(j\). A pair of electrodes is placed in the solution, \(\mathrm{d}\) metres apart. An electric potential, \(\mathrm{V}\) volts, is applied across the solution and an electric current \(\mathrm{I}\) is recorded. An electric charge \(\mathrm{Q}\) coulombs (\([\mathrm{A s}]\)) is passed through the solution. The electric current (unit \([\mathrm{A}]\)) is the rate of transport of charge, \(\mathrm{dQ} / \mathrm{dt}\).

The speed of ion \(\mathrm{i}\) through the solution \(\mathrm{v}_{\mathrm{i}}\) is given by the ratio of the distance travelled to the time taken. Thus \(\mathrm{v}_{\mathrm{i}}\) is a measure of the distance travelled in one second; \(\mathrm{v}=(\mathrm{a} / \mathrm{t})\left[\mathrm{ms}^{-1}\right]\). A more interesting property is the velocity of ion \(\mathrm{i}\) in an electric field gradient measured using the ratio, volt/metre (Or, \(\mathrm{V} / \mathrm{m}\)). Thus electric mobility \(\mathrm{u}_{\mathrm{i}}\) has the unit, \(\left[\mathrm{m} \mathrm{s}^{-1} / \mathrm{V} \mathrm{m} \mathrm{m}^{-1}\right]=\left[\mathrm{v}_{\mathrm{i}} \mathrm{E}^{-1}\right]=\left[\mathrm{m}^{2} \mathrm{~s}^{-1} \mathrm{~V}^{-1}\right]\).

The molar conductivity \(\Lambda\) of a salt solution is given by the ratio, \(\left(\kappa / c_{j}\right)\left\{=\left[S \mathrm{~m}^{2} \mathrm{~mol}^{-1}\right]\right\}\). In fact the majority of research publications describe the dependence of \(\Lambda\) on the concentration of salt in a specified solvent at defined \(\mathrm{T}\) and \(\mathrm{p}\). For salt solutions both cations and anions contribute to \(\Lambda\); the transport number of an ion \(j\) describes the fraction of current carried by the \(j\) ion. Thus \(\mathrm{t}_{\mathrm{j}}=\left|\mathrm{z}_{\mathrm{j}}\right| \, \mathrm{c}_{\mathrm{j}} \, \mathrm{v}_{\mathrm{j}} / \sum\left|\mathrm{z}_{\mathrm{i}}\right| \, \mathrm{c}_{\mathrm{i}} \, \mathrm{v}_{\mathrm{i}}\).

An electric current (i.e. a flow of electric charge) through a system is impeded by the electrical resistance. The ohm (symbol \(\Omega\)) is the unit of electrical resistance being the ratio of electric potential (unit = volt) to electric current (unit = ampere) Then, ohm = volt/ampere (d)

In other words, Electrical resistance/ohm=[electric potential gradient /volt]/[electric current/ampere] . From equation (c) ohm (symbol \(\Omega\))

\[=\mathrm{V} \mathrm{} \mathrm{A}^{-1}=\mathrm{m}^{2} \mathrm{~kg} \mathrm{~s}^{-3} \mathrm{~A}^{-2}\]

The property (electrical) resistance is a measure of the impedance to the flow of electric charge. This somewhat negative outlook is not consistent with the attitude of chemists who are interested in the ‘mechanism’ by which a system conducts electrical charge. Chemists prefer to discuss the property of electric conductance rather than resistance. The conductance is measured using the unit siemen, symbol S [6]. A key component of electrical circuits is the electrical capacitance measured using the unit farad, symbol \(\mathrm{F}\) [7].

If the composition of an electrical conductor is uniform, the electrical resistance is directly proportional to its length \(\ell\) and inversely proportional to its cross sectional area, \(\mathrm{a}\). The material forming the conductor is characterized by its resistivity, \(\rho\). Thus[8] resistance,

\[\mathrm{R}=\rho \, \ell / \mathrm{a}\]

Footnotes

[1] The electric charge on an electron \(=1.602 \times 10^{-19} \mathrm{C}=1.602 \times 10^{-19} \mathrm{~As}\). A given single wire carries a current of 1 ampere.
Then in one second, the electric charge carried by that wire \(= 1.602 \times 10^{-19} \mathrm{~C}\)
For one coulomb to pass a given point, the number of electrons passing \(=\frac{1}{1.602 \times 10^{-19}}\)
In other words \(6.24 \times 10^{18}\) electrons pass by.

[2] P.W. Atkins and L. Jones, Chemistry; Molecules, Matter and Change, W. H. Freeman, New York, 1997,p.658.

[3] \([\mathrm{V}] \equiv\left[\mathrm{kg} \mathrm{m}{ }^{2} \mathrm{~s}^{-3} \mathrm{~A}^{-1}\right]=\left[\mathrm{J} \mathrm{A}^{-1} \mathrm{~s}^{-1}\right]\)

[4] \([\Omega]=\left[\mathrm{VA}^{-1}\right]=\left[\mathrm{S}^{-1}\right]\)

[5] \(\begin{aligned}
&\kappa=\left[\mathrm{S} \mathrm{m}^{-1}\right]\\
&\left[\mathrm{A} \mathrm{} \mathrm{m}^{-2}\right]=\kappa \,\left[\mathrm{V} \mathrm{} \mathrm{m}^{-1}\right]
\end{aligned}\)

[6] \(\mathbf{S}=\Omega^{-1}\)

[7] \(\mathrm{F}=\mathrm{A}^{2} \mathrm{~s}^{4} \mathrm{~kg}^{-1} \mathrm{~m}^{-2}=\mathrm{As} \mathrm{} \mathrm{V}^{-1}=\mathrm{C} \mathrm{V}^{-1}\)

[8] \(\begin{gathered}
{[\mathrm{S}] \equiv\left[\Omega^{-1}\right] \equiv\left[\mathrm{A} \mathrm{V}^{-1}\right] \equiv\left[\mathrm{m}^{-2} \mathrm{~kg}^{-1} \mathrm{~s}^{3} \mathrm{~A}^{2}\right]} \\
{[\Omega]=[\Omega \mathrm{m}] \,[\mathrm{m}] \,[\mathrm{m}]^{-2}}
\end{gathered}\)