Download Topic0990 Electrical Units In attempting to understand the properties

Topic0990
Electrical Units
In attempting to understand the properties of chemical substances, chemists divide
chemistry into two parts. In one part, chemists are interested in understanding
intramolecular forces which hold molecules together. For example, using quantum
mechanics and associated theories of covalent bonding, chemists describe the
cohesive forces holding carbon, hydrogen and nitrogen atoms together in
cyanomethane, CH3CN. At ambient temperature and pressure, cyanomethane is a
liquid. In the second part of the sub-division of chemistry, chemists describe the
intermolecular forces [1] which hold assemblies of molecules together in, for
example, liquid and solid states; e.g. those forces which hold CH3CN molecules
together in the liquid state. Common experience tells us that intermolecular forces are
weaker than intramolecular forces. When we heat CH3CN( l ) at ambient pressure,
the liquid boils at a characteristic temperature to form a vapour. The intermolecular
separation dramatically increases but the covalent bonds within CH3CN do not break.
[Of course, these bonds break at very high temperatures - thermolysis.] Here the
emphasis centres on intermolecular cohesion. But this cannot be the whole story. If
cohesion is the only force operating, molecules would collapse into each other in
some nuclear catastrophe. This does not happen. Opposing the forces of cohesion
are repulsive forces. In fact everyday experience leads to the idea of "size"; 'size is
repulsive'.
Basic Physics
Molecules contain charged particles; protons (with positive electric charge) and
electrons (with negative electric charge-- by convention). Intermolecular forces are
understandable in terms of equations describing electrical interactions between
electrically charged particles. An SI base unit is the ampere; symbol = A [2].
The SI unit of electric charge is the coulomb (symbol = C) defined as A s [3].
Electric Current
An electric current I is driven through an electrical resistance R, by an electric
potential gradient across the resistance. An ammeter measures the electric current I.
The voltmeter records the electric potential gradient, ∆E across the resistance. The
property called resistance R is given by Ohm’s Law;
∆E = I ⋅ R .
(a)
In the IUPAC system the unit of resistance is ohm [symbol Ω ≡ V A −1 ]. The
electric potential difference is measured in volts, symbol V [4].
Electrical Capacitance
In a simple electric circuit, a small battery is connected across a parallel plate
capacitance. No current flows in this circuit. The battery produces a set of equal in
magnitude but opposite in sign electric charges on the two plates. A capacitance stores
electric charge. In practice the extent to which a capacitance stores charge depends on the
chemical substance between the two plates. This substance is characterised by its electric
permittivity; symbol = ε . Where a vacuum exists between the two plates, the electric
permittivity equals ε0 [5].
The permittivity of a liquid is measured by comparing capacitance C when the
gap between the plates is filled with this liquid and with capacitance C0 when the gap
is "in vacuo".
Then
or
ε r = ε / ε 0 = C / C0
(b)
C = ε r ⋅ Co
(c)
For all substances, εr is greater than unity. In other words, with increase in εr so the
electrical insulating properties of the system increase. At this stage, we have not
offered a molecular explanation of the properly called εr but we have indicated that
εr be can measured [6,7].
Intermolecular Forces and Energies
Molecule i and molecule j are separated by a distance r; we assert that r >> molecular
radii of molecules i and j. Our discussion centres on the assertion that a force (symbol
X) exists between the two molecules. Moreover, this force depends on the distance
of separation r.
Thus
X = f(r)
(d)
Ion-Interactions
The force X between two electric charges q1 and q2 distance r apart ‘in vacuo’ is
given by equation (e); (Couloumb's Law) [8].
X = q1 ⋅ q 2 / 4 ⋅ π ⋅ ε 0 ⋅ r 2
(e)
Two ions, i and j, have charge numbers zi and zj respectively [for K +, zj = + 1; for
SO42- , zj = -2]. For two ions ‘in vacuo’, the interionic force is given by equation (f).
Fij = ( z i e ) ⋅ ( z j e) / 4π ⋅ ε 0 ⋅ r 2
(f)
r
But pairwise potential energy, U ij = − ∫ Fij ⋅ dr
(g)
r =∞
Hence, U ij = ( z i e ) ⋅ ( z j e ) / 4π ⋅ ε 0 ⋅ r
(h)
Equation (h) yields the interaction potential energy between a pair of ions [9]. The
result is an energy expressed in joules. However, there are often advantages in
considering an Avogadro number (i.e. a mole) of such pairwise interactions.
U ij / J mol −1 = N A ⋅ ( z i e ) ⋅ ( z j e ) / 4π ⋅ ε 0 ⋅ r
(i)
We consider two classes of ion-ion interactions:
(i) Ions i and j have the same sign
For cation-cation and anion-anion pairwise interactions the force between the ions is
repulsive. The pairwise potential energy increases with decrease in ion-ion
separation. To bring two ions having the same charge closer together we have to do
work on the system, increasing the pairwise potential energy Uij.
(ii) Ions i and j have opposite signs
For this system, ( z i ⋅ z j ) < 0 . Ion-ion interaction is attractive and the potential energy
Uij decreases with decrease in rij. We write z i ⋅ z j to indicate the modulus of the
product of the charge numbers.
U ij / J mol −1 = − N A ⋅ z i ⋅ z j e 2 / 4π ⋅ ε 0 ⋅ r
(j)
Hence Uij has a (1/r) dependence on distance apart.
Electric field strength, E is the force exerted on unit charge at the point in
question [10]. At distance r from charge q,
E = q / 4π ⋅ ε 0 ⋅ r 2
(k)
Solvent Effects
An important topic in Chemistry concerns the effect of solvents on ion-ion
interactions. Here we assume that solvents are characterised by their relative
permittivities, εr. In a solvent the pairwise cation-anion interaction energy is given by
equation (l) .
U ij / J = − z i ⋅ z j e 2 / 4π ⋅ ε 0 ⋅ ε r ⋅ r
U ij / J mol -1 = − z i ⋅ z j ⋅ N A ⋅ e 2 / 4π ⋅ ε 0 ⋅ ε r ⋅ r
(l )
(m)
As commented above, εr is always greater than unity. Hence for a given system at
fixed distance apart r, Uij increases (becomes less negative) with increase in εr. With
increase in εr, the ions are increasingly insulated and so at given distance r the
stabilisation of the cation-anion pair is less marked.
Molecular Dipole Moments
A given molecule comprises an assembly of positive and negative charges. Consider
a point 0, distance r from this assembly. We are concerned with the electric field
strength at point 0, a short distance from the dipole moment. In the previous section
we assumed that this assembly is simply characterised by the electric charge (i.e.
z j ⋅ e for ion j). However, in those cases where the overall charge is zero, a
measurable electric field is detected at point 0. In 1912 Peter Debye showed that this
field could be accounted for as a first approximation by characterising a molecule by
its dipole moment. In the next approximation the electric field at 0 can also be
accounted by an additional contribution from a distribution of charges within a
molecule called a quadrupole, and in the next approximation by an additional
contribution from a distribution called an octupole [11].
In a homonuclear diatomic molecule such as H2 and Cl2, the positive nuclei are
embedded in charge clouds describing the distribution of negatively charged
electrons. For such molecules the "centres" of positive charges and negative charges
coincide. But for the molecule HCl the electron distribution favours the more
electronegative chlorine atom. Hence the centres of positive and negative charges,
magnitude +q and -q, are separated by a dipole length l . The molecule has a dipole
moment, a characteristic and permanent property of an HCI molecule. The
(molecular) dipole moment µ is given by the product ‘ q ⋅ l '. A dipole moment has
both magnitude and direction; it is a vector [12].
Footnotes
[1] The classic reference in this subject is: J. O. Hirschfelder, C. F. Curtiss and R. B.
Bird, Molecular Theory of Gases and Liquids, Wiley, New York 1954; corrected
printing ,1964.
[2] The ampere is that constant current flowing in two parallel straight conductors,
having negligible cross section, one metre apart in vacuo which produces a force
between each metre of length equal to 2 x 10-7 N.
[3] When a current of one A flows for one second, the total charge passed is one
coulomb. In practice, a current of 1 A is very high and the common unit is
milliampere (symbol: mA). The starter motor in a conventional car requires a peak
current of around 30 A.
Electric charge on a single proton, e = 1.602 x 10-19 C.
Faraday, F = N A ⋅ e = 9.649 x 10 4 C mol -1
[4] Just to keep up with the way the units are developing we note: electric current
electric potential gradient
coulomb C = A s
volt V = J A-1 s-1 = J C-1
(J = joule).
Thus volt expressed as J C-1 is energy per coulomb of electric charge passed. This
link between electric potential and energy is crucial.
electrical resistance
ohm Ω = V A-1
Ohm's Law is a phenomenological law.
[5] Continuing our concern for units.
electrical capacitance:
unit = farad F ≡ A s V-1
electric permittivity ε ; unit = F m-1
electric permittivity of a vacuum, ε0 = 8.854 x 10 -12 F m -1
relative permittivity ε r = ε / ε 0; unit = 1
Older literature calls εr, the "dielectric constant". But this property is not a constant
for a given substance such as water ( l ). Thus ε and εr depend on both temperature
and pressure [6]; ε and ε r for a given liquid depend on electric field strength and
frequency of AC current applied to the capacitance.
[7] The quantity (4 ⋅ π ⋅ ε 0 ⋅ 10 −7 ) −1 / 2 equals 2.998 x 10 8 m s -1 which is the speed of
light.
[8] We check
the units. If X is a force, the unit for X is newton (symbol N). Then the
right-hand side should simplify to the same unit. Electric charge is expressed in C [=
A s]; ε0 has units of F m -1 [= A s V-1 m -1].
Then X = [C].[C]/[1].[1].[A s V-1 m-1]. [m]2 But
[V] = [J A-1 s-1]
Then X = [A2 s2]/ [A s J -1 A s m] = [J m -1l = [N]
[9] We check that our units are correct. If Uij is an energy expressed in joules, the
terms on the right-hand side should reduce to joules.
U ij = [A s] ⋅ [A s]/[1] ⋅ [1] ⋅ [A s V -1 m -1 ] ⋅ [m]
= [A 2 s 2 ] /[ A s J -1 A s] = [J]
[10]
E = [C] /[1] ⋅ [1] ⋅ [A s V -1 m -1 ] ⋅ [m 2 ]
= [A s]/[A s V -1 m] = [V m -1 ]
= [J A -1 s −1 m -1 ] = [J m −1 ] /[A s] = [N C -1 ]
Thus electric field strength is expressed in V m-1 or N C-1; the latter is clearly a force
per unit charge.
[11] The classic text is:- P. Debye, Polar Molecules, Chemical Catalog Co., New
York 1929 (available as Dover paperback). We do not consider here interactions
involving quadrupoles, octupoles, etc. These molecular properties are reviewed by A.
D. Buckingham, Quart. Rev., 1959, 13, 183.
[12]
Thus
Dipole moment, µ = q ⋅ l = [C] ⋅ [m ]
µ = [C m], coulomb metre .
Dipole moments are normally quoted using the unit, debye. [The unit is named in
honour of Peter Debye.] 1 D = 3.336 x 10 -30 C m.