pH, soil pH, working principle of pH meter etc.

1. Soil Fertility and Nutrient Management
Experiential Learning, NRM Module-1, ACSS-451
Dr. Pabitra Kumar Mani
Assoc. Professor
Deptt. Agril. Chemistry and Soil Science
Bidhan Chandra Krishi Viswavidyalaya,
E-mail: pabitramani@gmail.com,
Website: www.bckv.edu.in
 : 91-33-25822132,
91-9477465968,
August 21, 2013

2. The Danish biochemist Soren Sorensen invented the pH scale in 1909.
When applied to the full cell formed from half cells present on inside and outside of glass
pH electrode, Nernst equation takes form:
pH inside of the electrode has constant value, thus it can be included in the potential
part of the equation:
This form of the equation describes behavior of the glass electrode used for pH measurements

4. The Ionization of Water Is Expressed by an Equilibrium Constant
The degree of ionization of water at equilibrium (Eqn 2–1) is small; at 25 °C only
about two of every 109 molecules in pure water are ionized at any instant. The
equilibrium constant for the reversible ionization of water (Eqn 2–1) is
In pure water at 25°C, the concentration of water is 55.5 M (grams of H2O in 1 L
divided by its gram molecular weight: (1,000 g/L)/(18.015 g/mol)) and is essentially
constant in relation to the very low concentrations of H and OH, namely, 1X 10-7 M.
Accordingly, we can substitute 55.5 M in the equilibrium constant expression (Eqn 2–
3) to yield
which, on rearranging, becomes
where Kw designates the product (55.5 M)(Keq), the ion product of water at 25 °C.
The value for Keq, determined by electrical-conductivity measurements of pure
water, is 1.8 X 10 -16 M at 25°C. Substituting this value for Keq in Equation gives the
value of the ion product of water:

5. Thus the product [ H+ ][ OH- ] in aqueous solutions at 25°C always equals 1x10 -14 M2. When
there are exactly equal concentrations of H+ and OH-, as in pure water, the solution is said to
be at neutral pH. At this pH, the concentration of H+ and OH- can be calculated from the ion
product of water as follows:

8. How a pH meter works
When one metal is brought in contact with another, a voltage difference
occurs due to their differences in electron mobility.
When a metal is brought in contact with a solution of salts or
acids, a similar electric potential is caused, which has led to the invention
of batteries.
Similarly, an electric potential develops when one liquid is brought
in contact with another one, but a membrane is needed to keep such
liquids apart.
A pH meter measures essentially the electro-chemical potential
between a known liquid inside the glass electrode (membrane) and an
unknown liquid outside. Because the thin glass bulb allows mainly the agile
and small hydrogen ions to interact with the glass, the glass electrode
measures the electro-chemical potential of hydrogen ions or the
potential of hydrogen.
To complete the electrical circuit, also a reference electrode is
needed. Note that the instrument does not measure a current but only an
electrical voltage, yet a small leakage of ions from the reference electrode is
needed, forming a conducting bridge to the glass electrode.

10. General Principles
Reference electrode | salt bridge | analyte solution | indicator electrode
Eref
Ej
Eind
Ecell = Eind – Eref + Ej
Reference cell :
a half cell having a known electrode
potential
Indicator electrode:
has a potential that varies in a
known way with variations in
the concentration of an analyte
A cell for potentiometric determinations.

11. Introduction
1.) Potentiometry

Use of Electrodes to Measure Voltages that Provide Chemical Information
-
Various electrodes have been designed to respond selectively to specific analytes

Use a Galvanic Cell
Unknown solution becomes a ½cell
Add Electrode that
transfers/accepts electrons from
unknown analyte
Connect unknown solution by salt
bridge to second ½-cell at fixed
composition and potential

Indicator Electrode: electrode that
responds to analyte and
donates/accepts electrons

Reference Electrode: second ½ cell at a
constant potential

Cell voltage is difference between the
indicator and reference electrode

12. Reference Electrodes
1.) Overview


Potential change only dependent on one ½ cell
concentrations
Reference electrode is fixed or saturated  doesn’t change!
{

 [ Fe 2 + ]  
0.05916

  − 0.222 − 0.05916 log[ Cl − ]
E cell = 0.771 −
log 
 [ Fe 3 + ]  
1





Potential of the cell
only depends on
[Fe2+] & [Fe3+]
Unknown solution of
[Fe2+] & [Fe3+]
}
Reference electrode,
[Cl-] is constant
Pt wire is indicator
electrode whose
potential responds
to [Fe2+]/[Fe3+]

13. Reference Electrodes
2.)
Silver-Silver Chloride Reference Electrode
Eo = +0.222 V
Activity of Cl- not 1E(sat,KCl) = +0.197 V

Convenient
-
Common problem is porous plug becomes clogged
Reduction potential

14. Reference Electrodes
3.)
Saturated Calomel Reference Electrode (S.C.E)
Eo = +0.268 V
Activity of Cl- not 1E(sat,KCl) = +0.241 V


Saturated KCl maintains constant [Cl-]
even with some evaporation
Standard hydrogen electrodes are
cumbersome
- Requires H2 gas and
freshly prepared Pt surface

15. Indicator Electrodes
1.)
Two Broad Classes of Indicator Electrodes

Metal Electrodes
- Develop an electric potential in response to a redox
reaction at the metal surface

Ion-selective Electrodes
- Selectively bind one type of ion to a membrane to
generate an electric potential
Remember an electric potential is generated by a separation of charge

16. Indicator Electrodes
2.)
Metal Electrodes
Platinum
Most common metal indicator electrode
Inert: does not participate in many chemical reactions
Simply used to transmit electrons
Other electrodes include Gold and Carbon
Metals (Ag, Cu, Zn, Cd, Hg) can be used to monitor their aqueous ions
Most metals are not useable
Equilibrium not readily established at the metal surface



Example:
E+o = +799 V
½ Reaction at Ag indicator electrode:
E(sat,KCl) = +0.241 V
½ Reaction at Calomel reference electrode:
Cell Potential from Nernst Equation:

 1 
0.05916

  − { 0.241}
E cell = E + − E − = 0.799 −
log 
 [ Ag + ]  
1





Cell voltage changes as a function of [Ag +]
Potential of Ag
indicator electrode

17. Typical electrode system for
measuring pH. (a) Glass
electrode (indicator) and
saturated calomel
electrode (reference)
immersed in a solution of
unknown pH.
(b) Combination probe
consisting of both an
indicator glass electrode
and a silver/silver chloride
reference. A second
silver/silver chloride
electrode serves as the
internal reference for the
glass electrode.
The two electrodes are
arranged concentrically with the internal reference in the center and the
external reference outside. The reference makes contact with the analyte
solution through the glass frit or other suitable porous medium.
Combination probes are the most common configuration of glass electrode
and reference for measuring pH.

18. The majority of pH
electrodes available
commercially are
combination electrodes
that have both glass H+
ion sensitive electrode
and additional reference
electrode conveniently
placed in one housing.

19. When cell is not at standard conditions,
use Nernst Equation
• In a chemical reaction such as:
aA + bB  cC + dD
c
d
aC .aD
o
o
∆G = ∆G + 2.3RT log a b = ∆G + 2.3RT log Q
a A .aB
Substitute
∆G = − nFE
∆G o = − nFE o
Where:
∆Go : Free energy change when all the reactants and
products are in their standard states (unit activity).
R : is the gas constant.
T : is the temperature in the absolute temperature
Q : Reaction Quotient

20. c
d
aC .aD
∆G = ∆G + 2.3RT log a b
a A .aB
o
[C ]c [ D]d
− nFE = −nFE o + 2.3RT log
[ A]a [ B]b
2.3RT
[C ]c [ D]d
E = Eo −
log
nF
[ A]a [ B]b
Where concentrations are substituted for activities
At 298 K the equation becomes
K
0.0591
[C ]c [ D]d
E = Eo −
log
n
[ A]a [ B]b
…… Nernst Equation
 At Equilibrium, ∆G = 0, E = 0. Hence
0.0591
0=E −
log K
n
o
0.0591
E =
log K
n
o

21. Derivation of the EMF equation
For the general rn.,
Red
⇋ Ox + ne-
The chemical potential are given by
µRed = µ0Red+ RTlnaRed
µOx = µ0Ox+ RTlnaOx
µRed - µOx = µ0Red - µ0Ox + RTln(aRed /aOx)
-ΔG = (µ0Red - µ0Ox) + RTln(aRed /aOx)
but , considering the electrical work associated with transfer of n no. Of electrons,
ΔG= -nFE
 μ0

− μ0


 Red
Ox  - RT ln aOx
E=
nF
nF a
Red
RT a Ox
∴E = E ln
nF a
Red
0
When aOx =aRed = 1, then
μ0

− μ0


Ox 
0
 Red
E =
nF
a
0.0591
so, E = E log Ox
n
a
Red
0
T=2980K
F=96000
Coulomb

22. In principle it should be possible to determine the H + ion activity or concn. of
a soln by measuring the potential of a Hydrogen electrode inserted in the
given soln. The EMF of a cell, free from liquid junction potential,
consisting of a Hydrogen electrode and a reference electrode, should be
given by,
E = E ref – RT/F ln aH+
E = E ref + 2.303 RT/F pH
∴pH = ( E- Eref )F/2.303 RT
R= 8.314 J/mol/°K
F= faraday constant ,96,485
T= Kelvin scale
So, by measuring the EMF of the Cell E obtained by combining the H
electrode with a reference electrode of known potential, Eref , the pH of the
soln. may be evaluated.
The electric potential at any point is defined as the work done in bringing a unit charge from
infinity to the particular point
Reduced state ⇋ Oxidised state + n Electron
M
= Mn+ + nE
E(+) = E0 – (RT/F) ln aMn+
Nernst Equation

23. Electrodes and Potentiometry :
pH Electrodes
1.) pH Measurement with a Glass Electrode


Glass electrode is most common ion-selective electrode
Combination electrode incorporates both glass and
reference electrode in one body
Ag(s)|AgCl(s)|Cl-(aq)||H+(aq,outside) H+(aq,inside),Cl-(aq)|AgCl(s)|Ag(s)
Outer reference
electrode
[H+] outside
(analyte solution)
[H+] inside
Inner reference
electrode
Glass membrane
Selectively binds H+
Electric potential is generated by [H+] difference across glass membrane

24. Electrodes and Potentiometry
pH Electrodes
2.) Glass Membrane

Irregular structure of silicate lattice
Cations (Na+) bind oxygen
in SiO4 structure

25. pH Electrodes
2.) Glass Membrane

Two surfaces of glass “swell” as they absorb water
- Surfaces are in contact with [H+]

26. pH Electrodes
2.) Glass Membrane

H+ diffuse into glass membrane and replace Na+ in hydrated gel
region
-
Ion-exchange equilibrium
Selective for H+ because H+ is only ion that binds
significantly to the hydrated gel layer
Charge is slowly carried
by migration of Na+
across glass membrane
E = constant − β (0.05916) pH
Potential is determined
by external [H+]
Constant and b are measured when electrode is calibrated with solution of known pH

27. the operation of a glass electrode is related to the situations existing at the
inner and outer surfaces of the glass membrane. Glass electrodes require
soaking in water for some hours before use and it is concluded that a
hydrated layer is formed on the glass surface, where an ion exchange
process can take place. If the glass contains sodium, the exchange process
can be represented by the equilibrium
The concn of the soln within the glass bulb is fixed, and hence on the inner
side of the bulb an equilibrium condition leading to a constant potential
is established. On the outside of the bulb, the potential developed will be
dependent upon the hydrogen ion concentration of the soln in which the
bulb is immersed.
Within the layer of 'dry' glass which exists between the inner and outer
hydrated layers, the conductivity is due to the interstitial migration of
sodium ions within the silicate lattice.

28. Most often used pH electrodes are glass
electrodes. Typical model is made of glass
tube ended with small glass bubble. Inside of
the electrode is usually filled with buffered
solution of chlorides in which silver wire
covered with silver chloride is immersed. pH
of internal solution varies - for example it can be
1.0 (0.1M HCl) or 7.0 (different buffers used by
different producers).
Active part of the electrode is the glass
bubble. While tube has strong and thick walls,
bubble is made to be as thin as possible.
Surface of the glass is protonated by both
internal and external solution till equilibrium
is achieved. Both sides of the glass are
charged by the adsorbed protons, this charge is
responsible for potential difference. This
potential in turn is described by the
Nernst equation and is directly proportional to
the pH difference between solutions on both
sides of the glass.

29. pH Electrodes
3.)
Calibration

A pH electrode should be calibrated with two or
more standard buffers before use.

pH of the unknown should lie within the range
of the standard buffers
Measured
voltage
is
correlated with a pH, which
is then used to measure an
unknown.

34. For soils containing predominantly negatively charged clays, dilution of
the soil solution by distilled water increases the absolute value of the
surface potential and changes the distribution of H+ ions between the
DDL and the bulk solution. The proportion of H+ ions in the DDL relative
to the bulk solution increases so that the measured pH, which is the bulk
solution pH, is higher than that of the natural soil. This effect is
especially noticeable in saline soils.
An alternative method is to shake the soil with 0.01 M CaCl2 solution until
equilibrium is attained. This solution, containing the most abundant
exchangeable cation in many soils, also has an ionic strength I
approximating that of the soil solution.
In this soil suspension, both H+ and Ca2+ exchange with other cations
in the DDL, but the ratio of H+/√Ca2+ in solution remains relatively
constant, even when the ratio of soil to equilibrating liquid changes. Thus,
pH measured in 0.01 M CaCl2 is a more accurate reflection of the
natural soil pH, as is demonstrated by the comparison between soil pH
measured in water, in 0.01 M CaCl2 , and in a soln. that has been shaken
with successive samples of soil to achieve equilibrium

35. The ionic distribution at a negatively charged clay surface

37. For soils containing predominantly negatively charged clays, dilution of
the soil solution by distilled water increases the absolute value of the
surface potential and changes the distribution of H+ ions between the
DDL and the bulk solution. The proportion of H+ ions in the DDL relative
to the bulk solution increases so that the measured pH, which is the bulk
solution pH, is higher than that of the natural soil. This effect is
especially noticeable in saline soils.
An alternative method is to shake the soil with 0.01 M CaCl2 solution until
equilibrium is attained. This solution, containing the most abundant
exchangeable cation in many soils, also has an ionic strength I
approximating that of the soil solution.
In this soil suspension, both H+ and Ca2+ exchange with other cations
in the DDL, but the ratio of H+/√Ca2+ in solution remains relatively
constant, even when the ratio of soil to equilibrating liquid changes. Thus,
pH measured in 0.01 M CaCl2 is a more accurate reflection of the
natural soil pH, as is demonstrated by the comparison between soil pH
measured in water, in 0.01 M CaCl2 , and in a soln. that has been shaken
with successive samples of soil to achieve equilibrium

38. I = 0.0127 x Ecw
Griffin and Jurinak, 1973
where Ci and zi are the concentration and charge, respectively, of ion i in a mixture
of ionic species i = 1 to n. The term ‘concentration’ refers to the mass of an ionic
species per unit volume of solution (e.g. mol/L).
An individual ion experiences weak forces due to its interaction with
water molecules (the formation of a hydration shell), and stronger electrostatic
forces due to its interaction with ions of opposite charge. Effectively, this means
that the ability of the ion to engage in chemical reactions is decreased,
relative to what is expected when it is present at a particular concentration
with no interactions. This effect is accounted for by defining the activity ai of ion i,
which is related to its concentration by the equation
where fi is the activity coefficient of the ion. Values of fi range from 0 to 1. In very
dilute solutions where the interaction effects are negligible, fi approaches 1, and
ai is approximately equal to Ci. There is extensive theory on the calculation of activity
coefficients, but all calculations make use of the ionic strength I, such as in the
Debye–Huckel limiting law (Atkins, 1982)

39. The difference in pH(ΔpH) between the sediment and
supernatant produced during soil pH determination is termed the
'suspension effect′ (McLean 1982). If the soil suspension is
allowed to settle, the pH as measured in the suspension liquid is
often higher than in the sediment layer.
The suspension effect is influenced by the extent to which
electrodes encounter clay and humus particles and the soil CO2 is in
equilibrium with atmospheric concentrations, and the magnitude of
liquid junction effects (Foth and Ellis 1988).
The suspension effect is minimised by measuring soil pH using
0.01 M CaCl2 or 1 M KCl solutions instead of water (Sumner 1994).
In addition, the use of CaCl2 or KCl solutions has the advantages of
(i) decreasing the effect of the junction potential of the calomel
reference electrode,
(ii) equalising the salt content of soils, and
(iii) Preventing dispersion of the soil (Tan 1995).

40. Junction Potential
1.) Occurs Whenever Dissimilar Electrolyte Solutions are in Contact



Develops at solution interface (salt bridge)
Small potential (few milli volts)
Junction potential puts a fundamental limitation on the
accuracy of direct potentiometric measurements
- Don’t know contribution to the measured voltage
Different ion mobility results in
separation in charge
Mobilties of ions in water at 25oC:
Na+ : 5.19 × 10 –8 m2/sV
K+ : 7.62 × 10 –8
Cl– : 7.91× 10 –8
Again, an electric potential is generated by a separation of charge

41. General relationship between soil pH and availability of plant nutrients in minimally and
moderately weathered soils; the wider the bar, the more availability.

42. Nitrogen availability is maximum between pH 6 and 8, because this is the most
favorable range for the soil microbes that mineralize the nitrogen in organic matter
and those organisms that fix nitrogen symbiotically. High phosphorus availability at
high pH-above 8.5-is due to sodium phosphates that have high solubility. In
calcareous soil, pH 7.5 to 8.3, phosphorus availability is re-duced by the presence
of calcium carbonate that represses the dissolution of calcium phosphates.
Maximum phosphorus availablity is in the range 7.5 to 6.5. Below pH 6.5,
increasing acidity is associated with increasing iron and aluminum in solution and the
formation of relatively insoluble iron and aluminum phosphates.
Note that potassium, calcium, and magnesium are widely available in
alkaline soils. As soil acidity increases, these nutrients show less availability as a
result of the decreasing CEC and decreased amounts of exchangeable nutrient
cations.
Iron and manganese availability increase with increasing acidity because of
their increased solubility.These two nutrients are frequently deficient in plants growing
in alkaline soils because of the insolubility of their compounds
Boron, copper, and zinc are leachable and can be deficient in leached, acid soils.
Conversely, they can become insoluble (fixed) and unavailable in alkaline soils.
In acid soils, molybdenum is commonly deficient owing to its reaction with iron to form
an insoluble compound. For plant nutrients as a whole, good overall nutrient
availability occurs near pH 6.5. In intensively weathered soils, nutrients such as B and
Zn may be deficient at lower pH than for minimally and moderately weathered soils. As
a consequence, a desirable pH is 5.5 in intensively weathered soils.

44. Ohm's Law States that the current I (amperes) flowing in a
conductor is directly proportional to the applied electromotive force
E (volts) and inversely proportional to the resistance R (ohms) of
the conductor
The reciprocal of the resistance is termed the conductance (G): this
is measured in reciprocal ohms (or Ω –), for which the name Siemens
(S) is used. The resistance of a sample of homogeneous material,
length l, and cross-section area a, is given by:
where ρ is a characteristic property of the material termed the
resistivity (formerly called specific resistance). In SI units, l and a will be
measured respectively in metres and square metres, so that ρ refers to a
metre cube of the material, and
The reciprocal of resistivity is the conductivity, κ (formerly specific
conductance), which in SI units is the conductance of a one metre cube
of substance and has the units Ω – m - , but if ρ is measured in Ωcm, then
κ will be measured in Ω - cm - '.

45. Electrical Conduction
• Ohm's Law:
V=IR
voltage drop (volts = J/C)
resistance (Ohms)
current (amps = C/s)
C = Coulomb
• Resistivity, ρ:
-- a material property that is independent of sample size and
geometry
surface area
RA
ρ=
l
• Conductivity, σ
of current flow
current flow
path length
1
σ=
ρ
45

46. Electrical Properties
• Which will have the greater resistance?
2l
D
l
2D
R1 =
2ρl
8ρl
= 2
D 2 πD
π 
2
ρl
ρl R1
R2 =
= 2 =
2
8
D
2D  π
π 
2 
• Analogous to flow of water in a pipe
• Resistance depends on sample geometry and size.
46

47. Electrical Conductivity
This is based on the principle that the conductivity, or ease with which
an electric current is carried through a solution, is proportional to the
quantity of ions (actually, the quantity of ionic charge) in solution.
To measure soil conductivity, the soil is
mixed with water until it has a paste like
consistency. A pair of electrodes,
commonly made of platinum metal, is
inserted into this "saturated paste." The
electrodes are characterized by their
surface area, A cm2 (usually about 1 cm2),
and their separation distance of L cm
(usually about 1 cm), as illustrated in
Figure. An alternating current is then
applied to the electrode pair, and the
conductance of the paste is measured in
units of reciprocal ohms (mho),or Siemens
Two-electrode cell used to measure
electrical conductivity of solutions and
saturated soil pastes

48. Conductivity Probe
• 2 metals in contact with
electrolyte solution
• Voltage is applied to
electrodes and resulting
current that flows btw
electrodes is used to
determine conductance
• Amount of current flowing
depends on:
– Solution conductivity
– Length, surface area,
geometry of electrodes

49. Conductivity Probe
• Apply an AC Voltage to Two Electrodes of Exact Dimensions
• Acids, Bases and Salts (NaCl) Dissolve in Solution and Act as
Current Carriers
• Current Flow is Directly Proportional to the Total Dissolved
Solids in Solution
• Physical Dimensions of a Conductivity Electrode are Referred to
as the Cell Constant
• Cell Constant is Length/Area Relationship
– Distance Between Plates = 1.0 cm
– Area of Each Plate = 1.0 cm x 1.0 cm
– Cell Constant = 1.0 cm-1

50. Cell Constant
• Cell constant:
– Measure of current
response of a sensor
conductive solution
– Due to sensor’s
dimensions and
geometry
– Units: cm-1 (length
divided by area)

51. But this conductance depends on A and L and is therefore a characteristic of the
measuring device as well as the solution. It is more useful to define a specific
conductance, termed the electrical conductivity or EC, by the equation:
EC, unlike conductance, is strictly a property of the solution, whereas L/A is a
characteristic of the electrode geometry and is called the cell constant.
For a saturated paste, an EC value greater than
4 milli Siemens/cm (mS cm-I) is diagnostic of a saline soil.
The current-carrying ability of a mole of cation or anion charge in solution is
termed the equivalent conductance. This value is measured in very dilute
solutions to ensure that the salts are fully dissociated into individual cations
and anions. The measured conductance is then assumed to result from the
summation of conductances contributed by each of the ions. As a result,
each ion can be assigned an equivalent conductance, listed in Table 8.2,
that represents its contribution (per mole of charge) to the measured (total)
conductivity.

52. Conduction & Electron Transport
• Metals (Conductors):
-- for metals empty energy states are adjacent to filled states.
-- thermal energy
excites electrons
into empty higher
energy states.
-- two types of band
structures for metals
empty
band
- partially filled band
- empty band that
overlaps filled band
partly
filled
band
Partially filled band
Energy
Overlapping bands
Energy
empty
band
filled states
filled
band
filled states
GAP
filled
band
filled
band
52

53. Energy Band Structures: Insulators
& Semiconductors
• Insulators:
• Semiconductors:
Energy
empty
conduction
band
filled states
GAP
filled
valence
band
filled
band
-- narrow band gap (< 2 eV)
-- more electrons excited
across band gap
Energy
empty
conduction
band
?
GAP
filled states
-- wide band gap (> 2 eV)
-- few electrons excited
across band gap
filled
valence
band
filled
band
53

54. It is apparent that H+ and OH- have anomalously high conductances, while
most cations and anions contribute about equally per mole of ionic charge to
the conductivity of solutions. This fact allows simple but approximate
empirical relationships to be established between the concentration of
electrolyte solutions and the conductivity, irrespective of ionic composition.

55. For strong electrolytes the molar conductivity
increases as the dilution is increased, but it appears to
approach a limiting value known as the molar conductivity at
infinite dilution(Λ∞). The quantity Λ∞ can be determined by
graphical extrapolation for dilute solutions of strong
electrolytes.
For weak electrolytes the extrapolation method cannot
be used for the determination of Λ∞ but it may be calculated
from the molar conductivities at infinite dilution of the respective
ions, use being made of the 'Law of Independent Migration of
Ions‘.
At infinite dilution the ions are independent of each other,
and each contributes its part of the total conductivity, thus:

56. THE BASlS OF CONDUCTIMETRIC TlTRATlONS
principle underlying conductimetric titrations, i.e. The
substitution of ions of one conductivity by ions of another
conductivity.

57. Definitions
Further definitions
J = σ ε <= another way to state Ohm’s law
J ≡ current density
current
I
=
=
surface area A
like a flux
ε ≡ electric field potential = V/
J = σ (V/ )
Electron flux
conductivity
voltage gradient
57

58. Example: Conductivity Problem
What is the minimum diameter (D) of the wire so that V < 1.5 V?
= m
100
I = 2.5 A
Cu wire -
+
V
100 m
πD 2
4
Solve to get
R=

V
=
Aσ I
< 1.5 V
2.5 A
6.07 x 107 (Ohm-m)-1
D > 1.87 mm
58

59. Filling of electronic band structure in various types of material at thermodynamic
Equilibrium. In metals and semimetals the Fermi level EF lies inside at
least one band. In insulator (electricity)insulators and semiconductors the Fermi
level is inside a band gap, however in semiconductors the bands are near enough to
the Fermi level to be Fermi-Dirac statistics thermally populated with electrons or
Electron holes.

60. The calomel reference electrode consists of a glass tube with a potassium
chloride (KCl) electrolyte which is in intimate contact with a mercuric chloride
element at the end of a KCL element. It is a fragile construction, joined by a
liquid junction tip made of porous ceramic or similar material. This kind of
electrode is not easily 'poisoned' by heavy metals and sodium.
The glass electrode consists of a sturdy glass tube with a thin glass bulb
welded to it. Inside is a known solution of potassium chloride (KCl)
buffered at a pH of 7.0. A silver electrode with a silver chloride tip makes
contact with the inside solution. To minimise electronic interference, the
probe is shielded by a foil shield, often found inside the glass electrode.
Most modern pH meters also have a thermistor temperature probe which
allows for automatic temperature correction, since pH varies somewhat with
temperature

61. The conductance of an electrolytic solution at any temperature
depends only on the ions present, and their concentration.
When a solution of an electrolyte is diluted, the conductance will
decrease, since fewer ions are present per millilitre of solution to
carry the current. If all the solution be placed between two
electrodes 1 cm apart and large enough to contain the whole of
the solution, the conductance will increase as the solution is
diluted. This is due largely to a decrease in inter-ionic effects
for strong electrolytes and to an increase in the degree of
dissociation for weak electrolytes.
The molar conductivity (Λ) of an electrolyte is defined as the
conductivity due to one mole and is given by:

62. Electrons in solids
• In a solid, there are so many electrons with energies very
near each other that ‘bands’ of states develop.
All we draw is the
“band diagram”
Solid
Isolated atoms

63. Conductivity: Comparison
• Room temperature values (Ohm-m)-1 = (Ω - m)-1
METALS
Silver
Copper
Iron
conductors
6.8 x 10 7
6.0 x 10 7
1.0 x 10 7
CERAMICS
-10
Soda-lime glass 10-10 -11
Concrete
10-9
Aluminum oxide <10-13
SEMICONDUCTORS
POLYMERS
Polystyrene
Silicon
4 x 10 -4
Polyethylene
Germanium 2 x 10 0
GaAs
10 -6
semiconductors
<10 -14
-15
10 -10-17
insulators
Selected values from Tables 18.1, 18.3, and 18.4, Callister & Rethwisch 8e.
63

64. Electrons in atoms
Electrons in an atom have particular energies
(quantized energy states) depending on which
orbital they are in.
Energy
H
B
1s
He
Pauli
exclusion
1s
principle
1p
1s

65. •
Bandthere are so manyRepresentation
Structure electrons with energies very
In a solid,
near each other that ‘bands’ of states develop.
Adapted from Fig. 18.3,
Callister & Rethwisch 8e.
65

66. Electron Energy Band Structures
Adapted from Fig. 18.2, Callister & Rethwisch 8e.
66

67. Definition of Conductivity
Energy
• The free-est electron (the
electron with the highest
energy) defines the position
Band gap
of the “Fermi level.”
– Above Ef, all available
Ef,
electron states in the
energy bands are empty Fermi
level
– Below Ef, they are all
filled.
Band gap
• If there is no gap between
filled and empty states, the
material is conductive.
Metal (Cu)
• If there is a gap, the material
is a semiconductor or
insulator.
Empty 4p
(conduction)
partially
filled 4s
(conduction)
Filled (valence)

69. Energy band structures for various metals
Energy
• Partially filled or empty
bands are called
‘conduction bands.’
• Any band that is totally filled
is considered to be a
“valence band.”
– We usually ignore ‘deep’
valence bands.
Empty 4p
(conduction)
Band gap
partially
filled 4s
(conduction)
Band gap
Filled 3d (valence)
filled 3d, 3p, 3s, 2p,
2s, 1p, 1s (valence)
Metal (Cu)
Deep valence-only
an issue for optical
properties