|
|
| Line 1: |
Line 1: |
| The '''conductivity''' (or '''specific conductance''') of an [[electrolyte]] solution is a measure of its ability to [[electrical conductivity|conduct electricity]]. The [[SI]] unit of conductivity is [[siemens (unit)|siemens]] per meter (S/m). | | The '''Moffat distribution''', named after the [[physicist]] [[Anthony Moffat]], is a [[continuous probability distribution]] based upon the [[Cauchy distribution|Lorentzian distribution]]. Its particular importance in [[astrophysics]] is due to its ability to accurately reconstruct [[point spread function]]s, whose wings cannot be accurately portrayed by either a [[Gaussian function|Gaussian]] or [[Cauchy distribution|Lorentzian]] function. |
|
| |
|
| Conductivity measurements are used routinely in many industrial and environmental applications as a fast, inexpensive and reliable way of measuring the ionic content in a solution.<ref name=Gray>{{cite book |last=Gray |first=James R. |title=Environmental Instrumentation and Analysis Handbook |editor=Down, R.D; Lehr, J.H. |publisher=Wiley |year=2004 |chapter=Conductivity Analyzers and Their Application |chapterurl=http://books.google.com/books?id=6jhELyGJOm0C&pg=PA491 |accessdate=2009-05-10 |pages=491–510 |isbn=978-0-471-46354-2}}</ref> For example, the measurement of product conductivity is a typical way to monitor and continuously trend the performance of [[water purification]] systems.
| | ==Characterisation== |
| | ===Probability density function=== |
| | The Moffat distribution can be described in two ways. Firstly as the distribution of a bivariate random variable (''X'',''Y'') centred at zero, and secondly as the distribution of the corresponding radii |
| | :<math>R=\sqrt{X^2+Y^2}.</math> |
| | In terms of the random vector (''X'',''Y''), the distribution has the [[probability density function]] (pdf) |
| | : <math> |
| | f(x,y; \alpha,\beta)=\left(\beta-1\right)\left(\pi\alpha^2\right)^{-1}\left[1+\left(\frac{x^2+y^2}{\alpha^2}\right)\right]^{-\beta} , \, |
| | </math> |
|
| |
|
| In many cases, conductivity is linked directly to the [[total dissolved solids]] (T.D.S.). High quality deionized water has a conductivity of about 5.5 μS/m, typical drinking water in the range of 5-50 mS/m, while sea water about 5 S/m<ref>{{cite web |url=http://www.lenntech.com/water-conductivity.htm |title=Water Conductivity |publisher=Lenntech|accessdate=5 January 2013}}</ref> (i.e., sea water's conductivity is one million times higher than that of deionized water).
| | where <math>\alpha</math> and <math>\beta</math> are [[astronomical seeing|seeing]] dependent parameters. In this form, the distribution is a reparameterisation of a [[bivariate Student distribution]] with zero correlation. |
|
| |
|
| Conductivity is traditionally determined by measuring the [[alternating current|AC]] [[electrical resistance|resistance]] of the solution between two [[electrode]]s. Dilute solutions follow [[Friedrich Kohlrausch|Kohlrausch]]'s Laws of concentration dependence and additivity of ionic contributions. [[Lars Onsager]] gave a theoretical explanation of Kohlrausch's law by extending [[Debye–Hückel equation|Debye–Hückel theory]].
| | In terms of the random variable ''R'', the distribution has density |
| | : <math> |
| | f(r; \alpha,\beta)=2r \frac{\beta-1}{\alpha^2} \left[1+\left(\frac{r^2}{\alpha^2}\right)\right]^{-\beta} . \, |
| | </math> |
|
| |
|
| ==Units== | | ===Differential equation=== |
| [[File:WiderstandWasser.PNG|thumb|right|Resistivity of pure water (in MΩ-cm) as a function of temperature]]
| | The pdf of the Moffat distribution is a solution to the following [[differential equation]]: |
| The [[SI]] unit of conductivity is [[Siemens (unit)|S]]/m and, unless otherwise qualified, it refers to 25 °C (standard temperature). Often encountered in industry is the traditional unit of μS/cm. 10<sup>6</sup> μS/cm = 10<sup>3</sup> mS/cm = 1 S/cm. The numbers in μS/cm are higher than those in μS/m by a factor of 100 (i.e., 1 μS/cm = 100 μS/m). Occasionally a unit of "EC" (electrical conductivity) is found on scales of instruments: 1 EC = 1 mS/cm. Sometimes encountered is a so-called [[mho]] (reciprocal of ohm): 1 mho/m = 1 S/m. Historically, mhos antedate Siemens by many decades; good vacuum-tube testers, for instance, gave transconductance readings in micromhos.
| | :<math>\left\{\begin{array}{l} |
| | \left(r^3+\alpha ^2 r\right) f'(r)+f(r) \left(-\alpha ^2+2 \beta r^2-r^2\right)=0, \\ |
| | f(1)=\frac{2 (\beta -1) \left(\frac{1}{\alpha ^2}+1\right)^{-\beta}}{\alpha ^2} |
| | \end{array}\right\} |
| | </math> |
|
| |
|
| The commonly used standard cell has a width of 1 cm, and thus for very pure water in equilibrium with air would have a resistance of about 10<sup>6</sup> ohm, known as a [[megohm]], occasionally spelled as "megaohm".<ref>{{cite web | url=http://physics.nist.gov/Pubs/SP811/sec09.html#9.3 Spelling unit names with prefixes | title=9 Rules and Style Conventions for Spelling Unit Names | publisher=NIST.gov | accessdate=2011-08-02}}</ref> [[Ultrapure Water|Ultra-pure water]] could achieve 18 megohms or more. Thus in the past megohm-cm was used, sometimes abbreviated to "megohm". Sometimes, a conductivity is given just in "microSiemens" (omitting the distance term in the unit). While this is an error, it can often be assumed to be equal to the traditional μS/cm. The typical conversion of conductivity to the total dissolved solids is done assuming that the solid is sodium chloride: 1 μS/cm is then an equivalent of about 0.6 mg of NaCl per kg of water.
| | ==References== |
|
| |
|
| Molar conductivity has the SI unit S m<sup>2</sup> mol<sup>−1</sup>. Older publications use the unit Ω<sup>−1</sup> cm<sup>2</sup> mol<sup>−1</sup>.
| | * [http://adsabs.harvard.edu/abs/1969A%26A.....3..455M A Theoretical Investigation of Focal Stellar Images in the Photographic Emulsion (1969) – A. F. J. Moffat] |
| | | {{ProbDistributions|multivariate}} |
| == Measurement ==
| | [[Category:Continuous distributions]] |
| {{main|Electrical conductivity meter}}
| | [[Category:Probability distributions]] |
| [[File:Conductimetrie-schema.png|thumb|right|Principle of the measurement]]
| |
| | |
| The [[electrical conductivity]] of a solution of an [[electrolyte]] is measured by determining the [[Electrical resistance|resistance]] of the solution between two flat or cylindrical [[electrode]]s separated by a fixed distance.<ref>{{cite book|last=Bockris|first=J. O'M.|coauthors=Reddy, A.K.N; Gamboa-Aldeco , M.|title=Modern Electrochemistry |publisher=Springer |year=1998 |edition=2nd. |url=http://books.google.com/?id=utDyTYpimkUC |accessdate=2009-05-10 |isbn=0-306-45555-2 }}</ref> An alternating voltage is used in order to avoid [[electrolysis]]. The resistance is measured by a [[Electrical conductivity meter|conductivity meter]]. Typical frequencies used are in the range 1–3 [[Hertz|kHz]]. The dependence on the frequency is usually small,<ref>Marija Bešter-Rogač and Dušan Habe, "Modern Advances in Electrical Conductivity Measurements of Solutions", Acta Chim. Slov. 2006, 53, 391–395 [http://acta.chem-soc.si/53/53-3-391.pdf (pdf)]</ref> but may become appreciable at very high frequencies, an effect known as the [[Debye–Falkenhagen effect]].
| |
| | |
| A wide variety of instrumentation is commercially available.<ref>{{cite book| last=Boyes| first=W.| title=Instrumentation Reference Book|publisher=Butterworth-Heinemann | year=2002| edition=3rd.|url=http://books.google.com/?id=sarHIbCVOUAC |accessdate=2009-05-10 |isbn=0-7506-7123-8 }}</ref> There are two types of cell, the classical type with flat or cylindrical electrodes and a second type based on induction.<ref>Gray, p 495</ref> Many commercial systems offer automatic temperature correction.
| |
| Tables of reference conductivities are available for many common solutions.<ref>{{cite web |url=http://myweb.wit.edu/sandinic/Research/conductivity%20v%20concentration.pdf |title=Conductivity ordering guide |publisher= EXW Foxboro |date=3 October 1999 |accessdate=5 January 2013}}</ref>
| |
| | |
| == Definitions ==
| |
| Resistance, ''R'', is proportional to the distance, ''l'', between the electrodes and is inversely proportional to the cross-sectional area of the sample, ''A'' (noted ''S'' on the Figure above). Writing ρ (rho) for the specific resistance (or [[resistivity]]),
| |
| :<math>R=\frac{ l}{A}\rho.</math>
| |
| In practice the conductivity cell is [[calibrated]] by using solutions of known specific resistance, ρ<sup>*</sup>, so the quantities ''l'' and ''A'' need not be known precisely.<ref>{{cite web|url=http://www.astm.org/Standards/D1125.htm|title=ASTM D1125 - 95(2005) Standard Test Methods for Electrical Conductivity and Resistivity of Water|accessdate=2009-05-12}}</ref> If the resistance of the calibration solution is ''R''<sup>*</sup>, a cell-constant, ''C'', is derived.
| |
| :<math>R^*=C \times \rho^*</math>
| |
| | |
| The specific conductance, κ (kappa) is the reciprocal of the specific resistance.
| |
| :<math>\kappa =\frac{1}{\rho}=\frac{C}{R}</math>
| |
| Conductivity is also [[Electrical conductivity meter|temperature-dependent]].
| |
| Sometimes the ratio of ''l'' and ''A'' is called as the cell constant, denoted as G<sup>*</sup>, and conductance is denoted as G. Then the specific conductance κ (kappa), can be more conveniently written as
| |
| :<math>\kappa =G^* \times G</math>
| |
| | |
| == Theory ==
| |
| The specific conductance of a solution containing one electrolyte depends on the concentration of the electrolyte. Therefore it is convenient to divide the specific conductance by concentration. This quotient is termed [[molar conductivity]], is denoted by Λ<sub>m</sub>
| |
| :<math>\Lambda_m=\frac{\kappa}{c}</math>
| |
| | |
| === Strong electrolytes ===
| |
| [[Strong electrolyte]]s are believed to [[dissociate]] completely in solution. The conductivity of a solution of a strong electrolyte at low concentration follows [[Kohlrausch's Law]]
| |
| :<math>\Lambda_m =\Lambda_m^0-K\sqrt{c} </math>
| |
| where <math>\Lambda_m^0</math> is known as the limiting molar conductivity, ''K'' is an empirical constant and ''c'' is the electrolyte concentration. (Limiting here means "at the limit of the infinite dilution".) In effect, the observed conductivity of a strong electrolyte becomes directly proportional to concentration, at sufficiently low concentrations i.e. when
| |
| <math>\Lambda_m^0 > K\sqrt{c} </math>
| |
| As the concentration is increased however, the conductivity no longer rises in proportion.
| |
| Moreover, Kohlrausch also found that the limiting conductivity of anions and cations are additive: the conductivity of a solution of a salt is equal to the sum of conductivity contributions from the cation and anion.
| |
| :<math>\Lambda_m^0= \nu_+ \lambda_+^0 + \nu_-\lambda_-^0</math>
| |
| where:
| |
| * <math>\nu_+</math> and <math>\nu_-</math> are the number of moles of cations and anions, respectively, which are created from the dissociation of 1 mole of the dissolved electrolyte;
| |
| * <math>\lambda_+^0</math> and <math>\lambda_-^0</math> are the limiting molar conductivities of the individual ions.
| |
| | |
| The following table gives values for the limiting molar conductivities for selected ions.
| |
| | |
| <blockquote>
| |
| {|class="wikitable"
| |
| |+Limiting ion conductivity in water at 298 K
| |
| |-
| |
| !Cations!!λ<sub>+</sub><sup>0</sup> /mS m<sup>2</sup>mol<sup>-1</sup> !!anions!!λ<sub>-</sub><sup>0</sup> /mS m<sup>2</sup>mol<sup>-1</sup>
| |
| |-
| |
| |H<sup>+</sup>||34.96||OH<sup>-</sup>||19.91
| |
| |-
| |
| |Li<sup>+</sup>||3.869||Cl<sup>-</sup>||7.634
| |
| |-
| |
| |Na<sup>+</sup>||5.011||Br<sup>-</sup>||7.84
| |
| |-
| |
| |K<sup>+</sup>||7.350||I<sup>-</sup>||7.68
| |
| |-
| |
| |Mg<sup>2+</sup>||10.612||SO<sub>4</sub><sup>2-</sup>||15.96
| |
| |-
| |
| |Ca<sup>2+</sup>||11.900||NO<sub>3</sub><sup>-</sup>||7.14
| |
| |-
| |
| |Ba<sup>2+</sup>||12.728||[[acetate|CH<sub>3</sub>CO<sub>2</sub><sup>-</sup>]]||4.09
| |
| |}
| |
| </blockquote>
| |
| An interpretation of these results was based on the theory of Debye and Hückel.<ref>{{cite book|last=Wright, M.R.|title=An Introduction to Aqueous Electrolyte Solutions|publisher=Wiley|year=2007|isbn=978-0-470-84293-5}}</ref>
| |
| :<math>\Lambda_m =\Lambda_m^0-(A+B\Lambda_m^0 )\sqrt{c} </math>
| |
| where ''A'' and ''B'' are constants that depend only on known quantities such as temperature, the charges on the ions and the [[dielectric constant]] and [[viscosity]] of the solvent. As the name suggests, this is an extension of the [[Debye–Hückel equation|Debye–Hückel theory]], due to [[Lars Onsager|Onsager]]. It is very successful for solutions at low concentration.
| |
| | |
| === Weak electrolytes ===
| |
| A weak electrolyte is one that is never fully dissociated (i.e. there are a mixture of ions and complete molecules in equilibrium). In this case there is no limit of dilution below which the relationship between conductivity and concentration becomes linear. Instead, the solution becomes ever more fully dissociated at weaker concentrations, and for low concentrations of "well behaved" weak electrolytes, the degree of dissociation of the weak electrolyte becomes proportional to the inverse square root of the concentration.
| |
| | |
| Typical weak electrolytes are [[weak acid]]s and [[weak base]]s. The concentration of ions in a solution of a weak electrolyte is less than the concentration of the electrolyte itself. For acids and bases the concentrations can be calculated when the value(s) of the [[acid dissociation constant]](s) is(are) known.
| |
| | |
| For a [[monoprotic acid]], HA, obeying the inverse square root law, with a dissociation constant ''K''<sub>a</sub>, an explicit expression for the conductivity as a function of concentration, ''c'', known as [[Ostwald's dilution law]], can be obtained.
| |
| :<math>\frac{1}{\Lambda_m}=\frac{1}{\Lambda_m^0}+\frac{\Lambda_m c}{K_a(\Lambda_m^0)^2}</math>
| |
| | |
| === Higher concentrations ===
| |
| Both Kohlrausch's law and the Debye-Hückel-Onsager equation break down as the concentration of the electrolyte increases above a certain value. The reason for this is that as concentration increases the average distance between cation and anion decreases, so that there is more inter-ionic interaction. Whether this constitutes [[ion-association]] is a moot point. However, It has often been assumed that cation and anion interact to form an [[ion-association|ion-pair]]. Thus the electrolyte is treated as if it were like a weak acid and a constant, ''K'', can be derived for the equilibrium
| |
| :A<sup>+</sup> + B<sup>-</sup> {{eqm}} A<sup>+</sup>B<sup>-</sup>; K=[A<sup>+</sup>][B<sup>-</sup>]/[A<sup>+</sup>B<sup>-</sup>]
| |
| Davies describes the results of such calculations in great detail, but states that ''K'' should not necessarily be thought of as a true [[equilibrium constant]], rather, the inclusion of an "ion-association" term is useful in extending the range of good agreement between theory and experimental conductivity data.<ref>{{cite book|last=Davies|first=C.W.|title=Ion Association| publisher = Butterworths|location=London|year=1962}}</ref> Various attempts have been made to extend Onsager's treament to more concentrated solutions.<ref>{{cite journal| last=Miyoshi| first=K.| year=1973|title=Comparison of the Conductance Equations of Fuoss - Onsager, Fuoss-Hsia and Pitts with the Data of Bis(2, 9-dimethyl-1, 10-phenanthroline)Cu(I) Perchlorate |journal=Bull. Chem. Soc. Japan |volume=46|issue=2|pages=426–430| doi=10.1246/bcsj.46.426}}</ref>
| |
| | |
| The existence of a so-called ''conductance minimum'' has proved to be a controversial subject as regards interpretation. Fuoss and Kraus suggested that it is caused by the formation of ion-triplets,<ref>{{cite journal|doi=10.1021/ja01304a001|last=Fuoss|first=R.M.|coauthors=Kraus, C.A.|year=1935|title=Properties of Electrolytic Solutions. XV. Thermodynamic Properties of Very Weak Electrolytes|journal=J. Amer. Chem. Soc.|volume=57|pages=1–4}}</ref> and this suggestion has received some support recently.<ref>{{cite journal |last=Weingärtner |first=H.|coauthors=Weiss, V.C.; Schröer, W.|year=2000| title=Ion association and electrical conductance minimum in Debye–Hückel-based theories of the hard sphere ionic fluid|journal=J. Chem. Phys.|volume=113|issue=2|pages=762-|doi=10.1063/1.481822|bibcode = 2000JChPh.113..762W }}</ref><ref>{{cite journal|last=Schröer |first=W.|coauthors=Weingärtner, H.|year=2004|title=Structure and criticality of ionic fluids|journal=Pure Appl. Chem.|volume=76|issue=1|pages=19–27|doi=10.1351/pac200476010019}} [http://media.iupac.org/publications/pac/2004/pdf/7601x0019.pdf pdf]</ref>
| |
| | |
| === Conductivity Versus Temperature ===
| |
| Generally the conductivity of a solution increases with temperature, as the mobility of the ions increases. For comparison purposes reference values are reported at an agreed temperature, usually 298 K (~25 C), although occasionally 20C is used. It is often necessary to take readings from a sample at some other temperature, where it would be inconvenient to wait for the sample to heat or cool. So called 'compensated' measurements are made at a convenient temperature but the value reported is a calculated value of the expected value of conductivity of the solution, as if it had been measured at the reference temperature (usually 298 K (~25 C), although occasionally 20C is used). Basic compensation is normally done by assuming a linear increase of conductivity versus temperature of typically 2% per degree.<ref>{{cite web|url=http://www.jenway.com/adminimages/A02_001A_Effect_of_temperature_on_conductivity.pdf |title=The effect of temperature on conductivity measurement |publisher=Bibby Scientific |accessdate=5 January 2013}}</ref>
| |
| Although generally satisfactory for room temperatures, and for purely comparative measurements, the further the measurement temperature is from the reference temperature, the less accurate such simply compensated measurements become. More sophisticated instruments allow programmable compensation functions, which can be specific to the ion species being measured, and which may include additional non-linear terms.
| |
| | |
| == Applications ==
| |
| Notwithstanding the difficulty of theoretical interpretation, measured conductivity is a good indicator of the presence or absence of conductive ions in solution, and measurements are used extensively in many industries.<ref>{{cite web|url=http://www.aquariustech.com.au/pdfs/tech-bulletins/Electrol_Condct_Thery.pdf|title=Electrolytic conductivity measurement, Theory and practice|publisher=Aquarius Technologies Pty Ltd.}}</ref> For example, conductivity measurements are used to monitor quality in public water supplies, in hospitals, in boiler water and industries which depend on water quality such as brewing. This type of measurement is not ion-specific; it can sometimes be used to determine the amount of [[total dissolved solids]] (T.D.S.) if the composition of the solution and its conductivity behavior are known.<ref name=Gray/>
| |
| It should be noted that conductivity measurements made to determine water purity will not respond to non conductive contaminants (many organic compounds fall into this category), therefore additional purity tests may be required depending on application.
| |
| | |
| Sometimes, conductivity measurements are linked with other methods to increase the sensitivity of detection of specific types of ions. For example, in the boiler water technology, the [[boiler blowdown]] is continuously monitored for "cation conductivity", which is the conductivity of the water after it has been passed through a cation exchange resin. This is a sensitive method of monitoring anion impurities in the boiler water in the presence of excess cations (those of the alkalizing agent usually used for water treatment). The sensitivity of this method relies on the high mobility of H<sup>+</sup> in comparison with the mobility of other cations or anions.
| |
| | |
| Conductivity detectors are commonly used with [[ion chromatography]].<ref>{{cite web|url=http://www.chromatography-online.org/ion-chromatography/Detectors-for-Ion-Exchange-Chromatography.html|title=Detectors for ion-exchange chromatography|accessdate=2009-05-17}}</ref>
| |
| | |
| == See also ==
| |
| {{commons category|Conductometry}}
| |
| *[[Debye–Falkenhagen effect]]
| |
| *[[Wien effect]]
| |
| *[[Svante Arrhenius]]
| |
| *[[Alfred Werner]] - coordination chemistry
| |
| *[[equivalence point|Conductimetric titration]] - methods to determine the equivalence point
| |
| | |
| == References ==
| |
| {{reflist}}
| |
| {{Refbegin}}
| |
| *{{cite journal|last=Friedman|first=Harold L.|title=Relaxation Term of the Limiting Law of the Conductance of Electrolyte Mixtures|journal=The Journal of Chemical Physics|year=1965|volume=42|issue=2|pages=462|doi=10.1063/1.1695956}}
| |
| {{Refend}}
| |
| == External links ==
| |
| *[http://www.coleparmer.co.uk/techinfo/techinfo.asp?htmlfile=Conductivity.htm&ID=78 Cole-Parmer Technical Library] Conductivity
| |
| *[http://bat4ph.com/Presentations/Knick/Powerpoint/%5B04%5D%20Conductivity.ppt Conductivity measurement] PowerPoint presentation
| |
| [[Category:Physical chemistry]] | |
![f(x,y;\alpha ,\beta )=\left(\beta -1\right)\left(\pi \alpha ^{2}\right)^{{-1}}\left[1+\left({\frac {x^{2}+y^{2}}{\alpha ^{2}}}\right)\right]^{{-\beta }},\,](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d3f2d64a27d793be7c4dc4ab079ebd0283f7ea2)
![f(r;\alpha ,\beta )=2r{\frac {\beta -1}{\alpha ^{2}}}\left[1+\left({\frac {r^{2}}{\alpha ^{2}}}\right)\right]^{{-\beta }}.\,](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf0999e78ec6f81835d6a200eca71670740ef775)
