formulasearchengine - User contributions [en]

Universal composability

2013-08-27T11:44:24Z

131.174.142.205:

'''Absolute electrode potential''', in [[electrochemistry]], according to an [[IUPAC]] definition,<ref>[http://goldbook.iupac.org/A00022.html IUPAC Gold Book - absolute electrode potential]</ref> is the [[electrode potential]] of a [[metal]] measured with respect to a universal reference system (without any additional metal–solution interface).

==Definition==
According to a more specific definition presented by Trasatti,<ref> Sergio Trasatti, "The Absolute Electrode Potential: an Explanatory Note (Recommendations 1986)", International Union of Pure and Applied Chemistry,
Pure & AppL Chem., Vol. 58, No. 7, pp. 955–66, 1986. http://www.iupac.org/publications/pac/1986/pdf/5807x0955.pdf (pdf)</ref> the absolute electrode potential is the difference in electronic energy between a point inside the metal ([[Fermi level]]) of an [[electrode]] and a point outside the [[electrolyte]] in which the electrode is submerged (an electron at rest in vacuum).

This potential is difficult to determine accurately. For this reason, [[standard hydrogen electrode]] is typically used for reference potential. The absolute potential of the SHE is 4.44 ± 0.02 [[volt|V]] at 25 [[°C]]. Therefore, for any electrode at 25 °C:

:<math>E^M_{\rm{(abs)}} = E^M_{\rm{(SHE)}}+(4.44 \pm 0.02)\ {\mathrm V}</math>

where:
:{{mvar|E}} is electrode potential
:V is [[volt]]
:''M'' denotes the electrode made of metal M
:(abs) denotes the absolute potential
:(SHE) denotes the electrode potential relative to the standard hydrogen electrode.

A different definition for the absolute electrode potential (also known as absolute half-cell potential and single electrode potential) has also been discussed in the literature.<ref> Alan L. Rockwood, "Absolute half-cell thermodynamics: Electrode potential", Physical Review A, Vol 33, No. 1, pp. 554–59, 1986.</ref> In this approach, one first defines an isothermal absolute single-electrode process (or absolute half-cell process.) For example, in the case of a generic metal being oxidized to form a solution-phase ion, the process would be

:M(metal) → M+(solution) + {{subatomic particle|electron|link=yes}}(gas)

For the [[hydrogen]] electrode, the absolute half-cell process would be

:{{sfrac|1|2}}H2 (gas) → [[hydron (chemistry)|H+]](solution) + {{subatomic particle|electron}}(gas)

Other types of absolute electrode reactions would be defined analogously.

In this approach, all three species taking part in the reaction, including the electron, must be placed in thermodynamically well-defined states. All species, including the electron, are at the same temperature, and appropriate standard states for all species, including the electron, must be fully defined. The absolute electrode potential is then defined as the Gibbs free energy for the absolute electrode process. To express this in volts one divides the Gibb’s free energy by the negative of Faraday’s constant.

Rockwood's approach to absolute-electrode thermodynamics is easily expendable to other thermodynamic functions. For example, the absolute half-cell entropy has been defined as the entropy of the absolute half-cell process defined above.<ref>Alan L. Rockwood, "Absolute half-cell entropy", Physical Review A, vol. 36, No. 3, pp. 1525–26, 1987.</ref> An alternative definition of the absolute half-cell entropy has recently been published by Fang et al.<ref>Zheng Fang, Shaofen Wang, Zhenghua Zhang, and Guanzhou Qiu, "The electrochemical Peltier heat of the standard hydrogen electrode reaction", Thermochimica Acta, Vol. 473, pp. 40–44, 2008.</ref> who define it as the entropy of the following reaction (using the hydrogen electrode as an example):

:{{sfrac|1|2}}H2 (gas) → H+(solution) + {{subatomic particle|electron}}(metal)

This approach differs from the approach described by Rockwood in the treatment of the electron, i.e. whether it is placed in the gas phase or in the metal.

==Determination==
The basis for determination of the absolute electrode potential under the Trasatti definition is given by the equation:

:<math>E^M{\rm (abs)} = \phi^M + \Delta ^M_S \psi</math>

where:
:{{math|''E''''M''(abs)}} is the absolute potential of the electrode made of metal M
:<math>\phi^M</math> is the electron [[work function]] of metal M
:<math>\Delta ^M_S \psi</math> is the [[Volta potential|contact (Volta) potential]] difference at the metal(''M'')–solution(''S'') interface.

For practical purposes, the value of the absolute electrode potential of the standard hydrogen electrode is best determined with the utility of data for an [[ideally polarizable electrode|ideally-polarizable]] [[mercury (element)|mercury]] (Hg) electrode:

:<math>E^\ominus {\rm (H^+/H_2)(abs)} = \phi^{\rm{Hg}} + \Delta ^{\rm{Hg}} _S \psi^\ominus_{\sigma=0} - E^{\rm{Hg}}_{\sigma=0}\rm{(SHE)}</math>

where:
:<math>E^\ominus {\rm (H^+/H_2)(abs)}</math> is the absolute standard potential of the hydrogen electrode
:{{math|1=''σ'' = 0}} denotes the condition of the [[point of zero charge]] at the interface.

The types of physical measurements required under the Rockwood definition are similar to those required under the Trasatti definition, but they are used in a different way, e.g. in Rockwood's approach they are used to calculate the equilibrium vapor pressure of the electron gas. The numerical value for the absolute potential of the standard hydrogen electrode one would calculate under the Rockwood definition is sometimes fortuitously close to the value one would obtain under the Trasatti definition. This near-agreement in the numerical value depends on the choice of ambient temperature and standard states, and is the result of the near-cancellation of certain terms in the expressions. For example, if a standard state of one atmosphere ideal gas is chosen for the electron gas then the cancellation of terms occurs at a temperature of 296 K, and the two definitions give an equal numerical result. At 298.15 K a near-cancellation of terms would apply and the two approaches would produce nearly the same numerical values. However, there is no fundamental significance to this near agreement because it depends on arbitrary choices, such as temperature and definitions of standard states.

==See also==
* [[Standard electrode potential]]
* [[Galvani potential]]
* [[Electrochemical potential]]

==References==
{{Reflist}}

[[Category:Electrochemistry]]
[[Category:Potentials]]

AW*-algebra

2013-05-23T15:23:18Z

131.174.142.211: fixed incorrect statement to a correct one

In [[population ecology]], '''Moran's theorem''' (or the Moran effect) states that the time [[correlation]] of two separate populations of the same species is equal to the correlation between the environmental variabilities where they live.

The theorem is named after [[Pat Moran (statistician)|Pat Moran]], who stated it in a paper on the dynamics of the [[Canadian lynx]] populations.<ref>Moran, P. A. P. 1953. The statistical analysis of the Canadian lynx cycle. II. Synchronization and meteorology. Australian Journal of Zoology 1: 291-298.</ref> It has been used to explain the synchronization of widely dispersed populations. It has the important consequence for [[conservation biology|conservation ecology]] that [[Minimum viable population|viability]] of spatially structured populations is lower than one would expect from the local populations: it increases the probability that several local populations go extinct simultaneously.<ref>Jörgen Ripa, Theoretical Population Ecology and Evolution Group, [http://equation-of-the-month.blogspot.co.uk/2012/02/moran-effect.html Equation of the month: the Moran effect]</ref>

In its original form it stated: If the two populations have population dynamics given by
:<math>N_1(t+1)=f(N_1(t))+\epsilon_1(t)</math>
:<math>N_2(t+1)=f(N_2(t))+\epsilon_2(t)</math>
where <math>N_i</math> is the population size of population <math>i</math>, <math>f</math> is a linear renewal function updating the populations in the same way, and <math>\epsilon_i</math> the environmental variabilities. Then <math>\rho_{N_1,N_2}=\rho_{\epsilon_1,\epsilon_2}</math>.

The original form assumed a strictly linear structure, but this assumption can be weakened to allow for non-linear functions. It has been suggested that the term "Moran effect" should be used for systems that do not strictly follow the original description.<ref>Esa Ranta, Veijo Kaitala, Per Lundberg, Ecology of Populations, Cambridge University Press, 2006 p. 78</ref> In the general case the correlations will be lower, and the accuracy of the Moran description depends on whether the populations tend to converge to an equilibrium state (good accuracy for low variance variability) or tend to oscillate (eventual breakdown of the correlation).<ref>T. Royama 2005. Moran effect on nonlinear population processes. Ecological Monographs 75:277–293. http://dx.doi.org/10.1890/04-0770</ref>

It has been tested experimentally in a number of cases, such as variation of fruit production,<ref>Rosenstock, T. S., Hastings, A., Koenig, W. D., Lyles, D. J. and Brown, P. H. (2011), Testing Moran's theorem in an agroecosystem. Oikos, 120: 1434–1440. doi: 10.1111/j.1600-0706.2011.19360.x</ref> acorn production,<ref>Ecology. 2013 Jan;94(1):83-93.
Large-scale spatial synchrony and cross-synchrony in acorn production by two California oaks.
Koenig WD, Knops JM.</ref> bird populations<ref>SÆTHER, B.-E., ENGEN, S., GRØTAN, V., FIEDLER, W., MATTHYSEN, E., VISSER, M. E., WRIGHT, J., MØLLER, A. P., ADRIAENSEN, F., VAN BALEN, H., BALMER, D., MAINWARING, M. C., MCCLEERY, R. H., PAMPUS, M. and WINKEL, W. (2007), The extended Moran effect and large-scale synchronous fluctuations in the size of great tit and blue tit populations. Journal of Animal Ecology, 76: 315–325. doi: 10.1111/j.1365-2656.2006.01195.x</ref> and coral reef fishes.<ref>Ecology. 2007 Jan;88(1):158-69. Spatial synchrony in coral reef fish populations and the influence of climate. Cheal AJ, Delean S, Sweatman H, Thompson AA.</ref>

==References==
{{reflist}}

[[Category:Population ecology]]
[[Category:Demography]]

Tightness of measures

2013-05-21T13:27:49Z

131.174.142.211: /* Compact spaces */

'''AMS-LaTeX''' is a collection of
[[LaTeX]] document classes and packages developed for the [[American Mathematical Society]] (AMS). Its additions to LaTeX include the typesetting of multi-line and other mathematical statements, document classes, and fonts containing numerous mathematical symbols.<ref>{{cite book
| url = http://www.ctan.org/tex-archive/info/mil/mil.pdf
| title = Math into LaTeX
| author = George Gratzer
| year = 1996
| isbn = 0-8176-3805-9
| accessdate = 2007-10-08
}}</ref>

It has largely superseded the
plain [[TeX]] macro package
'''AMS-TeX'''. AMS-TeX was originally written by [[Michael Spivak]], and was used by the AMS from 1983 to 1985.

The following code of the
LaTeX2e produces the AMS-LaTeX logo ([[Image:AMS-LaTeX.svg|80px|\AmS-\LaTeX]]):

<source lang=latex>
%%% -- AMS-LaTeX_logo.tex -------
\documentclass{article}
\usepackage{amsmath}

\begin{document}
\AmS-\LaTeX
\end{document}
</source>

The package has a suite of facilities to format multi-line equations. For example, the following
code,

<source lang=latex>
\begin{align}
y &= (x+1)^2 \\
&= x^2+2x+1
\end{align}
</source>

causes the equals signs in the two lines to be aligned with one another, like this:

:<math>
\begin{align}
y &= (x+1)^2 \\
&= x^2+2x+1
\end{align}
</math>

AMS-LaTeX also includes many flexible commands for formatting and numbering theorems, lemmas, etc. For example, one may use the environment <tt>theorem</tt>

<source lang=latex>
\begin{theorem}[Pythagoras] Suppose $a\leq b\leq c$ are the side-lengths of a right triangle.\\ Then $a^2+b^2=c^2$.\end{theorem}
\begin{proof}. . . \end{proof}
</source>

to generate

<blockquote>
'''Theorem''' (''Pythagoras'') ''Suppose'' <math>a\leq b\leq c</math> ''are the side-lengths of a right triangle. '' ''Then'' <math>a^2+b^2=c^2</math>. 
'''Proof'''. . . □
</blockquote>

==See also==
* [[AMSRefs]]

==References==
<references />

==External links==
*[http://www.ams.org/tex/ AMS TeX Resources]
*[http://www.tex.ac.uk/cgi-bin/texfaq2html?label=AMSpkg TeX FAQ on AMS packages]
*[http://www.tex.ac.uk/cgi-bin/texfaq2html?label=AMSpkg TeX FAQ on AMS-TeX]

[[Category:TeX]]

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