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[[File:Slope Field.png|thumb|right|250px|The slope field of dy/dx=x<sup>2</sup>-x-2, with the blue, red, and turquoise lines being (x<sup>3</sup>/3)-(x<sup>2</sup>/2)-2x+4, (x<sup>3</sup>/3)-(x<sup>2</sup>/2)-2x, and (x<sup>3</sup>/3)-(x<sup>2</sup>/2)-2x-4, respectively.]]
A '''radical ion''' is a [[Radical (chemistry)|free radical]] species that carries a [[charge (chemistry)|charge]].<ref>{{GoldBookRef|title=radical ion|file= R05073}}</ref> Radical [[ion]]s are encountered in [[organic chemistry]] as [[reactive intermediates]] and in [[mass spectrometry]] as gas phase ions. Positive radical ions are called radical cations whereas negative radical ions are called radical anions.
In [[mathematics]], a '''slope field''' (or '''direction field''') is a graphical representation of the solutions of a first-order [[differential equation]]. It is useful because it can be created without solving the differential equation analytically. The representation may be used to qualitatively visualize solutions, or to numerically approximate them.


==Definition==
==Notation==
===Standard case===
In organic chemistry, a radical ion is typically indicated by a superscript dot followed by the sign of the charge: <math>M^{\bullet +}</math> and <math>M^{\bullet -}</math>. In mass spectrometry, the sign is written first, followed by the superscripted dot:  <math>M^{+\bullet}</math> and <math>M^{-\bullet}</math>.<ref name="isbn0-9660813-2-3">{{cite book |author=Sparkman, O. David |title=Mass spectrometry desk reference |publisher=Global View Pub |location=Pittsburgh |year=2000 |isbn=0-9660813-2-3 |oclc= |doi= |page=53}}</ref>
The slope field is traditionally defined for the following type of differential equations
:<math>y'=f(x,y)</math>.
It can be viewed as a creative way to plot a real-valued function of two real variables <math>f(x,y)</math> as a planar picture. Specifically, for a given pair <math>x,y</math>, a vector with the components <math>[1, f(x,y)]</math> is drawn at the point <math>x,y</math> on the <math>x,y</math>-plane. Sometimes, the vector <math>[1, f(x,y)]</math> is normalized to make the plot better looking for a human eye. A set of pairs <math>x,y</math> making a rectangular grid is typically used for the drawing.


An [[Isocline]] (a series of lines with the same slope) is often used to supplement the slope field. In an equation of the form <math>y'=f(x,y)</math>, the isocline is a line in the <math>x,y</math>-plane plane obtained by setting <math>f(x,y)</math> equal to a constant.
== Radical anions ==
Many [[aromatic]] compounds can undergo [[one-electron reduction]] by [[alkali metal]]s. For example the reaction of [[naphthalene]] with [[sodium]] in an aprotic solvent yields the naphthalene '''radical anion''' - sodium ion salt. In a [[ESR spectrum]] this compound shows up as a [[quintet]] of quintets (25 lines). In the presence of a [[hydrogen ion|proton]] source the radical anion is protonated and effectively hydrogenated like in the [[Birch reduction]].


===General case of a system of differential equations===
The electron is transferred from the alkali metal ion to an unoccupied antibonding p-p п* orbital of the aromatic molecule.  This transfer is usually only energetically favorable if the aprotic solvent efficiently solvates the alkali metal ion.  Effectiveness for this is in the order [[diethyl ether]] < THF < [[dimethoxyethane|1,2-dimethoxyethane]] < [[HMPA]].  In principle any unsaturated molecule can form a radical anion, but the antibonding orbitals are only energetically accessible in more extensive conjugated systems.  Ease of formation is in the order [[benzene]] < [[naphthalene]] < [[anthracene]] < [[pyrene]], etc.  On addition of a proton source, the structure of the resulting hydrogenated molecule is defined by the charge distribution of the radical anion.  For instance, the anthracene radical anion forms mainly (but not exclusively) 9,10-dihydroanthracene.
Given a system of differential equations,
:<math>\frac{dx_1}{dt}=f_1(t,x_1,x_2,\ldots,x_n)</math>
:<math>\frac{dx_2}{dt}=f_2(t,x_1,x_2,\ldots,x_n)</math>
:::<math>\vdots</math>
:<math>\frac{dx_n}{dt}=f_n(t,x_1,x_2,\ldots,x_n)</math>
the slope field is an array of slope marks in the [[phase space]] (in any number of dimensions depending on the number of relevant variables; for example, two in the case of a first-order linear [[ordinary differential equation|ODE]], as seen to the right).  Each slope mark is centered at a point <math>(t,x_1,x_2,\ldots,x_n)</math> and is parallel to the vector


:<math>\begin{pmatrix} 1 \\ f_1(t,x_1,x_2,\ldots,x_n) \\ f_2(t,x_1,x_2,\ldots,x_n) \\ \vdots \\ f_n(t,x_1,x_2,\ldots,x_n) \end{pmatrix}</math>.
An example of a non-carbon radical anion is the [[superoxide]] anion, formed by transfer of one electron to an [[oxygen]] molecule.
The number, position, and length of the slope marks can be arbitrary.  The positions are usually chosen such that the points <math>(t,x_1,x_2,\ldots,x_n)</math> make a uniform grid. The standard case, described above, represents <math>n=1</math>. The general case of the slope field for systems of differential equations is not easy to visualize for <math>n>2</math>.


==General application==
A very effective way to remove any traces of water from [[THF]] is by [[reflux]] with [[sodium]] wire in the presence of a small amount of [[benzophenone]]. Benzophenone is reduced to the [[ketyl]] radical anion by sodium which gives the THF solution an intense blue color. However, any trace of water in THF will further reduce the ketyl to the colourless [[alcohol]]. In this way, the color of the THF signals the dryness and the [[distilled]] THF contains less than 10 [[Parts per million|ppm]] of water.<ref>[http://www.erowid.org/archive/rhodium/chemistry/equipment/benzophenone.ketyl.html The Benzophenone Ketyl Still Pot - [www.rhodium.ws&#93;<!-- Bot generated title -->]</ref> This treatment also effectively removes any peroxides in the THF.  Radical anions of this type are also involved in the [[Acyloin condensation]].
With computers, complicated slope fields can be quickly made without tedium, and so an only recently practical application is to use them merely to get the feel for what a solution should be before an explicit general solution is sought.  Of course, computers can also just solve for one, if it exists.


If there is no explicit general solution, computers can use slope fields (even if they aren’t shown) to numerically find graphical solutions. Examples of such routines are [[Euler's method]], or better, the [[Runge-Kutta methods]].
[[Cyclooctatetraene]] is reduced by elemental [[potassium]] all the way to the dianion because the 10 electron system is aromatic. [[Quinone]] is reduced to a [[semiquinone]] radical anion. [[Semidione]]s are derived from the reduction of dicarbonyl compounds.


==Software for plotting slope fields==
== Radical cations ==
Different software packages can plot slope fields.  
Cationic radical species do also exist but are much less stable. They appear prominently in mass spectrometry, and some compounds containing the [[dioxygenyl]] cation can be prepared in bulk.<ref>{{cite doi|10.1021/ic50013a036}}</ref> When a gas-phase molecule is subjected to [[electron ionization]] one electron is abstracted by an electron in the electron beam to create a radical cation M<sup>+</sup><sup>.</sup>. This species represents the [[molecular ion]] or parent ion and will tell the precise [[molecular weight]]. On a typical [[mass spectrum]] more signals show up because the molecular ion fragments into a complex mixture of ions and uncharged radical species. For example the [[methanol]] radical cation fragments into a [[methyl]] cation CH<sub>3</sub><sup>+</sup> and a [[hydroxyl]] radical. In [[naphthalene]] the unfragmented radical cation is by far the most prominent peak in the mass spectrum. Secondary species are generated from [[hydrogen ion|proton]] gain (M+1) and proton loss (M-1).  


===Example code in [[GNU Octave]]/[[MATLAB]] ===
[[Polaron]]s and [[bipolaron]]s are radical cations encountered in doped [[conducting polymer]]s.
<source lang="matlab">
Ffun = @(X,Y)X.*Y;              % function f(x,y)=xy
[X,Y]=meshgrid(-2:.3:2,-2:.3:2); % choose the plot sizes
DY=Ffun(X,Y); DX=ones(size(DY)); % generate the plot values
quiver(X,Y,DX,DY);              % plot the direction field
hold on;
contour(X,Y,DY,[-6 -2 -1 0 1 2 6]); %add the isoclines
title('Slope field and isoclines for f(x,y)=xy')
</source>
 
===Alternate example code in [[GNU Octave]]/[[MATLAB]] ===
<source lang="matlab">
funn = @(x,y)y-x;                % function f(x,y)=y-x
[x,y]=meshgrid(-2:0.5:2);        % intervals for x and y
slopes=funn(x,y);                % matrix of slopes
dy=slopes./sqrt(1+slopes.^2);    % normalize the line element...
dx=sqrt(1-dy.^2);                % ...magnitudes for dy and dx
quiver(x,y,dx,dy);              % plot the direction field
</source>
 
=== Example code for [[Maxima (software) | Maxima]] ===
 
/* field for y'=xy (click on a point to get an integral curve) */
plotdf( x*y, [x,-2,2], [y,-2,2]);
 
==Examples==
<gallery Caption="y' = xy">
Image:Slope_field_1.svg|Slope field
Image:Slope_field_with_integral_curves_1.svg|Integral curves
image:Isocline_3.png|Isoclines (blue), slope field (black), and some solution curves (red)
</gallery>
 
==See also==
*[[Examples of differential equations]]
*[[Vector field]]
*[[Laplace transform applied to differential equations]]
*[[List of dynamical systems and differential equations topics]]


==References==
==References==
* Blanchard, Paul; Devaney, Robert L.; and Hall, Glen R. (2002). ''Differential Equations'' (2nd ed.). Brooks/Cole: Thompson Learning. ISBN 0-534-38514-1
{{Reflist}}


==External links==
{{DEFAULTSORT:Radical Ion}}
* {{MathWorld |title = Slope field |urlname = SlopeField}}
[[Category:Reactive intermediates]]
* [http://www.math.psu.edu/cao/DFD/Dir.html Slope field plotter]
[[Category:Mass spectrometry]]


[[Category:Calculus]]
[[fr:Ion radicalaire]]
[[Category:Differential equations]]
[[it:Ione radicalico]]
[[Category:Articles with example MATLAB/Octave code]]
[[uk:Іон-радикал]]

Revision as of 03:25, 14 August 2014

A radical ion is a free radical species that carries a charge.[1] Radical ions are encountered in organic chemistry as reactive intermediates and in mass spectrometry as gas phase ions. Positive radical ions are called radical cations whereas negative radical ions are called radical anions.

Notation

In organic chemistry, a radical ion is typically indicated by a superscript dot followed by the sign of the charge: M∙+ and M∙−. In mass spectrometry, the sign is written first, followed by the superscripted dot: M+∙ and M−∙.[2]

Radical anions

Many aromatic compounds can undergo one-electron reduction by alkali metals. For example the reaction of naphthalene with sodium in an aprotic solvent yields the naphthalene radical anion - sodium ion salt. In a ESR spectrum this compound shows up as a quintet of quintets (25 lines). In the presence of a proton source the radical anion is protonated and effectively hydrogenated like in the Birch reduction.

The electron is transferred from the alkali metal ion to an unoccupied antibonding p-p п* orbital of the aromatic molecule. This transfer is usually only energetically favorable if the aprotic solvent efficiently solvates the alkali metal ion. Effectiveness for this is in the order diethyl ether < THF < 1,2-dimethoxyethane < HMPA. In principle any unsaturated molecule can form a radical anion, but the antibonding orbitals are only energetically accessible in more extensive conjugated systems. Ease of formation is in the order benzene < naphthalene < anthracene < pyrene, etc. On addition of a proton source, the structure of the resulting hydrogenated molecule is defined by the charge distribution of the radical anion. For instance, the anthracene radical anion forms mainly (but not exclusively) 9,10-dihydroanthracene.

An example of a non-carbon radical anion is the superoxide anion, formed by transfer of one electron to an oxygen molecule.

A very effective way to remove any traces of water from THF is by reflux with sodium wire in the presence of a small amount of benzophenone. Benzophenone is reduced to the ketyl radical anion by sodium which gives the THF solution an intense blue color. However, any trace of water in THF will further reduce the ketyl to the colourless alcohol. In this way, the color of the THF signals the dryness and the distilled THF contains less than 10 ppm of water.[3] This treatment also effectively removes any peroxides in the THF. Radical anions of this type are also involved in the Acyloin condensation.

Cyclooctatetraene is reduced by elemental potassium all the way to the dianion because the 10 electron system is aromatic. Quinone is reduced to a semiquinone radical anion. Semidiones are derived from the reduction of dicarbonyl compounds.

Radical cations

Cationic radical species do also exist but are much less stable. They appear prominently in mass spectrometry, and some compounds containing the dioxygenyl cation can be prepared in bulk.[4] When a gas-phase molecule is subjected to electron ionization one electron is abstracted by an electron in the electron beam to create a radical cation M+.. This species represents the molecular ion or parent ion and will tell the precise molecular weight. On a typical mass spectrum more signals show up because the molecular ion fragments into a complex mixture of ions and uncharged radical species. For example the methanol radical cation fragments into a methyl cation CH3+ and a hydroxyl radical. In naphthalene the unfragmented radical cation is by far the most prominent peak in the mass spectrum. Secondary species are generated from proton gain (M+1) and proton loss (M-1).

Polarons and bipolarons are radical cations encountered in doped conducting polymers.

References

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fr:Ion radicalaire it:Ione radicalico uk:Іон-радикал

  1. ↑ Template:GoldBookRef
  2. ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  3. ↑ The Benzophenone Ketyl Still Pot - [www.rhodium.ws]
  4. ↑ Template:Cite doi
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