Hamilton–Jacobi–Bellman equation: Difference between revisions

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The '''van 't Hoff factor <math> i </math>''' (named after [[J. H. van 't Hoff]]) is a measure of the effect of a solute upon [[colligative properties]] such as [[osmotic pressure]], relative lowering in [[vapor pressure]], [[Boiling-point elevation|elevation of boiling point]] and [[freezing point depression]]. The van 't Hoff factor is the ratio between the actual concentration of particles produced when the substance is dissolved, and the [[concentration]] of a substance as calculated from its mass.  For most non-[[electrolyte]]s dissolved in water, the van' t Hoff factor is essentially 1.  For most [[ionic compound]]s dissolved in water, the van 't Hoff factor is equal to the number of discrete ions in a [[formula unit]] of the substance.  This is true for [[ideal solution]]s only, as occasionally [[Ion-association|ion pairing]] occurs in solution.  At a given instant a small percentage of the ions are paired and count as a single particle.  Ion pairing occurs to some extent in all electrolyte solutions.  This causes deviation from the van 't Hoff factor.  The deviation for the van 't Hoff factor tends to be greatest where the ions have multiple charges.


==Dissociated solutes==
The degree of dissociation is the fraction of the original solute molecules that have [[dissociation (chemistry)|dissociated]]. It is usually indicated by the Greek symbol <math> \alpha </math>. There is a simple relationship between this parameter and the van't Hoff factor. If a fraction <math> \alpha </math> of the solute dissociates into <math> n </math> ions, then


:::<math> i = \alpha n + (1- \alpha) = 1 + \alpha (n - 1) </math>
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For example, the dissociation
 
:::KCl {{Eqm}} K<sup>+</sup> + Cl<sup>-</sup>
 
yields <math> n = 2 </math> particles, so that <math> i = 1 + \alpha </math>
 
==Associated solutes==
Similarly, if a fraction <math>\alpha</math> of <math> n </math> moles of solute associate to form one mole of an n-mer (dimer, trimer, etc.), then
 
:::<math> i =1 - (1 - \frac{1}{n})\alpha </math>
 
For the dimerisation of acetic acid in benzene
 
:::2CH<sub>3</sub>COOH {{Eqm}} (CH<sub>3</sub>COOH)<sub>2</sub>
 
2 moles of acetic acid associate to form 1 mole of dimer, so that
:::<math>i=1-(1-\frac{1}{2})\alpha = 1-\frac{\alpha}{2}</math>
 
== Physical significance of <math>i</math> ==
 
[[File:Carboxylic acid dimers.png|thumb|Dimerization of a carboxylic acid by formation of [[hydrogen bond]]s (shown as dotted lines).]]
* When solute particles associate in solution, <math>i</math> is less than 1. For example, [[carboxylic acid]]s such as ethanoic acid ([[acetic acid]]) or [[benzoic acid]] form [[Dimer (chemistry)|dimers]] in benzene, so that the number of solute particles is half the number of acid molecules.
 
* When solute particles [[dissociate]] in solution, <math>i</math> is greater than 1. (''e.g.'' sodium chloride in water, [[potassium chloride]] in water, [[magnesium chloride]] in water)
* When solute particles neither dissociate nor associate in solution, <math>i</math> equals 1. (''e.g.'' Glucose in water)
The value of <math>i</math> is ;
<math>i</math> = the actual number of particles in solution after dissociation ÷ the number of formula units initially dissolved in solution.
Means the number of particles per formula unit of the solute when a solution is dilute.  
* Association of Molecules (i<1) <math>i = 1 - \frac{n-1}{n}\alpha</math>
 
== See also ==
* [[Colligative properties]]
* [[Solutions]]
* [[Raoult's law]]
* [[Van 't Hoff equation]]
* [[Dissociation (chemistry)]]
* [[Osmotic coefficient]]
 
[[Category:Physical chemistry]]
[[Category:Dimensionless numbers]]

Revision as of 19:13, 4 March 2014


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