User:Milton Beychok/Sandbox: Difference between revisions

From Citizendium
Jump to navigation Jump to search
imported>Milton Beychok
imported>Milton Beychok
No edit summary
Line 1: Line 1:
'''Henry's law''' is one of the [[gas laws]], formulated by the British chemist, William Henry, in 1803.  It states that:


:''At a constant temperature, the amount of a given gas dissolved in a given type and volume of liquid is directly proportional to the [[partial pressure]] of that gas in equilibrium with that liquid.''
==Formula and Henry constant==
A formula for Henry's Law is:
:<math> e^{p\,} = e^{kc\,} \,</math>
where:
:<math>e\,</math> is approximately 2.7182818, the base of the [[natural logarithm]] (also called [[Euler's number]])
:<math>p\,</math> is the partial pressure of the [[solute]] above the [[solution]]
:<math>c\,</math> is the [[concentration]] of the solute in the solution (in one of its many units)
:<math>k\,</math> is the Henry's Law constant, which has units such as L·atm/mol, atm/([[mol fraction]]) or Pa·m<sup>3</sup>/mol.
Taking the [[natural logarithm]] of the formula, gives us the more commonly used formula:<ref>[http://www.udel.edu/pchem/C443/Lectures/Lecture33.pdf University of Delaware physical chemistry lecture]</ref> <ref name=Mortimer>{{cite book|author=Robert G. Mortimer|title=Physical Chemistry|edition=Second Edition|publisher=Academic Press|year=2000|id=ISBN 0-12-508345-9}}</ref><ref>[http://www.800mainstreet.com/9/0009-006-henry.html Online Introductory Chemistry: Solubiltiy of gases in liquids]</ref>
:<math> p = kc \,</math>
Some values for ''k'' include:
:[[oxygen]] (O<sub>2</sub>) : 769.2 L·atm/mol &nbsp; &nbsp; &nbsp;
:[[carbon dioxide]] (CO<sub>2</sub>) : 29.4 L·atm/mol &nbsp; &nbsp; &nbsp;
:[[hydrogen]] (H<sub>2</sub>) : 1282.1 L·atm/mol &nbsp; &nbsp;
when these gases are dissolved in [[water]] at 298 [[kelvin]]s.
'''As shown in Table 1 below, there are other forms of Henry's Law each of which defines the constant ''k'' differently and requires different dimensional units'''.<ref name=SmithandHarvey>{{cite journal|author=Francis L. Smith and Allan H. Harvey |year=2007 |month=September |title=Avoid Common Pitfalls When Using Henry's Law |journal=CEP (Chemical Engineering Progress) |volume= |issue= |pages= |issn=0360-7275}}</ref> The form of the equation presented above is consistent with the example numerical values presented for oxygen, carbon dioxide and hydrogen and with their corresponding dimensional units.
Note that the unit of concentration was chosen to be [[molarity]]. Hence the dimensional units: ''L'' is liters of solution, ''atm'' is the partial pressure of the gaseous solute  above the solution (in atmospheres of absolute pressure), and ''mol'' is the moles of the gaseous solute in the solution. Also note that the Henry's Law constant, ''k'',  varies with the solvent and the temperature.
===Other forms of Henry's law===
There are various other forms Henry's Law which are discussed in the technical literature.<ref name=SmithandHarvey/><ref name=UArizona>[http://www.chem.arizona.edu/~salzmanr/103a004/nts004/l41/l41.html University of Arizona chemistry class notes]</ref><ref name="multiple">[http://www.henrys-law.org An extensive list of Henry's law constants, and a conversion tool]</ref>
{| class="wikitable"
|+ '''Table 1: Some forms of Henry's law and constants (gases in water at 298 K), derived from <ref name="multiple"/>
! equation: || <math> k_{H,pc} = \frac{p_{gas}}{c_{aq}}</math> || <math> k_{H,cp} = \frac{c_{aq}}{p_{gas}} </math> || <math> k_{H,px} = \frac{p_{gas}}{x_{aq}} </math> || <math> k_{H,cc} = \frac{c_{aq}}{c_{gas}} </math>
|-
! dimension: || <math>\left[\frac{\mathrm{L}_{soln} \cdot \mathrm{atm}}{\mathrm{mol}_{gas}}\right]</math> || <math> \left[\frac{\mathrm{mol}_{gas}}{\mathrm{L}_{soln} \cdot \mathrm{atm}}\right] </math> || <math>\left[\frac{\mathrm{atm} \cdot (\mathrm{mol}_{water}+
\mathrm{mol}_{gas})}{\mathrm{mol}_{gas}}\right]</math> || <math>\left[ \text{dimensionless} \right]</math>
|-
|align=center| [[Oxygen|O<sub>2</sub>]] ||align=center| 769.23||align=center| 1.3 E-3 ||align=center| 4.259 E4 ||align=center| 3.180 E-2
|-
|align=center| [[Hydrogen|H<sub>2</sub>]] ||align=center| 1282.05 ||align=center| 7.8 E-4 ||align=center| 7.099 E4 ||align=center| 1.907 E-2
|-
|align=center| [[CO2|CO<sub>2</sub>]] ||align=center| 29.41 ||align=center| 3.4 E-2 ||align=center| 0.163 E4 ||align=center| 0.8317
|-
|align=center| [[Nitrogen|N<sub>2</sub>]] ||align=center| 1639.34  ||align=center| 6.1 E-4 ||align=center| 9.077 E4 ||align=center| 1.492 E-2
|-
|align=center| [[Helium|He]] ||align=center| 2702.7 ||align=center| 3.7 E-4||align=center| 14.97 E4 ||align=center| 9.051 E-3
|-
|align=center| [[Neon|Ne]] ||align=center| 2222.22 ||align=center| 4.5 E-4 ||align=center| 12.30 E4 ||align=center| 1.101 E-2
|-
|align=center| [[Argon|Ar]] ||align=center| 714.28 ||align=center| 1.4 E-3 ||align=center| 3.955 E4 ||align=center| 3.425 E-2
|-
|align=center| [[Carbon monoxide|CO]] ||align=center| 1052.63  ||align=center| 9.5 E-4 ||align=center| 5.828 E4 ||align=center| 2.324 E-2
|}
where:
:<math>c_{aq}\,</math> = [[mole (unit)|moles]] of gas per [[liter]] of solution
:<math>\mathrm{L}_{soln}\,</math> = liters of solution
:<math>p_{gas}\,</math> = partial pressure of gas above the solution, in [[atmosphere (unit)|atmospheres]] of [[absolute pressure]]
:<math>x_{aq}\,</math> = mole fraction of gas in solution = moles of gas per total moles ≈ moles of gas per mole of water
:<math>\mathrm{atm}\,</math> = atmospheres of absolute pressure
As can be seen by comparing the equations in the above table, the Henry's Law constant <math>k_{H,pc}</math> is simply the inverse of the constant <math>k_{H,cp}</math>. Since all <math>k_{H}</math> may be referred to as the Henry's Law constant, readers of the technical literature must be quite careful to note which version of the Henry's Law equation is being used.<ref name=SmithandHarvey/>
It should also be noted the Henry's Law is a limiting law that only applies for ''dilute enough'' solutions. The range of concentrations in which it applies becomes narrower the more the system diverges from non-ideal behavior. Roughly speaking, that is the more chemically ''different'' the solute is from the solvent.
It also only applies for solutions where the solvent does not [[chemical reaction|react chemically]] with the gas being dissolved.  A common example of a gas that does react with the solvent is [[carbon dioxide]], which rapidly forms hydrated carbon dioxide  and then [[carbonic acid]] (H<sub>2</sub>CO<sub>3</sub>) with water.
===Temperature dependence of the Henry constant===
When the temperature of a system changes, the Henry constant will also change.<ref name=SmithandHarvey/> This is why some people prefer to name it Henry coefficient. There are multiple equations assessing the effect of temperature on the constant. A simple example is <ref name="multiple"/>, which is a form of the [[van 't Hoff equation]]:
:<math> k(T) = k(T_\Theta) \cdot e^{ \left[ -C \cdot \left( \frac{1}{T}-\frac{1}{T_\Theta}\right)\right]}\, </math>
where
:'''''k''''' for a given temperature is the Henry's Law constant (as defined in the first section of this article), identical with '''''k<sub>H,pc</sub>''''' defined in Table 1,
:'''''T''''' is in kelvins,
:the index <math>\Theta</math> ([[theta]]) refers to the standard temperature (298 K).
The above equation is an approximation only and should be used only when no better experimentally derived formula for a given gas exists.
The following table lists some values for constant ''C'' (dimension of kelvins) in the equation above:
{|class="wikitable"
|+ '''Table 2: Values of ''C'''''
|'''Gas''' || align=center| [[Oxygen|O<sub>2</sub>]] || align=center| [[Hydrogen|H<sub>2</sub>]] ||  align=center| [[CO2|CO<sub>2</sub>]] || align=center| [[Nitrogen|N<sub>2</sub>]] || align=center| [[Helium|He]] || align=center| [[Neon|Ne]] || align=center| [[Argon|Ar]] || align=center| [[Carbon monoxide|CO]]
|-
| '''''C''''' || align=center| 1700  ||align=center| 500 || align=center| 2400 || align=center| 1300|| align=center| 230 || align=center| 490|| align=center|  1300 || align=center| 1300
|}
Because solubility of gases is decreasing with increasing temperature, the partial pressure a given gas concentration has in liquid must increase. While heating water (saturated with nitrogen) from 25 °C to 95 °C the solubility will decrease to about 43% of its initial value. Partial pressure of CO<sub>2</sub> in seawater doubles with every 16 K increase in temperature.<ref>Takahashi, T. et al (2002). ''Global sea-air CO<sub>2</sub> flux based on climatological surface ocean CO<sub>2</sub> and seasonal biological and temperature effects'', Deep-Sea Research (Part II, Topical Studies in Oceanography) '''49''', 9-10, pp. 1601-1622.</ref>
The constant ''C'' may be regarded as:
:<math> C = \frac{\Delta_{solv}H}{R} = \frac{-d \ln\left(k(T)\right)}{d(1/T)}</math>
where
:<math> \Delta_{solv}H \,</math> is the  [[enthalpy of solution]]
:<math>R</math> is the  [[gas constant]].
==Henry's law in geophysics==
In [[geophysics]] a version of Henry's law applies to the solubility of a [[noble gas]] in contact with [[silicate]] melt.  One equation used is
:<math>\rho_m/\rho_g=e^{-\beta(\mu_{{\rm ex},m}-\mu_{{\rm ex},g})}\,</math>
where:
:subscript m = melt
:subscript g = gas phase
:<math>\rho</math> = the densities of the solute gas in the melt and gas phase
:<math>\beta=1/k_BT</math> an inverse temperature scale
:<math>k_B</math> = the [[Boltzmann constant]]
:<math>\mu_{{\rm ex},m}</math> and <math>\mu_{{\rm ex},g}</math> = the excess [[chemical potential]] of the solute in the two phases.
==Henry's law versus Raoult's law==
Both Henry's law and [[Raoult's law]] state that the vapor pressure of a component, ''p'', is proportional to its concentration.
:Henry's law:  <math> p = k \,x</math>
:Raoult's law: <math> p = p^\star\,x</math>
where:
:<math>\,x</math> is the mole fraction of the component
:<math>\,k</math> is the Henry constant &nbsp; (Note that the numerical value and dimensions of this constant change when mole fractions are used rather than molarity, as seen in Table 1)
:<math>p^\star</math> is the equilibrium vapor pressure of the pure component.
A nearly pure component (i.e., either the solvent or the solute gas) approximatly obeys Raoult's law and a dilute component approximately obeys Henry's law. In other words, in a dilute solution, the solute gas approximately obey's Henry's law and the solvent approximately obeys Raoult's law.<ref name=Mortimer/>
==References==
{{reflist}}

Revision as of 03:08, 18 February 2008

Retrieved from "https://citizendium.org/wiki/index.php?title=User:Milton_Beychok/Sandbox&oldid=388989"