Henry's law

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In chemistry, Henry's law is one of the gas laws, formulated by William Henry. 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.

Contents

A formula for Henry's Law is:

e^{p\,} = e^{kc\,} \,

where:

p\, the partial pressure of the solute above the solution
c\, the concentration of the solute in the solution (in one of its many units)
k\, the Henry's Law constant, which has units such as L·atm/mol, atm/(mol fraction) or Pa·m3/mol.

Taking the natural logarithm of the formula, gives us the more commonly used formula:[1]

p = kc \,

Some values for k include:

oxygen (O2) : 769.2 L·atm/mol      
carbon dioxide (CO2) : 29.4 L·atm/mol      
hydrogen (H2) : 1282.1 L·atm/mol    

when these gases are dissolved in water at 298 kelvin.

Note that in the above, 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.

As discussed in the next section, there are other forms of Henry's Law each of which defines the constant k differently and requires different dimensional units. The form of the equation presented above is consistent with the given example numerical values for oxygen, carbon dioxide and hydrogen and with their corresponding dimensional units.

There are various other forms Henry's Law which are discussed in the technical literature.[2][3]

Table 1: Some forms of Henry's law and constants (gases in water at 298K), derived from [3]
equation: k_{H,pc} = \frac{p_{gas}}{c_{aq}} k_{H,cp} = \frac{c_{aq}}{p_{gas}} k_{H,px} = \frac{p_{gas}}{x_{aq}} k_{H,cc} = \frac{c_{aq}}{c_{gas}}
dimension: \left[\frac{L_{soln} \cdot atm}{mol_{gas}}\right] \left[\frac{mol_{gas}}{L_{soln} \cdot atm}\right] \left[\frac{atm \cdot (mol_{water}+ mol_{gas})}{mol_{gas}}\right] \left[ dimensionless \right]
O2 769.23 1.3 E-3 4.259 E4 3.180 E-2
H2 1282.05 7.8 E-4 7.099 E4 1.907 E-2
CO2 29.41 3.4 E-2 0.163 E4 0.8317
N2 1639.34 6.1 E-4 9.077 E4 1.492 E-2
He 2702.7 3.7 E-4 14.97 E4 9.051 E-3
Ne 2222.22 4.5 E-4 12.30 E4 1.101 E-2
Ar 714.28 1.4 E-3 3.955 E4 3.425 E-2
CO 1052.63 9.5 E-4 5.828 E4 2.324 E-2

where:

c_{aq}\, = moles of gas per liter of solution
L_{soln}\, = liters of solution
p_{gas}\, = partial pressure of gas above the solution, in atmospheres of absolute pressure
x_{aq}\, = mole fraction of gas in solution = moles of gas per total moles ≈ moles of gas per mole of water
atm\, = atmospheres of absolute pressure

As can be seen by comparing the equations in the above table, the Henry's Law constant kH,pc is simply the inverse of the constant kH,cp. Since all kH 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.

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 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 (H2CO3) with water.

When shifting temperature of the system, the Henry constant will not remain changeless. This is why some people prefer to name it Henry coefficient. There are multiple equations assessing the behaviour of the constant. A simple example is [3], which is a form of the van't Hoff equation:

k_{H,cp} = k_{H,cp,\Theta} \cdot e^{ \left[ C \cdot \left( \frac{1}{T}-\frac{1}{T_\Theta}\right)\right]}\,

where

T is in kelvins
the index Θ (Theta) refers to the standard temperature (298K).

The following table lists some values for constant C (dimension of kelvins) in the equation above:

Table 2: Values of C
Gas O2 H2 CO2 N2 He Ne Ar CO
C 1700 500 2400 1300 230 490 1300 1300

It can be seen, that the solubility of gases is decreasing with increasing temperature. While heating water (saturated with nitrogen) from 25°C to 95°C the solubility will decrease to about 43% of its initial value. This can be verified when heating water in a pot. Small bubbles evolve and rise, long before the water reaches boiling temperature. The constant C may be regarded as:

C = \frac{-\Delta_{solv}H}{R} = \frac{d \cdot ln \left(k_{H,cp}\right)}{d(1/T)}

where

\Delta_{solv}H \, is the enthalpy of solution
R is the gas constant.

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

\rho_m/\rho_g=e^{-\beta(\mu_{{\rm ex},m}-\mu_{{\rm ex},g})}\,

where:

subscript m = melt
subscript g = gas phase
ρ = the number densities of the solute gas in the melt and gas phase
β = 1 / kBT an inverse temperature scale
kB = the Boltzmann constant
μex,m and μex,g = the excess chemical potential of the solute in the two phases.

Both Henry's law and Raoult's law relate the vapor pressure of a component to its concentration. It is possible (and sometimes more convenient) for either law to write the concentration in terms of mole fractions x. Note however that the numerical value of k as well as its dimensions change when mole fractions are used rather than molarity (as seen in the Table 1).

Henry's law: p = k_{H,x} \,x
Raoult's law: p = p * x

The difference is that p* is the equilibrium vapor pressure of the pure component whereas the Henry constant kH is a value that differs from p*. It must be determined experimentally from the mixtures, not the pure compound.

If the solution is ideal (which it seldom is), both components follow Raoult's law over the entire composition range. In most systems, the laws can only be applied in a very limited concentration range at extreme ends of the range. In that case, the minority component (the solute) follows Henry's law, but the solvent still follows Raoult's law. The Gibbs-Duhem equation can be used to prove that this is so.

  1. ^ University of Delware physical chemistry lecture
  2. ^ University of Arizona chemistry class notes
  3. ^ a b c An extensive list of Henry's Law constants, and a conversion tool

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