Behavior of real gases

How to modify the ideal gas law to represent real gases ? Compressibility factor

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Section summary
1. Mixture of ideal gas
2. Partial pressures
3. Dalton's Law

1. Ideal gas

Perfect Gas law / Avogadro Law

A mixture of ideal gas adopts the same behavior as a pure ideal gas, the equation of state of the ideal gas is thus still valid for the whole gas :

PV = nRT

P = absolute pressure of the gas
V = volume of the gas
n = quantity of the gas (in moles)
T = absolute temperature of the gas
R = perfect gas constant

This equation is however an ideal representation of the behavior of gases, considering there is no interactions in between molecules, and essentially valid at low pressure. It is thus required to correct it to represent the actual behavior of gases. This correction is done thanks to the introduction of the compressibility factor

2. Compressibility factor

The compressibility factor is defined as the ratio in between the volume of the real gas and the volume of the ideal gas. It is noted Z :

Z = V/Vid

With :

V = volume of real gas in m3
Vid = volume of ideal gas in m3

As the volume of ideal gas can be expressed thanks to the perfect gas law as n.R.T/P, Z can also be written the following way :

Z = (P.V) / (n.R.T)

In practice, the compressibility factor Z can be found in diagrams in the literature, typically on diagram said of Amagat. It is important to understand here that the compressibility factor allows to represent the gas, but also the liquid phase, for very low Z.

At low pressure, Z = 1, we are close to the ideal gas. When the fluid is changing phase, Z changes suddenly very strongly, from high Z to low Z in the case of a condensation.

3. Real gas law

The real gas law is a generalization of the ideal gas law thanks to the compressibility factor.

The equation of real gas law is :

P.V = Z.n.R.T

The most common relations calculated for an ideal gas (see this page), can then be adapted thanks to the compressibility factor.

3.1 Molar volume of a gas

Vm = Z*(RT/P)

3.2 Density of a gas

ρ = (P.M) / (Z.R.T)

3.3 Mass flow rate and volume flow rate

Qv = Qm*(Z.R.T) / (P.M)

3.4 Correction of volume flowrate of real gases

Qv2 = Qv1 * (Z2/Z1) * (T2/T1) * (P1/P2)

4. Law of corresponding states

4.1 Pure substances

Having a graph of Amagat to determine Z is a method but is not the simpler one. Indeed, a specific diagram is required per substance, which is not always available. Another method was then developed after scientists observed that the properties of pure substances are identical when they are at the same reduced conditions. The reduced conditions being the ratio in between the actual pressure and the critical pressure, and the actual temperature and the critical temperature. 2 substances having the same reduced conditions are in corresponding states.

Reduced pressure and temperature calculation


PR = reduced pressure
P = actual pressure
PC = critical pressure of the substance (same unit as P)
TR = reduced temperature
T = actual temperature in K
TC = critical temperature of the substance in K

As this law is actually using 2 coordinates, the law of corresponding state is said to be at 2 parameters.

Once PR and TR are determined in the conditions of the study, it is possible to use abacus to determine the compressibility factor Z. The most common graph has been developped by Reid and Sherwood in their book "The properties of Gases and Liquids". The following graph is an extension of this work at higher reduced pressure, found in Wikipedia (Daniele Pugliesi)

Law of corresponding state for Z calculation

4.2 Mixtures

It is also possible to adapt the law of corresponding states to mixtures. To do so, a pseudo critical pressure of the mixture must be calculated by weighing the critical pressures of each components by their molecular weight. The reasoning is then the same as for the pure substances.

Reduced coordinates for mixtures


PCM = pseudo critical pressure of the mixture
yi = molar fraction of component i in the mixture
PCi = critical pressure of component i (same unit at PCM)
TCM = pseudo critical temperature of the mixture
TCi = critical temperature of component i (same unit at TCM)

Once calculated, Z is used in the equation of state of real fluids (see paragraph 3).

4.3 Law of corresponding states at 3 parameters

The law of corresponding state is a modelization, as a consequence, it creates some approximation and results tend to represent the reality but are not quite accurate in certain cases. In a bid to improve this, a modification of the law of corresponding states, introducing a 3rd parameter has been proposed. Different proposals were done but the theory of Pitzer, introducing the acentric factor is the most widespread.

Acentric factor calculation


ω = acentric factor
PS = saturation pressure of the substance at TR = 0.7

Note that to calculate PS, one must 1st calculate what is the temperature corresponding to TR=0.7, determine T, then calculate PS(T).

Calculation of Z corrected by the acentric factor


Z0 = compressibility factor obtained from the law of corresponding states at 2 parameters
ω = acentric factor
Z1 = correction factor, found on abacus

Note that this method can be used but requires to use a lot of abacus. Thus other equation of states, fitting better with computer calculations were later developed.

It is possible to express the partial pressure of one of the gas constituting the mixture thanks to the molar fraction of this gas in the mixture.

Partial pressure as a function of molar ratio

With :

yi = molar ratio of gas i in the mixture

Pi = P.yi expresses the Dalton law, which means that the partial pressure of an ideal gas in a mixture is calculated by multiplying the total pressure by the molar fraction.

Note that, if only the mass ratio is known, it is possible to convert to the molar ratio thanks to the molar mass of each components.

yimol = yimass.M/Mi

With :

M = average molar mass of the mixture
Mi = molar mass of the gas i