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Section summary |
---|

1. Mixture of ideal
gas |

2. Partial pressures |

3. Dalton's 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**

With

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

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/V _{id}**

V = volume of real
gas in m^{3}

V_{id} = 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.

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.

Vm = Z*(RT/P)

ρ = (P.M) / (Z.R.T)

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

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

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.

With

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)

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.

With

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).

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.

With

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).

With

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**.

With :

y_{i} = molar ratio of gas i in the mixture

**P _{i} = P.y_{i} 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.

y_{imol} = y_{imass}.M/M_{i}

With :

M = average molar mass of the mixture

M_{i} = molar mass of the gas i