Gas compressibility factor, Z

What is the gas compressibility factor?

The gas compressibility factor is the ratio of a gas's molar volume to that of an ideal gas at constant temperature and pressure. The compressibility factor for an ideal gas is unity, which is typically written as Z = PV / RT.

The compressibility factor is used as a correction factor to ideal behavior. Thus, v [real] = Z v[ideal] is used to calculate the actual volume, v [real], as the product of the compressibility factor and the ideal gas volume, all at the same pressure and temperature.

Where is the problem?

The gas compressibility factor plays an important role in predicting the volume and pressure of gases in gas reservoirs in the presence of impurity gases.

Z for a mixture of gases

Some impurities such as nitrogen and carbon dioxide often exist in appreciable amounts in natural gases. The Z-factor for non-hydrocarbon components of natural gas in certain corresponding states differ markedly from those of hydrocarbons. This makes the non-hydrocarbon and hydrocarbon components not additive. The z-factor for the gas mixture by introducing a correction factor is given below. To account for the non-additive behavior of volumes of hydrocarbon and nonhydrocarbon gases, correction functions come from the correlation of data (Z-factors) generated by the Peng Robinson equation of state.?

Equation:?

Z mixture = c {n x Z nitrogen + (1-n) Z hydrocarbon},

Where: Z mixture = Actual Z factor for gas mixture, Z nitrogen = Z factor for N2 in the mixture, Z hydrocarbon = Z factor for hydrocarbon gas, n = Mole fraction of N2 in gas mixture

?c is an arbitrary factor to account for the fact that volumes of hydrocarbon and non-hydrocarbon are not additive. The values of factor c are available in charts for different mixtures at certain pressures and temperatures.

The root of the problems: Nonideality of real gases

The gas compressibility factor tells us how much the real gas deviates from the ideal gas at a given pressure and temperature. As said above, it is simply defined as the ratio of the molar volume of a gas to the molar volume of an ideal gas at the same temperature and pressure, Z = V Actual / V Ideal.?Air is very nearly an ideal gas where Z=1.0.

In general, deviation from ideal behavior becomes more significant the closer a gas is to a phase change, the lower the temperature or the larger the pressure.

Compressibility factor values are usually obtained by calculation from equations of state (EOS).

To express a relationship between the variables P, V, and T, the Z factor is introduced to ideal gas laws, PV = n Z RT

This can be derived as since Z = V Actual / V Ideal

at the same temperature T and pressure P

V Actual = Z x V ideal. Multiply both sides by P,

PV actual = Z x PV ideal

PV ideal = n RT,

Therefore, PV actual = Z x n RT or Z = PV actual / n RT

where p is the pressure, n is the number of moles of gas, T is the absolute temperature, and R is the gas constant.

Another way Z is expressed is since Z = PV actual / n RT?

?Z = P x V x Molar mass of the gas / [ Mass of gas x RT = [P] / [Density of the gas x T x R specific], R specific = R / Molar mass

Z = P / [Rho x T x R specific]

Here comes van der Walls’s contribution

Van der Waals equation

The ideal gas law treats gas molecules as point particles that interact with their containers but not each other, meaning they neither take up space nor change kinetic energy during collisions (i.e. all collisions are perfectly elastic). The ideal gas law states that volume (V) occupied by n moles of any gas has a pressure (P) at temperature (T) in kelvins given by the following relationship, where R is the gas constant:

PV = n RT

To account for the volume that a real gas molecule takes up, the van der Waals equation replaces V in the ideal gas law with (Vm-b), where Vm is the molar volume of the gas and b is the volume that is occupied by one mole of the molecules. This leads to P(Vm-b) = RT

The second modification made to the ideal gas law accounts for the fact that gas molecules do interact with each other (they usually experience attraction at low pressures and repulsion at high pressures) and that real gases therefore show different compressibility than ideal gases. Van der Waals provided for intermolecular interaction by adding to the observed pressure P in the equation of state a term

[ P+ a*(1/Vm^2)] [ (Vm-b)] = RT

Where a is a constant whose value depends on the gas. The van der Waals equation is therefore written as

?[ P+ a*(n^2/V^2)] [ (V-nb)] = RT.

Where Vm is the molar volume of the gas, R is the universal gas constant, T is temperature, P is pressure, and V is volume. When the molar volume Vm is large, b becomes negligible in comparison with Vm, a/Vm2 becomes negligible concerning P, and the van der Waals equation reduces to the ideal gas law, PVm=RT.

Theorem of corresponding states

The principle of Corresponding States (PCS) was stated by van der Waals and reads: “Substances behave alike at the same reduced states.”?

Reduced properties

Reduced properties are used to define corresponding states. Reduced properties provide a measure of the “departure” of the conditions of the substance from its own critical conditions and are defined as follows:

Pr = P/Pc, Tr = T/Tc, and Vr = V/Vc P, T, and V are temperature, pressure, and volume. Subscript c stands for critical point and r stands for reduced temperature and pressure. If Pr = Tr = Vr = 1, the substance is at its critical condition. If we are beyond critical conditions, Tr > 1, Pr > 1 and Vr > 1. By the same token, if all the conditions are subcritical, Tr < 1, Pr < 1 and Vr < 1. Critical conditions become the scaling factor by which substances can be compared among each other in terms of their “departure from criticality” or reduced properties.

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Compressibility factor Zc at the critical point

The compressibility factor at the critical point is defined as

Zc= Pc*Vc*Molar mass/ RTc, where the subscript c indicates the critical point, is predicted to be a constant independent of substance by many equations of state; the Van der Waals equation e.g. predicts a value of 3/8=0.375

To summarize,

??Zc and Zr serve different purposes in the context of the theory of corresponding states. - Zc (critical compressibility factor): Represents the universal value for all gases at their critical point, indicating a common behavior near the critical point. It is a characteristic property of a gas at its critical conditions. - Zr (reduced compressibility factor): Represents the dimensionless quantity obtained by dividing the actual compressibility factor (Z) of a gas by its critical compressibility factor (Zc) at a reduced temperature and pressure. Zr is the same for all gases at the same reduced temperature and pressure.

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