Effective Pressure in van der Waals Gases
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Real gases differ from ideal gases in two ways. First, they have finite size. Secondly, there are forces acting between the particles or molecules in a real gas. The microscopic description of these forces has a consequence for the macroscopic Equation of State. Previously, two Equations of States for gases were introduced:
- the Ideal Gases \[ PV =nRT \label{ideal}\]
- and the van der Waals (VdW) gas \[\left( P + a \dfrac{n^2}{V^2} \right) (V-nb) = nRT \label{vdw}\]
The VdW Equation of State is one many proposed to describe non-ideal gas behavior (i.e., real gases) and differs from the Ideal Gas Equation of State by the introduction of two adjustable parameters, \(a\), and \(b\), that correct for the size of the gas particles and intermolecular attractive forces in the ideal gas equation of state, respectively (Table A8).
Mapping Real Gas properties onto the Ideal Gas Law
Since the ideal gas law (Equation \(\ref{ideal}\)) is an extremely useful relationship, we do not want to throw it away just because it does not work for describing real systems. We can "remap" the VdW equation of State to an effective Ideal gas Equation of State is we introduce effective volume and pressures instead of the measured volume and pressure.
\[ P_{eff}V_{eff} =nRT \label{eff}\]
where the "effective pressure" (\(P_{eff}\)) of a VdW gas can be defined as
\[ P_{eff} = P_{actual} + \dfrac{n^a}{V^2}\]
and the effective volume (\(V_{eff}\)) can be defined
\[V_{eff}= V_{actual}-nb\]
Notice that the actual pressure predicted from the Ideal Gas Law (\(P\)) is reduced proportional to the square of the gas density.
\[P_{actual} = P_{eff} - \dfrac{n^a}{V^2} \]
This makes sense since the intermolecular forces (that are ignore in Ideal Gases) will "pull" the gas closer together to reduce the obseved pressure. The stronger the intermolecular forces, the greater \(a\) in Equation \(\ref{vdw}\) and the greater this real gas effect.
Fugacities
The fugacity of a real gas is an effective partial pressure which replaces the mechanical partial pressure in an accurate computation of the chemical equilibrium constant. Fugacities are determined experimentally or estimated direclty from proposed Equations of Sates (e.g., Equation \(\ref{ideal}\) and \(\ref{vdw}\)) that are closer to reality than an ideal gas. The word "fugacity" is derived from the Latin for "fleetness", which is often interpreted as "the tendency to flee or escape". The concept of fugacity was introduced by Gilbert N. Lewis in 1901. The ideal gas pressure and fugacity are related through the dimensionless fugacity coefficient \(\phi\).
\[ \phi = \dfrac{f}{P}\]
So fugacity is related to how non-ideal a gas is. If you can find an expression that describes the non-ideality it can be used to derive the fugacity.
Fugacity is a simple hack that allows us to model the behavior of real gases within the mathematical framework of ideal gas thermodynamics.
Effective Concentrations
The VdW gas would obey the ideal gas equation of state if only we used the effective volume and pressure instead of the measured volume and pressure (Equation \(\ref{eff}\)). Alternatively, we can map real gas properties into the Ideal Gas Law differently:
\[ PV =n_{effective}RT \label{eff2}\]
in this case the non-ideal behavior is combined into an "effective number" which correlates to an effective concentration (i.e., density):
\[ \rho_{eff} = \dfrac{n_{eff}}{V}\]
This "effective concentration" changes as a function of pressure.
\[\rho_{eff} = \rho_{eff}(P)\]
That is, change the pressure of the system and the "effective density" changes. This is a guiding principle behind other efforts to apply ideal behavior to real systems.
Contributors
Delmar Larsen