Equations of state for a real gas

TABLE 5.29 Van der Waals Constants for Gases The van der Waals equation of state for a real gas is ... 

We need an equation of state for a real gas to calculate ASm. i. The modified Berthelot equation is often used for pressures near ambient and was used in the original reference to calculate the correction to ideal behavior for N gas.h This equation is as follows... 

Real gases and vapors have intermolecular interactions. Recall that one equation of state for a real gas is the van der Waals equation, which is expressed in terms of two parameters, a and b. (a) For each of the following pairs of gases, decide which substance has the larger van der Waals a parameter ... 

From the ideal gas equation, it is found that for 1 mole of gas, PV/KT = 1, which is known as the compressibility factor. For most real gases, there is a large deviation from the ideal value, especially at high pressure where the gas molecules are forced closer together. From the discussions in previous sections, it is apparent that the molecules of the gas do not exist independently from each other because of forces of attraction even between nonpolar molecules. Dipole-dipole, dipole-induced dipole, and London forces are sometimes collectively known as van der Waals forces because all of these types of forces result in deviations from ideal gas behavior. Because forces of attraction between molecules reduce the pressure that the gas exerts on the walls of the container, van der Waals included a correction to the pressure to compensate for the "lost" pressure. That term is written as w2a/V2, where n is the number of moles, a is a constant that depends on the nature of the gas, and V is the volume of the container. The resulting equation of state for a real gas, known as van der Waals equation, is written as... 

We have shown from kinetic theory that the ideal gas equation has the correct form. The behavior of most real gases does not deviate drastically from the behavior predicted by this equation. So the best way of writing an equation of state for a real gas is to insert a correction factor into the ideal gas equation.1 This results in... 

For an ideal gas, satisfying the equation PV = RT under all conditions, dV/dT)p is equal to V/T it follows, therefore, from equation (22.2), that the Joule-Thomson coefficient is always zero. For a real gas, however, this coefficient is usually not zero even at very low pressures, when ideal behavior is approached in other respects. That this is the case may be seen by making use of an equation of state for a real gas. 

When the system has N identical spheres in the volume of V, there are N /2 pairs of excluded-volume interaction. Then, the change in the total free energy due to the excluded volume is AA/k T = (N /2)vJV. The change per sphere, (AA/k T)/N, is proportional to the density N/V. At low concentrations, the excluded volume is negligible. As the concentration increases and Nv approaches V, the effect becomes stronger. The same effect appears in the van der Waals equation of state for a real gas The correction to the volume is equal to the excluded volume. 

Attempts to improve on this model try to allow for the nonzero, but short-range, attractive forces, and also for the finite range of the repulsive force, which, essentially, gives a molecule its size. One well-known attempt to improve on the simple model is that of van der Waals, who wrote the equation of state for a real gas in the form ... 

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