A Real Gas Approximation van der Waals Equation

Simultaneously the following relation will also be fulfilled 

Nemst s theorem is applicable only to equilibrium systems. From the third law of thermodynamics, it follows that absolute zero is unattainable because, according to eq. (3.5.27), if near a temperature of absolute zero, a small amount of heat is taken off a system (AT - 0), a large enough (in a limit infinite) entropy change will take place this contradicts Nemst s theorem. 

Notice that at the aspiration of the temperature to absolute zero, the thermal heat capacities and Cp will also aspire to zero. 

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The virial equation is a general equation for describing real gases. The van der Waals equation is an approximate equation of state fora real gas the parameter a represents the role of attractive forces and the parameter b represents the role of repulsive forces. 

Figure 3-2. The pressure of a real gas as a function of temperature and volume approximation based on the van der Waals equation.

Tests of this prediction against experimental critical-point data of Table 2.4 reveal large deviations (e.g., an approximately 20% error even in the most favorable case of He) that reflect serious quantitative defects of the Van der Waals description. This is but one of many indications that the Van der Waals equation, although a distinct improvement over the ideal gas equation, is still a significantly flawed representation of real fluid properties. 

It has been already pointed out that van der Waals equation applies, 1 e reproduces experimental facts at least approximately, for the homogeneous system consisting either of liquid or of gas By the introduction of the Maxwell assumption considered above, it is possible to use van der Waals equation, to obtain information regarding pressures and volumes of heterogeneous systems in the following way The two points A and B both lie on the hypothetical and the real isotherms To each of these points we can apply the van der Waals equation Further, the points A and B represent the values which are the limits... 

The van der Waals equation can describe the real gas behavior in sufficient approximation over a wide range of temperatures and pressures. It takes into account the molecular size as well as the molecular interaction forces by the introduction of two additional terms (i) As the attractive forces tend to hold the molecules together, the pressure is lower than the ideal value. To account for this, the pressure is augmented by an attractive force term the inter-... 

The assumption known as the law of corresponding states asserts that the compressibility factor Z should be a function only of the reduced temperature Ti and the reduced pressure Pi, which is approximately correct for many real gases. It is seen that, for a van der Waal gas, the minimum value of Vr = 1/3, which can be achieved only at infinite pressure. From the equation of state written for the reduced temperature and pressure, we can derive the equivalent formula of compressibility as... 

Provided V is not too small, the first term in the equation above is approximately equal to the pressure exerted by an ideal gas nRT/ V — nb) nRT/V = Pideai- The equation given above for P predicts that the pressure exerted by a real gas will be less than that of an ideal gas. Figure 6-23 illustrates why. Because of attractive forces, molecules near the container walls are attracted toward the molecules behind them as a result, the gas exerts less force on the container walls. The term rP-afV takes into account the decrease in pressure caused by intermolecular attractions. In 1873, the Dutch physicist Johatmes van der Waals reasoned that the decrease in pressure caused by intermolecular attractions should be proportional to the square of the concentration, and so the decrease in pressure is represented in the form n a/V. The proportionality constant, a, provides a measure of how strongly the molecules attract each other. 

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