- Real gases are subject to the effects of molecular volume (intermolecular repulsive force) and intermolecular attractive forces. - The behavior of a real gas approximates that of an ideal gas as the pressure approaches zero. - The effects of non-ideal behavior are best seen when the PV product is plotted as a function of P. You should be able to identify the regions of such a plot in which attractive and repulsive forces dominate. - Each real gas has its own unique equation of state. Various general equations of state have been devised in which adjustable constants are used to approximate the behavior of a particular gas. - The most well-known equation of state is that of van der Waals. Although you need not memorize this equation, you should be able to explain the significance of its terms. The "ideal gas laws" as we know them do a remarkably good job of describing the behavior of a huge number chemically diverse substances as they exist in the gaseous state under ordinary environmental conditions, roughly around 1 atm pressure and a temperature of 300 K. Real gases are subject to the effects of molecular volume (intermolecular repulsive force) and intermolecular attractive forces. The behavior of a real gas approximates that of an ideal gas as the pressure approaches zero. The effects of non-ideal behavior are best seen when the PV product is plotted as a function of P . You should be able to identify the regions of such a plot in which attractive and repulsive forces dominate. Each real gas has its own unique equation of state. Various general equations of state have been devised in which adjustable constants are used to approximate the behavior of a particular gas. The most well-known equation of state is that of van der Waals. So we must reformulate the ideal gas equation of state as a relation that is true only in the limiting case of zero pressure: \[\lim_{P \rightarrow 0} PV=nRT \label{6.6.2}\] So what happens when a real gas is subjected to a very high pressure? The outcome varies with both the molar mass of the gas and its temperature, but in general we can see the the effects of both repulsive and attractive intermolecular forces: Repulsive forces : As a gas is compressed, the individual molecules begin to get in each other's way, giving rise to a very strong repulsive force acts to oppose any further volume decrease. As long as the energy of thermal motion dominates this attractive force, the substance remains in the gaseous state, but at sufficiently low temperatures the attractions dominate and the substance condenses to a liquid or solid. The universal attractive force described above is known as the dispersion , or London force. There may also be additional (and usually stronger) attractive forces related to charge imbalance in the molecule or to hydrogen bonding. These various attractive forces are often referred to collectively as van der Waals forces . As you can see in this plot for methane, some of this balance does remain as the pressure is increased. The van der Waals Equation of State How might we modify the ideal gas equation of state to take into account the effects of intermolecular interactions? The first and most well known answer to this question was offered by the Dutch scientist J.D. van der Waals (1837-1923) in 1873. The ideal gas model assumes that the gas molecules are merely points that occupy no volume; the " V " term in the equation is the volume of the container and is independent of the nature of the gas. The other effect that van der Waals needed to correct for are the intermolecular attractive forces. These are ignored in the ideal gas model, but in real gases they exert a small cohesive force between the molecules, thus helping to hold the gas together and reducing the pressure it exerts on the walls of the container. Because this pressure depends on both the frequency and the intensity of collisions with the walls, the reduction in pressure is proportional to the square of the number of molecules per volume of space, and thus for a fixed number of molecules such as one mole, the reduction in pressure is inversely proportional to the square of the volume of the gas. The smaller the volume, the closer are the molecules and the greater will be the effect. The van der Walls equation replaces the \(P\) term in the ideal gas equation with \(P + (a / V^2)\) in which the magnitude of the constant a increases with the strength of the intermolecular attractive forces. The complete van der Waals equation of state can be written as Figure \(\PageIndex{5}\): van der Waal's equation of state for a real gas Although most students are not required to memorize this equation, you are expected to understand it and to explain the significance of the terms it contains. You should also understand that the van der Waals constants \(a\) and \(b\) must be determined empirically for every gas. This can be done by plotting the P-V behavior of the gas and adjusting the values of \(a\) and \(b\) until the van der Waals equation results in an identical plot. The constant a is related in a simple way to the molecular radius; thus the determination of \(a\) constitutes an indirect measurement of an important microscopic quantity. Table \(\PageIndex{1}\): van der Waals constants for some gases Substance molar mass (g) a (L 2 -atm mole –2 ) b (L mol –1 ) hydrogen H 2 2 0.244 0.0266 helium He 4 0.034 0.0237 methane CH 4 16 2.25 0.0428 water H 2 O 18 5.46 0.0305 nitrogen N 2 28 1.39 0.0391 carbon dioxide CO 2 44 3.59 0.0427 carbon tetrachloride CCl 4 154 20.4 0.1383 The van der Waals equation is only one of many equations of state for real gases. More elaborate equations are required to describe the behavior of gases over wider pressure ranges. These generally take account of higher-order nonlinear attractive forces, and require the use of more empirical constants.