Hello. Today we'll talk about a real gases, and the Van der Waals equation that describes them. The ideal gas model allowed us to obtain many regularities in the behavior of molecular and thermodynamic systems. For non dense gases, at sufficiently high temperatures, it works well. Recall the main features of the ideal gas model, molecules are points particles, there is no interaction between them. However, many physical phenomena occurring in real gases owe their origin precisely to the interaction of molecules. Thus, with increasing density of the gas, deviates even more from the ideal. At high enough densities, the intermolecular interaction leads the condensation, that is, to a gas-liquid phase transition. Of course, phenomena of this kind, where the interaction between the molecules plays a crucial role, cannot be described in the frames of the ideal gas model. At the same time, quantitative accounting of the interaction of molecules, which would lead to an exact equation of state, is practically impossible due to the complexity of the interaction itself. Hence, the numerous attempts to obtain the real gas equation of state, which would fully take into account the main qualitative features of this interaction. In the textbooks, you can find mention of numerous approaches. The most successful model apparently, must be recognized as the model proposed by Van der Waals. Speaking of success here, we mean the transparency of physical interpretation, as well as the accuracy in describing the transmission of a gas to a liquid state. Most physical handbooks, usually list the constants of the Van der Waals equation related to various real gases. It shows how popular it is. We proceed to the derivation of the Van der Waals equation. Consider the main features of the intermolecular interaction. They can be described by the so-called Lennard-Jones Potential. Did approximates of the intermolecular potential energy, between a pair of neutral atoms and molecules. At large distances, it corresponds to a direction, and at small distances, it corresponds to sharp repulsion. Recall the ideal gas equation of state, PV equals vRT. Lets try to fix it, given the mentioned long-range attraction and short range repulsion. Let's start with the repulsion. As you can see, it is very sharp, so these size can be considered as the characteristic size of molecules. Even with strong compression, molecules will be at this distance from each other, so that an a molecule of one mole, will occupy a characteristic volume b. These constant depends on the type of molecules. New mole will occupy the volume V_i equal vb, so-called intrinsic volume, and it is almost impossible to compress the gas more strongly. Mathematically, this can be taken into account by replacing V here by V minus V_i, or by V minus vb. We come to the equation, P times V minus vb equals vRT, or P equals vRT, over V minus vb. It can be seen that as v tends to vb, the pressure tends to infinity, and this is just what we need. Now, let's deal with the attraction. A molecule located in the center of the gas, experiences the attraction from the side of one-half of molecules, for example, from the left, the same as from the side of molecules to the right. A molecule near the wall, is attracted only by molecules on the one side. This force is proportional to the density of the gas, f proportional to n. Indeed, if we double the density, the force of attraction will double. This force decreases with distance, and mainly acts on the molecules of the wall layer. We can assume that a certain average force f, proportional to the density m, acts on each molecule of the wall layer, and to number of such molecules per area S, is N_w equal nwS, where w is the characteristic thickness of the wall layer, determined by the radius effection of the attractive forces. The attracting force acting on nw molecules, is a F_i equal fnwS, or per unit area, P_i equals F_i over S of fnw. Note that P_i is proportional to n squared, since f is proportional to n, or we can say that P_i is proportional to N over V squared, or v over V squared. The proportionality coefficient is denoted by the letter a. This is the second, and last constants of the model, which is determined by the type of molecules. Thus P_i equals a, v over V squared. This force slows down the molecules hitting the wall and thus reduces the pressure. To understand how much, let's recall how we derived the expression for the pressure of an ideal gas, and then make a correction. We believed that the change in the momentum of molecules reflected from the wall is due to direction force of the wall, and per unit area came to the equation one-third nm, V squared equals P. Now, we understand that parts of the momentum of the incident molecules, is taken away by the force P_i. We come to the equation one-third nm, V squared equals P plus P_i. We see that for a real gas, we should replace P by P plus P_i, or by P plus a, v over V, squared. We come to the equation, P plus a, v over V, squared times V equals RT. We see that the pressure is indeed reduced. P equals vRT over V minus a, v over V, squared. Making both obtained corrections to the ideal gas equation of state, we obtain the desired real gas equation of state, or the Van der Waals equation; P plus a, v over V, squared times V minus vb equals vRT. For one mole of gas, it takes the form P plus a over V squared times V minus b equals RT. Here, the constant a, characterizes the large range attraction, while b, the short range repulsion of finite molecular size. We'll analyze the obtained equation of state, and isotherms corresponding to it, in our next lecture.