Vite Cervantes Isaac Gael The Virial equation Introduction Gases are a fundamental part of many industries. They have been used in overtime and have application in almost all the industry. Is for the importance of them that we try to explain their comportment. Throughout history, been exist a lot of math models try to find the real comportment of the gases. Is here where the virial equation appears. Development The virial equation comes from the virial´s theorem (going towards the equation of state1) this theorem talks about the interaction between molecules. The virial equation serves us to find thermodynamic properties. For use the virial equation is necessary find someone coefficients which are more difficult to find when extending the equation. In conclusion the Virial equation is a very important and difficult state equation. The virial theorem Was announced by Rudolf Clausius at a conference of the Lower Rhine Association for Natural and Medical Sciences entitled "On a Mechanical Theorem Applicable to Heat", in which he announced to the world the discovery of a theorem today known as the virial theorem in 1870. In classical mechanics, the virial theorem is a general equation that relates the average total kinetic energy of a system with its average potential energy. The virial theorem allows to calculate the average total kinetic energy including very complex systems in which it is very difficult to obtain a solution. This average total kinetic energy is related to the temperature of the system through the equipartition theorem. An example of its many applications is the use of the virial theorem to calculate the Chandrasekhar limit for the stability of white dwarf stars. Real gases and the virial equation Why gases are classified as real or ideal is due to deviations in gas behavior from the laws known as ideal gas laws. this deviation can be pointed out like a compressibility factor. where: πππ ππ = πππππ π£πππ’ππ =π π = πΆπππ π‘πππ‘ ππ πππππ πππ π π π = ππππππππ‘π’ππ π = ππππ π π’ππ π = πΆππππππ π ππππππ‘π¦ ππππ‘ππ When a gas has an ideal comportment the value of z = 1, this says the product of the molar volume for the pressure is directly proportional to the temperature. In the stadist mechanic this compressibility factor is expressed how a powers serie of molar volume: πππ π΅ πΆ π· =1+ + 2+ 3+β― π π ππ ππ ππ This equation describes the non-ideal comportment is caused by intermolecular forces. How it was commented find these coefficients of the Virial is complicated and this difficulty increases as the equation lengthens because each coefficient considers the interaction between more and more molecules. If the coefficients are called fi where the subscript is the coefficient number when fn this coefficient the interactions between n molecules are considered. Beginning to describe these coefficients we start with the second coefficient “B” this is function of the temperature and we know that it talks about the interaction between a pair of molecules. This graph shows how this coefficient changes as the temperature changes. And shows how these coefficients depend of the nature of the compound that is, it is generally independent between each substance When the value of B is close to zero or is 0, if we truncate the virial equation to this second coefficient (or all the coefficients different of “A” are equal to zero) it will be indicating that the gas has an ideal comportment. Reduced properties Remembering a little that the molecules of a gas are very disorderly and have great movement (compared to an incompressible flow or a solid), exerting pressure on them begins to cause the molecules to group together and if this pressure is the enough, the gas can be liquefied (become liquid). But exist a temperature at which this is not possible. This temperature is called the critical temperature. And this last temperature is in turn the critical pressure. It is discovered that if the ratio of the relationship of a given temperature between the critical temperature is obtained, in turn also with the pressure n and the critical pressure and some properties such as the compressibility factor were considered, they would be shared between different substances. In other words, different gases meeting these criteria have the same compressibility factor. This relationship between a variable p, T or V and its critical variable is called the reduced property. Thanks to this it has been possible to create diagrams based on these relationships. This graph shows how the Z factor changes as the reduced temperature and pressure change. In this case, the reduced temperature is indicated as the lines inside the diagram (the colored lines). And as can be seen at high temperatures as well as at low temperatures, it will begin to have ideal comportment. Conclusion In a conclusion the virial equation serves to describe the comportment of the gases. Not only the ideal gases but also the real gases another of its characteristics is the capacity to find thermodynamical properties and this equation is the basis for many simulation programs. Although it has many positive characteristics, its complexity causes truncated solutions to be used, and many times the calculations can be wrong or simply far from reality. 1 An equation of state is an algebraic relationship between pressure, temperature, and molar volume. Reference Betancourt, F. F. (2004). Desarrollo de una nueva ecuación utilizando el potencial de Lennard-Jones, Distrito Federal, México: Instituto Politécnico Nacional Guijón, M. (2006). Estudio de la Rigidez Neumática de Suspenciones para Vehículos, Terrasa, Catalunya: Universitat Politècnica de Catalunya Departament de Mecànica de Fluids Collins, GW (1978). The Virial Theorem in Stellar Astrophysics. Pachart Press. Carrión, J. (2011) Ecuación De Estado Del Virial, Perú, Universidad Nacional.
