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By Robert B. Israel

Arthur S. Wightman (introduction)

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24) This equation is used frequently in meteorology. 7 Thermodynamic energy law of ideal gases The first law of thermodynamics discussed previously is given directly by the principle of energy conservation. This law for ideal gases may also be derived from the kinetic theory of gases. According to the assumptions applied for ideal gases, a molecule of ideal gas may be considered as a solid body. If the N molecules in an ideal gas of unit mass possess the translational velocity ˆ + vyj y ˆ + vzj ˆz vj = vxj x (j = 1, 2, 3, · · · , N ) , the translational kinetic energy of these molecules is given by Kj = mj 2 2 2 (v + vyj + vzj ) + Krj + Kvj , 2 xj in which mj indicates the mass of molecule j, and Krj and Kvj are the rotational and vibrational kinetic energy.

But the meteorological turbulences are treated as a random process in current numerical models with the parameterizations using the grid-point data which do not really include the subgrid-scale information. The effects of subgrid-scale physical or chemical processes are represented also by parameterizations approximately in the numerical models. Thus, the meteorological turbulences form the major error source for the numerical predictions. It is generally difficult to estimate the strength of this error source, as the errors depend essentially on the unknown circulation patterns.

The ideal-gas equation and van der Waals equation may be considered as the approximations of these equations. The van der Waals equation of state will be derived in Chapter 6 using the partition function obtained from the maximum entropy principle in classical thermodynamics. Here, we discuss the general expression of the state equation using the thermodynamic energy law. Let a cylinder with a piston at one end to be filled by c mole of a single gas. As the gas expands slowly, it accepts heat from outside in order to keep a constant pressure.

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