Expansion of ideal gas

The former case is realized in nature in the expansion of ideal gases and the mixture of dilute solutions the most important temperature measuring instrument, the air thermometer, depends upon this fact. 

The right side of Eq. (1.64) shows the energy effect due to the expansion of volume at a constant pressure process. For a mixture of ideal gases, the internal energy is a function of temperature only, and hence Eq. (1.64) and PV=RT yields... 

With mixtures of ideal gases the mass expansion coefficient can be found from the thermal equation of state... 

So, for both the spontaneous expansion of an ideal gas and the spontaneous mixing of ideal gases, there is no change in internal energy (or enthalpy) but an increase in entropy It seems possible that increases in entropy underlie spontaneous processes. We will soon see that the characteristic feature of a spontaneous process is that it causes the entropy of the universe to increase. 

Real gases follow the ideal-gas equation (A2.1.17) only in the limit of zero pressure, so it is important to be able to handle the tliemiodynamics of real gases at non-zero pressures. There are many semi-empirical equations with parameters that purport to represent the physical interactions between gas molecules, the simplest of which is the van der Waals equation (A2.1.50). However, a completely general fonn for expressing gas non-ideality is the series expansion first suggested by Kamerlingh Onnes (1901) and known as the virial equation of state ... 

No tables of the coefficients of thermal expansion of gases are given in this edition. The coefficient at constant pressure, l/t)(3 0/3T)p for an ideal gas is merely the reciprocal of the absolute temperature. For a real gas or liquid, both it and the coefficient at constant volume, 1/p (3p/3T),, should be calculated either from the equation of state or from tabulated PVT data. 

Ideal Gases.—The state of unit mass of an ideal gas, undergoing adiabatic compression or expansion, is completely defined by the equations... 

For reactions between ideal gases, l-apT=0, and AH and the activation energy are independent of pressure. For reactions in the condensed phases, thermal expansivity is small and 1 — apT as 1. Hence,... 

In the latter units "impetus is numerically equal to the volume that unit weight of the explosion products, if ideal gases, would occupy on isothermal expansion at (Tv) to a pressure of 1 atm. The "work done in this expansion would depend on the conditions. Work done would equal impetus only if the expansion were against an external pressure of 1 atm thruout Refs 1) Dunkle s Syllabus (1957-1958), p 257 2) Dunkle,private communication,... 

None of these is far from = 0.003663, which is therefore commonly taken as the expansion coefficient for gases especially as ihc value for hydrogen, commonly used in the standard gas thermometer, is very near it. If the pressure as well as the volume is allowed to vary, the behavior of the ideal gas must be expressed by the Boyle-Charlcs law or the ideal gas law and the behavior of a real gas by one of the other equations of stale. See also Ideal Gas Law. 

The results of early experiments showed that the temperature did not change on the expansion of the gas, and consequently the value of the Joule coefficient was zero. The heat capacity of the gas is finite and nonzero. Therefore, it was concluded that (dE/dV)Tn was zero. Later and more-precise experiments have shown that the Joule coefficient is not zero for real gases, and therefore (dE/dV)Ttheoretical concepts of the ideal gas. 

For gases, this may be positive or negative, depending on conditions. Note that it is zero for an ideal gas. It applies directly to the Joule expansion, an adiabatic expansion of gas confined in a portion of a container to fill the entire container. 

A similar situation arises for work, w. Frame 9 discusses expansion work performed by ideal gases. Finally the First Law of Thermodynamics (Frame 8) can be written ... 

The volume coefficient of expansion p may be determined from tables of properties for the specific fluid. For ideal gases it may be calculated from (see Prob. 7-3)... 

The formulation of Section 9.5.1 has served to remove the chemistry from the field equations, replacing it by suitable jump conditions across the reaction sheet. The expansion for small S/l, subsequently serves to separate the problem further into near-field and far-field problems. The domains of the near-field problems extend over a characteristic distance of order S on each side of the reaction sheet. The domains of the far-field problems extend upstream and downstream from those of the near-field problems over characteristic distances of orders from to /. Thus the near-field problems pertain to the entire wrinkled flame, and the far-field problems pertain to the regions of hydrodynamic adjustment on each side of the flame in essentially constant-density turbulent flow. Either matched asymptotic expansions or multiple-scale techniques are employed to connect the near-field and far-field problems. The near-field analysis has been completed for a one-reactant system with allowance made for a constant Lewis number differing from unity (by an amount of order l/P) for ideal gases with constant specific heats and constant thermal conductivities and coefficients of viscosity [122], [124], [125] the results have been extended to ideal gases with constant specific heats and constant Lewis and Prandtl numbers but thermal conductivities that vary with temperature [126]. The far-field analysis has been... 

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