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Adiabatic process, Poisson equation Theoretical material.docx
Theoretical material
An adiabatic process occurs without transfer of heat or mass of substances between a thermodynamic system and its surroundings. In an adiabatic process, energy is transferred to the surroundings only as work. The adiabatic process provides a rigorous conceptual basis for the theory used to expound the first law of thermodynamics, and as such it is a key concept in thermodynamics.
Some chemical and physical processes occur so rapidly that they may be conveniently described by the term "adiabatic approximation", meaning that there is not enough time for the transfer of energy as heat to take place to or from the system.
By way of example, the adiabatic flame temperature is an idealization that uses the "adiabatic approximation" so as to provide an upper limit calculation of temperatures produced by combustion of a fuel. The adiabatic flame temperature is the temperature that would be achieved by a flame if the process of combustion took place in the absence of heat loss to the surroundings.
In meteorology and oceanography, the adiabatic cooling process produces condensation of moisture or salinity and the parcel becomes oversaturated. Therefore, it is necessary to take away the excess. There the process becomes a pseudo-adiabatic process in which the liquid water/salt that condenses is assumed to be removed as soon as it is formed, by idealized instantaneous precipitation. The pseudoadiabatic process is only defined for expansion, since a parcel that is compressed becomes warmer and remains undersaturated.
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Well, the first to way to relate temperature and volume is to equate W to pressure multiplied to delta V. W = - n CV ∆T, but W = pdV Thus, pdV = - n CV dT We have to write dV and dT in the expression since we are dealing with very small interval of time. In an adiabatic process, pressure, volume and temperature all change. In the other processes, one of them stayed the same. In isobaric the pressure didn’t change, in isochoric the volume didn’t change, and isothermic the temperature didn’t change, so only 2 out of 3 changed. But in an adiabatic process, all variables change. The next thing to do is to use the ideal gas equation, pV = nRT, and p = nRT/V, then you can plug that information from the previous equation.
R ln V + c1 + CV lnT + c2 = 0 where c1 and c2 are the integration constants Combining constants R ln V + CV lnT = c3 where c3 is the combined integration constant but R = CP - CV, so
where
or
dividing ln on both sides gives
Other variations of Poisson’s Equations are:
Gamma, γ, depends on the kind of gas present
For monoatomic gas, γ =1.7 For diatomic gas, γ = 1.4. |
For triatomic gas, γ = 1.3)
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