Adiabatic Expansion of the Ideal Gas

The adiabatic expansion of the ideal gas, c.f.. Sect. 4.1.3, is in fact an isentropic expansion in manal equilibrium with the environment. The ideal gas, i.e., the system, is in equilibrium with regard to compression energy. We start again with 

Here we do not change the amount of matter, therefore dn = 0 and we do not introduce the amount of matter in the variable set. Further, we put dS— 0. So no entropy may be exchanged with the environment. Or if the gas generates entropy, this entropy must be just removed from the gas into the environment to keep the entropy of the gas constant. It does not matter, how the entropy is kept constant. But if the gas and environment do not generate entropy, this indicates an equilibrium process and that all of the manal energy lost during expansion V )d V is taken 

We emphasize that the Const, in Eq. (5.19) emerges by a sloppy way of doing the calculation. Subsequently, we illustrate the calculation more accurately. In the first step, we set the total differentials of the energy in the two variable sets equal, i.e., dU T, V) = df/(5, V). Ifwehave/7= 0, then according to the law of Sc/iwarz, we obtain 9C /9V = 0, or that C = f(T) exclusively. In our calculation, we put more restrictive C as a constant. Thus, we obtain 

The integration of Eq. (5.20) with dS = 0 is often done by variable separation, thereby forgetting that the constant of integration is a function of the entfopy and moreover of the mol number in general. 

we examine another method to calculate the isentropic expansion of the ideal gas. Recall that the entropy of an ideal gas is 

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This equation is well known and often used to calculate initial fragment velocity, but its application can result in gross overestimation. Assuming adiabatic expansion of the ideal gas, it can be derived that ... 

An adiabatic expansion of the ideal gas is made from V2 to a volume V2. Work is done by the system, and the temperature drops to a lower temperature T, since no heat is added to the system. In this adiabatic process, q = 0, and... 

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