Entropy Gain in an Ideal Gas

It could be that Eq. (2.62) is not satisfactory by inspection and common sense. We consider a system as shown in Eig. 2.6. Here the total volume Vu,t is constant and is the sum of the individual volumes Vi ofgas(l)and V2 of gas (2), i.e., Vtot—Vi—+2 = 0. When we start with equal amounts of gas with the volume at the left side Vi smaller than the volume at the right side V2, it is suggestive that the pressure on the left side is higher than the pressure on the right side. In general, this may not be true, because the pressure is dependent also on the temperature of the respective gas. However, we assume pi p2. 

When we allow the plunger to move freely the volume at the left side will increase and the volume at the right side will decrease correspondingly. Therefore, we naturally feel that the process will continue until both pressures become equal. This means that the condition pi(5i, Ei) = P2 Si, V2) from Eq. (2.62) is fulfilled, but not necessarily the condition of equal temperature. 

In fact, the calculation of maximum entropy finds the most possible maximum entropy. We have not given a constraint for the entropy in Eqs. (2.60) and (2.61). However, in a nonequilibrium process the entropy should increase. Thermodynamics does not provide an answer as to which way the entropy would increase. We 

We inspect now an arbitrary system together with an environment system. The arbitrary system should have a certain energy dependent on the entropy, the volume, and the mol number, as usual UsysiSsys, Vsys, risys)- The environment should have an energy dependent on certain variable that we will specify later as UenviXi, X2, - - -). The energy of both systems should be constant, i.e.. 

For illustration we may think that the system consists of an ideal gas. The system and its environment is presented schematically in Fig. 2.7. 

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