Good morning, Tarea 6
Good morning everybody. If you have still questions or doubts how to solve Tarea 06. Please post them, here. I will give feed back.
Work for an ideal gas in a 3D container (corrected)
Using the differential al work calculated on exercise 6.2: 1. How can we use the result to obtain DW for an ideal gas in a 3D container? What is the interpretation of the parameter "a" in that case. Solution. The differential of work calculated on exercise 6.2 is Here, d represents the dimension of the box. So The parameter a is the lenght, according to equation (67). Because we are working on 3 dimensions, a³=V . This might come from the definition of entropy*. 2. What do we get for DW for d=3 and n --> infinity? And how is this result related to the expected form DW = N k_B T dV/V? Solution. On the last expression we get DW for three dimensional case. Now, applying the limit on infinity, we expect that the differential of work would become on well potential one. So,
Good morning.
ResponderBorrarIn the part 6.1, is not clear at all for me how can we relate the mean potential energy using the equipartition theorem. Should we use the intregral form and solve?
In the part 6.2 I got confused about the intrinsic variables because, as I see, they do not depend on N, but in the logarithm argument they are multiplied by N (at least I got a wrong answer). So, they aren't extensive, right?
If you compute x times the derivative of the potential you get up to a constant factor again the potential. If you do the same with x_j for the whole Hamiltonian, you get, again up to a constant factor, the potential energy of that particle. Then the partition theorem can be applied to get the mean potential energy for each particle
BorrarYes, they do not depend on N. But when taking the termodynamic limit you should change eventually change them together with N. This is how it works for the ideal gas. The volume is a predefined parameter of the system, when taking the thermodynamic limit one chooses to increase the volume along with the number of particles such that V/N remains constant. In case considered in 6.2 you should find out how you have to change "eta" and "a" such that the log of the partition function, i.e. ln(Omega) is extensive, that is to say "N^{-1} ln(Omega)" becomes constant in the thermodynamic limit.
BorrarOh I forgot, you may also check my comment to José Salvador's question.
BorrarThere I explain the issue of extensive, intensive and other kinds of variables in more detail.
Oh, I see it, I will check it. Thank you, professor
Borrar