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, professor.
ResponderBorrarConcerning to the grand canonical ensemble I got a doubt: Can we use the equipartion theorem? And in case we cannot, is it because the energy is not a constant value?
Hmm, I am not sure. I have never seen that people are using or deriving the equipartition theorem within the grand canonical ensemble.
ResponderBorrarAs a statement the equipartition theorem (or virial theorem) relates the expectation values of certain observables, where the expectation value is taken with respect to the probability distribution rho(p,q) of the CANÓNICAL ENSEMBLE. That means that temperature and external parameters are known and the number of particles is fixed.
In the GRAND CANÓNICAL ENSEMBLE, the difference is not the energy (the energy isn't fixed in the canonical ensemble, neither). It is the number of particles. But due to that the probability distribution in the grand canonical ensemble is a different one.
Unfortunately, I had a sign-error in the calculation of the chemical potential on page 55 or 56. I corrected this error and added a figure where the chemical potential is shown for the ideal gas. If you download the lecture notes again, you will get this corrected and extended version.
ResponderBorrar