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## Main Question or Discussion Point

Hi all, I am slightly confused with regard to some ideas related to the GCE and CE. Assistance is greatly appreciated.

Since the GCE's partition function is different from that of the CE's, are all state variables that are derived from the their respective partition functions still equal in general for the same substances (for example, ideal gases)?

Or do they differ between ensembles depending on the substance considered? In this case, how do I tell whether or not a substance's state functions change?

For example, denoting the GCE's partition function as ##\tilde{Z}## and the CE's as ##Z##,

$$U_{gce} = -( \partial_{\beta } \tilde{Z} ) + \mu \langle N \rangle$$

$$U_{ce} = -( \partial_{\beta } Z)$$

$$F_{gce} = -k_B T \ln{\tilde{Z}} + \mu \langle N \rangle $$

$$F_{ce} = - k_B T \ln{Z} $$

are these equivalent in general?

Also, more specifically, if I am considering some ideal gas in a GCE, can I still use things like ##PV = nRT##, ##U = \frac{3}{2}k_B T##, the Sackur-Tetrode equation etc ...?

Finally, the way I learnt to derive a CE partition function for an ideal gas was to start with a single-particle partition function, before generalising to an N-indistinguishable particle case. If I wanted a GCE partition function for an ideal gas, it doesn't make sense at all to start with a single particle case. How then, do I derive the GCE partition function for the ideal gas with ##\tilde{Z}##?

$$\tilde{Z} = \sum_{i,j} e^{\beta (- E_i + \mu N_j)}$$

Since the GCE's partition function is different from that of the CE's, are all state variables that are derived from the their respective partition functions still equal in general for the same substances (for example, ideal gases)?

Or do they differ between ensembles depending on the substance considered? In this case, how do I tell whether or not a substance's state functions change?

For example, denoting the GCE's partition function as ##\tilde{Z}## and the CE's as ##Z##,

$$U_{gce} = -( \partial_{\beta } \tilde{Z} ) + \mu \langle N \rangle$$

$$U_{ce} = -( \partial_{\beta } Z)$$

$$F_{gce} = -k_B T \ln{\tilde{Z}} + \mu \langle N \rangle $$

$$F_{ce} = - k_B T \ln{Z} $$

are these equivalent in general?

Also, more specifically, if I am considering some ideal gas in a GCE, can I still use things like ##PV = nRT##, ##U = \frac{3}{2}k_B T##, the Sackur-Tetrode equation etc ...?

Finally, the way I learnt to derive a CE partition function for an ideal gas was to start with a single-particle partition function, before generalising to an N-indistinguishable particle case. If I wanted a GCE partition function for an ideal gas, it doesn't make sense at all to start with a single particle case. How then, do I derive the GCE partition function for the ideal gas with ##\tilde{Z}##?

$$\tilde{Z} = \sum_{i,j} e^{\beta (- E_i + \mu N_j)}$$