Reif Ch7, Decomposition of partition function

[msaleh87]

Homework Statement

For a system A consists of two parts A' and A'' which interact only weakly with each other, if the states of A' and A'' are labeled respectively by r and s, then a state of A can be specified by the pair of numbers r,s and its corresponding energy [itex]E_{rs}[/itex] is simply additive, i.e.,
[itex]E_{rs}[/itex] = [itex]E^{'}_{r}[/itex] + [itex]E^{''}_{s}[/itex]

The partition function Z for the total system A is a sum over all states labeled by rs, i.e.,


Z=[itex]\sum_{r,s}e^{-\beta(E^{'}_{r}+E^{''}_{s})}[/itex] = [itex]\sum_{r,s}e^{-\beta E^{'}_{r}} \ e^{E^{''}_{s}}[/itex] = ([itex]\sum_{r}e^{-\beta E^{'}_{r}}[/itex])([itex]\sum_{r}e^{-\beta E^{''}_{s}}[/itex]) = [itex]Z^{'}Z^{''}[/itex]


My question is: how the sum of product [itex]\sum ()()[/itex] is converted to product of sum ([itex]\sum[/itex])([itex]\sum[/itex]), they are not generally equal

Thanks

 

[vela]

Do you understand why you can do this?
$$\sum_r \big[\sum_s e^{-\beta E_r}e^{-\beta E_s}\big] = \sum_r \big[ e^{-\beta E_r}\sum_s e^{-\beta E_s}\big] $$