# Reif Ch7, Decomposition of partition function

## 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 $E_{rs}$ is simply additive, i.e.,
$E_{rs}$ = $E^{'}_{r}$ + $E^{''}_{s}$

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

Z=$\sum_{r,s}e^{-\beta(E^{'}_{r}+E^{''}_{s})}$ = $\sum_{r,s}e^{-\beta E^{'}_{r}} \ e^{E^{''}_{s}}$ = ($\sum_{r}e^{-\beta E^{'}_{r}}$)($\sum_{r}e^{-\beta E^{''}_{s}}$) = $Z^{'}Z^{''}$

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

Thanks

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vela
Staff Emeritus
$$\sum_r \big[\sum_s e^{-\beta E_r}e^{-\beta E_s}\big] = \sum_r \big[ e^{-\beta E_r}\sum_s e^{-\beta E_s}\big]$$