Partition function of adsorbing molecules

Question

I have the following example:

A surface, having $N_0$ adsorption centers, has $N \le N_0$ gas molecules adsorbed on it. Disregarding interactions between the adsorbed molecules. An adsorption center having no adsorbed molecule on it is referred to as state 0 and has energy $\epsilon_{0}=0$, while one having an adsorbed molecule of energy $\epsilon_{i}$ is denoted by i. Each of the $N_{0}$ centers can assume any of the states 0,1,2.... independently of other adsorption centers.

a)compute canonical partition function $Q_1$, for one single adsorption center. b) Compute grand canonical partition function $Z_{1}$ for a single adsorption center, considering the center can also be unoccupied and assuming that thechemical potential of the adsorbing molecule is $\mu$ and their fugacity is $z=e^{\beta \mu}$.

a) $Q_{1}=\sum_{i=1}^{\infty}e^{-\beta\epsilon_{i}}$

for be I have the solution but I don't understand it at all:

b) $Z_{1}=\sum_{i=0}^{\infty}(e^{\beta\mu})^{i}*e^{-\beta\epsilon_{i}}= (e^{\beta\mu})^{0}*e^{-\beta\epsilon_{0}}+\sum_{i=1}^{\infty}(e^{\beta\mu})^{i}*e^{-\beta\epsilon_{i}}=1+zQ_{1}$

The problem is the last step I have no glue what is going on here.Okay I understand where the 1 is originating from but else... I tried figuring out the result by geometrical series and stuff. But this seems like pure magic to me. Or could it just be that the provided solution is wrong?? Please can anybody help!!

Answer 1

I believe there are two errors in the provided solution. I think the most important error is not in the last step, but in the first step. If I understand correctly, every adsorption site can take one and only one molecule. Adsorbing a molecule then comes with a factor $e^{\beta \mu}$, regardless of the energy level the molecule ends up on.

So the partition function is a sum of two scenarios: (a) no molecule at all is absorbed, resulting in a Boltzmann factor $e^{0} = 1$; (b) one molecule is adsorbed onto the $i$th energy level, losing its chemical potential but gaining energy $\epsilon _i$, resulting in a Boltzmann factor $e^{\beta (\mu - \epsilon _i) }$

Hence, \begin{equation} Z_1 = 1 + \sum _{i=1} ^{\infty} e ^{\beta (\mu - \epsilon _i)} = 1 + z \sum _{i=1} ^{\infty} e ^{- \beta \epsilon _i}. \end{equation} This is still not quite the same as $1 + zQ_1$, because the $i=0$ term in $Q_1$ is missing, which would bring in $z$ as another term. This term would be present if the molecule could adsorb onto the site in a state of zero energy, but the level $\epsilon _0$ is said to actually correspond to no adsorption at all. So this might be another error.