Question 23.2: Consider a vessel containing a gas consisting of N independe...

This problem asks us to use the Grand-Canonical ensemble to study this system. The Grand-Canonical ensemble is typically used to study systems in equilibrium. The system described in this problem contains two simple systems: the gas in the threedimensional region and the two-dimensional adsorption sites. We will focus on each of these systems separately. Then, we can relate the two systems by imposing the condition of equality of the chemical potentials. In Part (a), we are asked to derive an expression for λ, or the fugacity, in terms of P and T, and in Part (b), we are asked to derive an expression for <N> , the average number of atoms adsorbed onto a wall. Since this is an equilibrium situation, where atoms are in equilibrium between the gas phase and the adsorbed phase, the use of the Grand-Canonical ensemble is most appropriate. Recall that the Grand-Canonical ensemble was discussed in detail in Part III.

Part (a): Calculate the Fugacity, λ, Using the Grand-Canonical Ensemble

Because the gas atoms can adsorb onto the walls of the vessel, the number of gas atoms in the three-dimensional gas region does not remain constant. As discussed in Part III, the Grand-Canonical partition function is given by:

\Xi(\underline{ V }, T, \mu)=\sum_{N=0}^{\infty} Q(N, \underline{V}, T ) \lambda^{N}, \text { where } \lambda=\exp (\beta \mu)                          (21)

For a collection of N independent, indistinguishable atoms, we know that:

Q(N, \underline{ V }, T)=\frac{[q(\underline{V}, T)]^{N}}{N !}                          (22)

where q is the atomic partition function.
Substituting Eq. (22) in Eq. (21) yields:

\Xi(\underline{ V }, T, \mu)=\sum_{N=0}^{\infty} \frac{[q(\underline{V}, T) \lambda]^{N}}{N !}                          (23)

where \sum_{N=0}^{\infty} \frac{[q(\underline{V}, T) \lambda]^{N}}{N !}=e^{q \lambda}. Using the last result in Eq. (23), we obtain:

\Xi(\underline{ V }, T, \mu)=\exp (q \lambda)                        (24)

In Part III, we showed that:

P \underline{ V }=k_{B} T \ln \Xi                      (25)

Combining Eq. (25) with Eq. (24) yields:

\lambda=\frac{P \underline{V}}{k_{B} T q}                        (26)

For a monoatomic ideal gas, we know that:

q=\left(\frac{2 \pi m k_{B} T}{h^{2}}\right)^{3 / 2} \underline{ V } g_{e 1}                         (27)

Using Eq. (27) in Eq. (26), we obtain:

\lambda=P\left(k_{B} T\right)^{-5 / 2}\left(\frac{h^{2}}{2 \pi m}\right)^{3 / 2} \frac{1}{g_{e 1}}                          (28)

where λ is not a function of V.

Part (b): Calculate <N>

We are asked to calculate the average number of atoms in the adsorbed state. Naturally, the focus is the adsorbing wall, which we regard as being in equilibrium with the gas. Accordingly, μ and T are the same for both the gas and the adsorbed atoms. In fact, each adsorbing site acts independently, is separately in equilibrium with the gas, and has a Grand-Canonical partition function derived in Part III. Specifically:

\Xi_{s i t e}=\sum_{N=0}^{1} Q(N, \underline{ V }, T) \lambda^{N}                         (29)

and

Q(N, \underline{ V }, T)=\sum_{j} e^{-\beta \underline{ E }_{j}}

where the summation is over all possible states corresponding to N and V. In our case, there exists only one energy level for a given N and V, that is, -Nε. Therefore, Eq. (29) becomes:

\Xi_{s i t e}=\sum_{N=0}^{1} \exp (\beta N \varepsilon) \lambda^{N}                          (30)

In other words:

\Xi_{\text {site }}=e^{0} \lambda^{0}+e^{\beta \varepsilon} \lambda=1+\lambda e^{\beta \varepsilon}                          (31)

The average number of adsorbed atoms per site was derived in Part III.
Specifically:

<N>_{\text {site }}=k T\left(\frac{\partial \ln \Xi_{s i t e}}{\partial \mu}\right)_{T, \underline{ V }}=k T\left[\frac{\partial}{\partial \mu} \ln \left(1+e^{\beta \varepsilon} e^{\beta \mu}\right)\right]_{T, \underline{ V }}                          (32)

<N>_{\text {site }}=k T\left(\frac{\beta e^{\beta \varepsilon} e^{\beta \mu}}{1+e^{\beta \varepsilon} e^{\beta \mu}}\right)=\frac{e^{\beta \varepsilon} e^{\beta \mu}}{1+e^{\beta \varepsilon} e^{\beta \mu}}=\frac{1}{1+e^{-\beta \varepsilon} e^{-\beta \mu}}=\frac{1}{1+\lambda^{-1} e^{-\beta \varepsilon}}                          (33)

It then follows that for n adsorbing sites, the average number of adsorbing atoms is given by:

<N>=<N>_{\text {site }} n=\frac{n}{1+\lambda^{-1} e^{-\beta \varepsilon}}                         (34)

At this stage, we impose the condition of thermodynamic equilibrium between the gas and the adsorbing wall. Accordingly, using the expression for λ (see Eq. (28)) in Eq. (34), we obtain the desired expression for the average number of adsorbed atoms as a function of T and P. Specifically:

<N>=\frac{n}{1+\left[\left(\frac{2 \pi m}{h^{2}}\right)^{3 / 2} g_{e 1}\right] P^{-1}(k T)^{5 / 2} e^{-\beta \varepsilon}}                           (35)

At fixed temperature, <N> varies with pressure in a manner that is expected intuitively. At high pressures, P \gg 1, P^{-1}<<1, \text { and }<N>\sim n. Physically, at high pressures, the gas density is high and the atoms frequently approach and adsorb onto the adsorbing sites, so that <N>\sim n. At low pressures, P \ll 1, P^{-1} \gg 1, and <N>\sim n The effect of temperature is also apparent. At high temperatures, T \rightarrow \infty, e^{-\beta \varepsilon} \sim 1, \text { and }(k T)^{5 / 2} \gg 1, \text { so that }<N>\sim 0, because the adsorbed atoms are easily released from the adsorbing sites. At low temperatures, T \rightarrow 0, e^{-\beta \varepsilon} \sim 0, and (k T)^{5 / 2} \rightarrow 0 \text {, so that }<N>\sim n. Clearly, at high temperatures, a high pressure is required to keep the adsorbing sites filled, while only a modest pressure is required at low temperatures.
With respect to the temperature dependence, the atoms can be thought of as possessing kinetic energy in the three-dimensional phase and zero kinetic energy in the two-dimensional phase. When adsorbed onto the two-dimensional wall, an atom has no kinetic energy but gains -ε of energy through its interaction with the wall. When the temperature is very low, the kinetic energy of an atom is so small relative to ε, that the atom prefers to gain ε of energy by adsorbing onto the wall. On the other hand, when the temperature is high, the atom has far more kinetic energy than ε and, therefore, prefers to access the three-dimensional gas phase.