\sect{Adsorption of polar molecules to surface} Consider a 2D adsorbing surface in equilibrium with a 3D gas of atoms that has temperature $T$ and chemical potential $\mu$.On the surface there are $M$ sites ,each site can absorb at most one atom . At the adsorption site an atom forms an electric dipole $d$ that can be oriented at any direction away from the surface (see figure). In presence of an electric field $\mathcal{E}$ perpendicular to the surface the dipole has energy $-\mathcal{E}dcos(\theta)$ where $|\theta|<\pi/2$ is the angle between $d$ and $\mathcal{E}$. (a) Calculate the grand partition function $\mathcal{Z}(\beta,\mu,\mathcal{E})$ (b) Derive the average number $N$ of absorbed atoms. (c) Use the formal approach to define the average polarization $D$ as the expectation value of a system observable. Derive the state equation for $D$. (d) What are the results in the limit $\mathcal{E}\rightarrow0$ and in particular what is the ratio $D/N$. Explain how this result can be obtained without going through the formal derivation. \begin{center} \includegraphics[scale=0.45]{A34.eps} \end{center} \ \\ \ \\ OLD: \sect{Baruch's A43.} 1. Consider a 3-dimensional gas of atoms with a chemical potential $\mu$ that can adsorb on any of M sites on a surface; at each site at most one atom can be adsorbed. At the adsorption site an atom forms an electric dipole ${\bf d}$ that can be oriented at any direction {\em away} from the surface (see figure). In presence of an electric field ${\bf E}$ perpendicular to the surface the dipole has energy $-Ed\cos \theta$ where $|\theta | \leq \pi/2$ is the angle between ${\bf d}$ and ${\bf E}$. \begin{itemize} \item [(a)] Evaluate the number N of adsorbed atoms. Normalize the phase space of each adsorbed atom to 1. Derive the limit $E\rightarrow 0$ and explain why is the result finite, in spite of the adsorption energy being $0$ at $E=0$. \item [(b)] Find the average electric dipole perpendicular to the surface.\\ \end{itemize} \begin{center} \includegraphics[scale=0.45]{A34.eps} \end{center} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%