The chemical potentials for free electrons in metals A and B...

The chemical potentials for free electrons in metals A and B differ by $1.1 \mathrm{eV}$ when the metals are not in contact. When cubic blocks of these metals are brought into contact, electrons move from A to B until the chemical potential is uniform. How many electrons are transferred, if the contact area is $1 \mathrm{~cm}^2$ and the junction can be modelled as a parallel plate capacitor with gap $10^{-10} \mathrm{~m}$ ? Is this a significant fraction of the total number of free electrons in the metal blocks?

Key Concepts

Scale of Charge Transfer Relative to Total Free Electrons

Even though a measurable number of electrons may transfer to balance the chemical potentials, this number is extremely small compared to the total number of free electrons in the bulk of the metal. This illustrates that the correction needed to equilibrate the potentials involves only a tiny fraction of the overall electron population, having negligible effect on the bulk properties of the metals.

Charge Transfer and Capacitance Calculation

Using the parallel plate capacitor model, the capacitance is given by C = ??A/d, where ?? is the permittivity of free space, A is the contact area, and d is the gap between the metal surfaces. The voltage difference (which in this context is related to the chemical potential difference) drives the transfer of electrons, and the total transferred charge is given by Q = CV. Dividing this charge by the charge of an electron provides the number of electrons transferred.

Chemical Potential Equilibration

When two metals with different chemical potentials (essentially different Fermi levels) are brought into contact, electrons will flow from the metal with the higher energy level to the one with the lower energy level until equilibrium is reached. This process is driven by the tendency of the system to minimize its total free energy, resulting in a uniform chemical potential across both metals.

Capacitor Modeling of the Junction

The junction formed by the contact between the two metals can be modeled as a parallel plate capacitor, where the small gap between the metals behaves like the dielectric separating the two plates. The key parameters in this model are the area of the contact and the separation distance, which determine the capacitance of the junction.

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