Question

# In a classical model of the Hydrogen atom, the electron is moving around the proton on a circular orbit with radius r. Assume that a uniform external electric field is

applied so that the orbit of the electron is displaced by a small, compared to the radius of the orbit, distance Ax perpendicular to the plane of the orbit, and assume that the radius of the orbit remains unchanged. (a) Show that the electric field strength required to maintain the displacement of the orbit is given approximately by the expression below, where e is the magnitude of the charge of the electron: E \approx \frac{e}{4 \pi \epsilon_{0} r^{3}} \Delta x (b) Assume that in hydrogen gas (H2) each atom in a molecule does not interact with the other atom in the same molecule, or with any other atoms in neighbouring molecules.Assume that when a uniform external electric field is applied to the gas, all atoms are oriented such that the plane of the electron orbit is perpendicular to the field. Assume that the hydrogen gas is a dielectric, and that its polarisation is proportional to the magnitude of the external field. Under these assumptions, and using the result in part(a) above, show that the electric susceptibility Xe of the hydrogen gas H2 with a number of N/V molecules per unit volume, is given by: \chi_{e} \approx 8 \pi \frac{N}{V} r^{3} (c) Calculate, using the result in part (b) above, the electric susceptibility of hydrogen gas at a pressure of 105 Nm2 and a temperature of 293 K. Assume that hydrogen behaves as an ideal gas, which allows you to use the ideal gas equation. Use the Van de Waal's radius of the hydrogen atom, 120x10-12 m.

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