Electron gas equations for electron flow in diodes

By introducing the equations of gas motion and the notions of temperature and pressure of an electron gas, Hahn
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arrived at certain modifications of the equations of electron current in diodes obtained by Langmuir
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. In deriving his results Hahn assumed that the temperature of the electron gas remains constant: T = const.; (1) moreover, he assumed the law of perfect gases between the pressure and temperature: n = k n T, (2) where n is the number of electrons per unit volume and k is Boltzmann's constant, to apply to an electron cloud.