Does an electron release energy while roaming around the nucleus in a single energy level? I don't really get why the Rutherford theory that the electrons are orbiting around the nucleus is rejected.
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Yes, the electrons do orbit around the nucleus, but there's a catch. Rutherford said that the electrons could orbit at any distance from the nucleus. But this, would cause the electrons to lose energy as they revolved around the nucleus, as electrons are nothing but accelerated charges, and according to Maxwell, accelerated charges radiate energy.
This is where Neils Bohr and his atomic model comes in. He said that electrons can only revolve in those orbits where the angular momentum of the electron is an integral multiple of the Plank's constant by two time pi.
$$mvr = nh/2\pi$$
where $m$ is the mass of the electron, $v$ is its velocity, $r$ is the radius of the electron's orbit, and $h$ is Plank's constant.
He argued that only these radii for an electron to revolve without radiating energy was possible, as in these radii, the matter waves that the revolving electrons form are standing waves. Standing waves do not radiate energy. This is why Bohr's theory proved correct over Rutherford's theory.
Classical electromagnetic theory demands that if a charge experiences an acceleration (including accelerations perpendicular to the velocity that don't change its speed, like those experienced in uniform circular motion) it must radiate energy in the form of electromagnetic waves, which would make atoms unstable. That means that the Rutherford theory is inconsistent with reality.
Real atoms don't do this, of course. The full answer is provided by quantum mechanics (in which electrons are described by three-dimensional wave functions that sit in the resonant positions of a wave equation known as the Schrödinger equation), though there is a simplified model (the Bohr model, which thinks of electrons as classical particles which must follow additional restrictions on the orbits they can take) that is also good enough on that score. Either way, there is a lowest possible energy level, and once electrons get there they cannot radiate away any energy because there simply is no dynamical state of lower energy that they could possibly reach.