Atoms

In the Bohr model, the centripetal force required for an electron to revolve in a circular orbit is provided by the electrostatic force of attraction between the electron and the nucleus. The centripetal force is given by: where is the mass of the electron, is its speed, and is the radius of the orbit. The electrostatic force is given by Coulomb's law: where is the charge of the electron, and is Coulomb's constant. Equating these two forces: From this, we can solve for and in terms of , the principal quantum number. Using Bohr's quantization condition: Substitute into the force equation: Solving for gives: Now, the time period is the time taken for one complete revolution, and it is related to the velocity and the radius by: Substituting and into this equation: Thus, the time period of revolution of the electron in the -th orbit is proportional to .

The potential energy of an electron revolving in a hydrogen atom is negative, not positive. This is because the electron is bound to the nucleus by the electrostatic force, and the potential energy is given by: where is the potential energy, is Coulomb's constant, is the electron charge, and is the distance between the electron and the nucleus. Since the electron is bound, is negative.
However, the total energy of a charged particle, such as the electron in a hydrogen atom, is the sum of its kinetic energy and potential energy. For the electron in a hydrogen atom, the total energy is negative because the magnitude of the potential energy is greater than the kinetic energy in magnitude.
Thus, the assertion is false (the potential energy is negative), while the reason is true (the total energy of a charged particle is always negative in bound states, but the reason is about total energy being negative, not positive).

Therefore, the correct answer is (D): Assertion is false but the reason is true.