In the ground state of the H-atom the electron has a total energy of -13.6 eV. What are (a) its kinetic energy, and (b) its potential energy if the electron is one Bohr radius from the central nucleus.
To find the kinetic energy and potential energy of the electron in the ground state of the hydrogen atom, we can use the formula for the total energy:
E = Kinetic Energy + Potential Energy
Given that the total energy of the electron is -13.6 eV and it is in the ground state, we can assume that this total energy is equivalent to the potential energy of the electron, as there is no energy contribution from higher energy levels (excited states). Therefore:
(a) To find the kinetic energy, we can subtract the potential energy from the total energy:
Kinetic Energy = Total Energy - Potential Energy
= -13.6 eV - (-13.6 eV)
= -13.6 eV + 13.6 eV
= 0 eV
So, the kinetic energy of the electron in the ground state of the hydrogen atom is 0 eV.
(b) To find the potential energy, we need to consider the potential energy due to the attraction between the electron and the nucleus. The formula for the potential energy of an electron at a distance "r" from the nucleus of a hydrogen atom is given by:
Potential Energy = - (k * e^2) / r
Where:
- k is the electrostatic constant (8.9875 × 10^9 N m^2 C^-2)
- e is the elementary charge (1.602176634 × 10^-19 C)
- r is the distance between the electron and the nucleus (in meters)
We are given that the electron is one Bohr radius away from the nucleus. The Bohr radius for hydrogen is approximately 5.29 x 10^-11 meters.
Potential Energy = - (k * e^2) / r
= - (8.9875 × 10^9 N m^2 C^-2 * (1.602176634 × 10^-19 C)^2) / (5.29 x 10^-11 m)
≈ -2.18 x 10^-18 J
To convert the potential energy from joules to electron volts (eV), we can use the conversion factor 1 eV = 1.60218 x 10^-19 J:
Potential Energy = -2.18 x 10^-18 J / (1.60218 x 10^-19 J/eV)
≈ -13.6 eV
So, the potential energy of the electron in the ground state of the hydrogen atom, when it is one Bohr radius away from the nucleus, is approximately -13.6 eV.