The kinetic energy (KE) of an electron in a Bohr orbit is given by the magnitude of the total energy, but with a positive sign.
Here's a breakdown:
The total energy (E) of an electron in the nth Bohr orbit is given by:
E = - (Z2e4m) / (8ε02h2n2)
where:
- Z is the atomic number (number of protons in the nucleus)
- e is the elementary charge (magnitude of the electron's charge)
- m is the mass of the electron
- ε0 is the permittivity of free space
- h is Planck's constant
- n is the principal quantum number (orbit number)
According to the Virial Theorem, for a Coulomb potential (like the one between the electron and the nucleus), the kinetic energy is related to the potential energy (V) and total energy (E) by:
2KE + V = 0
Also, the total energy (E) is the sum of kinetic energy (KE) and potential energy (V):
E = KE + V
From these relationships we can say: V = 2E, and KE = -E.
Therefore, kinetic energy can be expressed as:
KE = (Z2e4m) / (8ε02h2n2)
It's important to note:
- The kinetic energy is always a positive value.
- The total energy is negative, indicating a bound state (the electron is bound to the nucleus).
- As the principal quantum number (n) increases, the kinetic energy decreases, and the electron is less tightly bound to the nucleus.
In simpler terms: The kinetic energy of an electron in a Bohr orbit is equal to the magnitude of its total energy but expressed as a positive value.
