Te ground state energy of hydrogen atom is –13.6 eV. What are the Kinetic and potential energies of the electron in this state

The ground state energy of a hydrogen atom being -13.6 eV is a fundamental concept in quantum mechanics, particularly in the study of atomic structure. To understand the kinetic and potential energies of the electron in this state, we can use the principles of classical mechanics and quantum theory.

Understanding Energy Components in the Hydrogen Atom

In the hydrogen atom, the total energy (E) of the electron is the sum of its kinetic energy (K) and potential energy (U). The relationship can be expressed as:

E = K + U

For the hydrogen atom, the potential energy is given by the formula:

U = - (k * e^2) / r

Where:

• k is Coulomb's constant (approximately 8.99 x 10^9 N m²/C²).
• e is the elementary charge (approximately 1.6 x 10^-19 C).
• r is the distance between the electron and the nucleus (for the ground state, this is the Bohr radius, about 5.29 x 10^-11 m).

Calculating Potential Energy

In the ground state of hydrogen, the potential energy can be calculated using the known values. The potential energy at the Bohr radius can be derived as follows:

U = - (k * e^2) / r

Substituting the values:

• k ≈ 8.99 x 10^9 N m²/C²
• e ≈ 1.6 x 10^-19 C
• r ≈ 5.29 x 10^-11 m

After performing the calculation, we find that the potential energy U is approximately -27.2 eV.

Determining Kinetic Energy

In a stable orbit, the kinetic energy of the electron can be related to the potential energy. For a hydrogen atom, the kinetic energy is half the magnitude of the potential energy:

K = -1/2 U

Substituting the value of U:

K = -1/2 (-27.2 eV) = 13.6 eV

Summary of Energies

Now, we can summarize the energies of the electron in the ground state of the hydrogen atom:

• Total Energy (E): -13.6 eV
• Kinetic Energy (K): 13.6 eV
• Potential Energy (U): -27.2 eV

In conclusion, the kinetic energy of the electron in the ground state of the hydrogen atom is 13.6 eV, while the potential energy is -27.2 eV. This relationship highlights the balance of forces acting on the electron and is a key aspect of atomic physics.