The kinetic and potential energy of electron present in second excited state of hydrogen is

The kinetic and potential energy of an electron in the second excited state of hydrogen can be understood through the principles of quantum mechanics and the Bohr model of the atom. In this model, the energy levels of an electron in a hydrogen atom are quantized, meaning the electron can only occupy certain discrete energy levels. The second excited state corresponds to the principal quantum number n = 3.

Energy Levels in Hydrogen

In the Bohr model, the total energy (E) of an electron in a hydrogen atom is given by the formula:

E = - (13.6 eV) / n²

Here, n is the principal quantum number. For the second excited state (n = 3), we can calculate the total energy:

Calculating Total Energy

• For n = 3:
• E = - (13.6 eV) / (3²) = - (13.6 eV) / 9 = -1.51 eV

Kinetic and Potential Energy Relationship

In a hydrogen atom, the total energy is the sum of kinetic energy (K) and potential energy (U). According to the principles of classical mechanics, the kinetic energy of the electron is related to its potential energy by the equation:

K = - (1/2) U

In the context of the hydrogen atom, the potential energy of the electron in the electric field created by the proton is given by:

U = - (k * e²) / r

Where k is Coulomb's constant, e is the charge of the electron, and r is the radius of the orbit. For the Bohr model, the radius of the nth orbit is:

r = n² * (a₀)

Where a₀ is the Bohr radius (approximately 0.529 Å). For n = 3, the radius becomes:

r = 3² * a₀ = 9 * a₀

Calculating Kinetic and Potential Energy

Using the relationship between kinetic and potential energy, we can find both:

• Since E = K + U, we can express U as:
• U = 2E = 2 * (-1.51 eV) = -3.02 eV
• Then, using K = - (1/2) U:
• K = - (1/2) * (-3.02 eV) = 1.51 eV

Summary of Energies

To summarize, for the electron in the second excited state (n = 3) of hydrogen:

• Total Energy (E): -1.51 eV
• Kinetic Energy (K): 1.51 eV
• Potential Energy (U): -3.02 eV

This relationship illustrates how the kinetic and potential energies are interrelated in quantum systems, particularly in the context of the hydrogen atom. The negative potential energy indicates that the electron is bound to the nucleus, while the positive kinetic energy reflects its motion within that potential well.