Atomic Structure

1. Energy of the Electron in Bohr Hydrogen Atom:

The energy of an electron in the -th orbit of a Bohr hydrogen atom is given by the equation:

Where:

• is the energy of the electron in the -th orbit (in eV).
• 13.6 eV is the energy of the electron in the first orbit (n = 1).
• is the principal quantum number (for the first orbit, ).

For , the energy of the electron is , which is the energy in the ground state. The given energy is , so we can infer that the electron is in the second orbit where . This is confirmed by the following calculation:

2. Bohr’s Quantization Condition for Angular Momentum:

According to Bohr’s model, the angular momentum of the electron in the -th orbit is quantized and is given by:

Where:

• is the angular momentum of the electron.
• is the principal quantum number (in this case, for the second orbit).
• is the reduced Planck's constant, , where

3. Calculating the Angular Momentum:

Substitute and into the equation for angular momentum:

4. Conclusion:

• The angular momentum of the electron in the second orbit of the Bohr hydrogen atom is .

Analyzing the assertion and reason given for a hydrogen atom's electron:

Assertion (A): The potential energy of an electron revolving in any stationary orbit in a hydrogen atom is positive.

Reason (R): The total energy of a charged particle is always positive.

To evaluate these statements, consider the concept of potential and total energies in atomic physics:

• The potential energy (PE) of an electron in a hydrogen atom is negative because it is related to the attractive force between the negatively charged electron and the positively charged nucleus. The potential energy is calculated as PE = -ke²/r, where k is Coulomb's constant, e is the elementary charge, and r is the radius of the orbit. This energy is negative due to the attractive interaction.
• The total energy (E) of an electron in its orbit is also negative in a bound system like the hydrogen atom. Total energy is the sum of kinetic energy (KE) and potential energy, with E = KE + PE. In stable, bound systems, the total energy is less than zero, which means it is negative.

Conclusion: Given these insights:

• Assertion (A) is false. The potential energy of an electron in a stationary orbit in a hydrogen atom is negative, not positive.
• Reason (R) is false. The total energy of a charged particle in a bound system like a hydrogen atom is not always positive; it is usually negative.

Thus, the correct option is:

Assertion (A) is false and Reason (R) is also false.