A hydrogrn atom is in state with energy -1.51ev.In the Bohr model ,what is the angular momentum of electron in the atom,with respect to an axis of nucleus.

To determine the angular momentum of an electron in a hydrogen atom using the Bohr model, we first need to understand the relationship between energy levels and angular momentum. In the Bohr model, the energy levels of the hydrogen atom are quantized, and each energy level corresponds to a specific angular momentum value.

Understanding Energy Levels in the Bohr Model

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

• E_n = -13.6 eV / n²

Here, E_n is the energy of the electron at level n, and n is the principal quantum number, which can take positive integer values (1, 2, 3, ...). The negative sign indicates that the electron is bound to the nucleus.

Finding the Principal Quantum Number

Given that the energy of the electron is -1.51 eV, we can set up the equation:

• -1.51 eV = -13.6 eV / n²

To find n, we rearrange the equation:

• n² = 13.6 eV / 1.51 eV
• n² ≈ 9.01
• n ≈ 3

This means the electron is in the third energy level (n = 3).

Calculating Angular Momentum

In the Bohr model, the angular momentum (L) of the electron is quantized and is given by the formula:

• L = nħ

Where ħ (h-bar) is the reduced Planck's constant, approximately equal to 1.055 x 10^-34 J·s. Now, substituting n = 3 into the equation:

• L = 3ħ

Now we can calculate L:

• L = 3 × 1.055 x 10^-34 J·s
• L ≈ 3.165 x 10^-34 J·s

Final Result

Thus, the angular momentum of the electron in the hydrogen atom, in the state with energy -1.51 eV, is approximately:

• L ≈ 3.165 x 10^-34 J·s

This value reflects the quantized nature of angular momentum in atomic systems, illustrating how the electron's motion is constrained by the principles of quantum mechanics.