Show tthat different orbits in bohrs atom are not equally spaced

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show tthat different orbits in bohrs atom are not equally spaced

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LectureNotes said show tthat different orbits in bohrs atom are not equally spaced

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How Bohr’s Model Describes Atomic Orbits

Niels Bohr proposed a model for the hydrogen atom in which an electron moves in specific, quantized orbits around the nucleus. Two significant results from Bohr’s model that help us see that orbits are not equally spaced are:

  1. The Radii of Electron Orbits: In the Bohr model for hydrogen, the radius of the n^\text{th} orbit is given by

    r_n = n^2 a_0

    where a_0 \approx 0.529 \times 10^{-10}\,\text{m} is the Bohr radius and n is a positive integer (1, 2, 3, \dots). Notice that r_n increases with n^2, so the spacing between successive orbits grows larger and is not a constant difference.

  2. The Energy Levels: The energy of the n^\text{th} energy level in a hydrogen-like atom is given by:

    E_n = -\frac{m e^4}{8 \varepsilon_0^2 h^2} \,\frac{1}{n^2}

    Here, m is the mass of the electron, e is the charge of the electron, h is Planck’s constant, and \varepsilon_0 is the permittivity of free space. Since E_n \propto \frac{1}{n^2}, the difference in energy between consecutive levels (E_{n+1} - E_n) varies with n and is not a fixed amount. This explains why the energies get closer together as n increases.

Demonstration via Energy Differences

To explicitly show the non-constant spacing in energies, consider the difference in energy between two consecutive levels n and n+1:

\Delta E = E_{n+1} - E_{n} = -\frac{m e^4}{8 \varepsilon_0^2 h^2}\left(\frac{1}{(n+1)^2} - \frac{1}{n^2}\right).

Because

\frac{1}{(n+1)^2} - \frac{1}{n^2}

is not a constant for different values of n, the energy levels move closer together as n increases.

Demonstration via Radius Differences

Similarly, the radius difference between two successive orbits r_{n+1} and r_{n} is:

r_{n+1} - r_n = (n+1)^2 \, a_0 - n^2 \, a_0 = (2n + 1)\,a_0.

This expression grows with n, so the spacing between successive radii is not a fixed value.

Key Points

  • Orbits are defined by quantum numbers (n=1,2,3,\ldots).
  • Radius of orbit grows as n^2 → Differences in orbital radii increase with n.
  • Energy of orbit scales as 1/n^2 → Differences in orbital energies decrease as n increases.

Summary Table

Quantity Formula Spacing Between Levels
Radius (r_n) r_n = n^2 a_0 r_{n+1} - r_n = (2n + 1)a_0 (not constant)
Energy (E_n) E_n = -\frac{m e^4}{8 \varepsilon_0^2 h^2} \frac{1}{n^2} E_{n+1} - E_n = -\frac{m e^4}{8 \varepsilon_0^2 h^2}\bigl(\tfrac{1}{(n+1)^2}-\tfrac{1}{n^2}\bigr) (not constant)

Because both radius and distinct energy level spaces change with n, the orbits in Bohr’s model are not equally spaced.

@LectureNotes