Question

Explain in details the Bohr's Module for the atom?

Answer 1

Niels Bohr proposed the Bohr Model of the Atom in 1915. Because the Bohr Model is a modification of the earlier Rutherford Model, some people call Bohr's Model the Rutherford-Bohr Model.

The modern model of the atom is based on quantum mechanics. The Bohr Model contains some errors, but it is important because it describes most of the accepted features of atomic theory without all of the high-level math of the modern version. Unlike earlier models, the Bohr Model explains the Rydberg formula for the spectral emission lines of atomic hydrogen.

The Bohr Model is a planetary model in which the negatively-charged electrons orbit a small, positively-charged nucleus similar to the planets orbiting the Sun (except that the orbits are not planar). The gravitational force of the solar system is mathematically akin to the Coulomb (electrical) force between the positively-charged nucleus and the negatively-charged electrons.

Main Points of the Bohr Model

• Electrons orbit the nucleus in orbits that have a set size and energy.
• The energy of the orbit is related to its size. The lowest energy is found in the smallest orbit.

• Radiation is absorbed or emitted when an electron moves from one orbit to another.

Bohr Model of Hydrogen

The simplest example of the Bohr Model is for the hydrogen atom (Z = 1) or for a hydrogen-like ion (Z > 1), in which a negatively-charged electron orbits a small positively-charged nucleus. Electromagnetic energy will be absorbed or emitted if an electron moves from one orbit to another.

Only certain electron orbits are permitted. The radius of the possible orbits increases as n², where n is the principal quantum number. The 3 ? 2 transition produces the first line of the Balmer series. For hydrogen (Z = 1) this produces a photon having wavelength 656 nm (red light).

Problems with the Bohr Model

• It violates the Heisenberg Uncertainty Principle because it considers electrons to have both a known radius and orbit.
• The Bohr Model provides an incorrect value for the ground state orbital angular momentum.
• It makes poor predictions regarding the spectra of larger atoms.
• It does not predict the relative intensities of spectral lines.
• The Bohr Model does not explain fine structure and hyperfine structure in spectral lines.
• It does not explain the Zeeman Effect.
• Electrons in atoms orbit the nucleus.
• The electrons can only orbit stably, without radiating, in certain orbits (called by Bohr the "stationary orbits"^[5]) at a certain discrete set of distances from the nucleus. These orbits are associated with definite energies and are also called energy shells or energy levels. In these orbits, the electron's acceleration does not result in radiation and energy loss as required by classical electromagnetics. The Bohr model of an atom was based upon Planck's quantum theory of radiation.
• Electrons can only gain and lose energy by jumping from one allowed orbit to another, absorbing or emitting electromagnetic radiation with a frequency ? determined by the energy difference of the levels according to the Planck relation:

  {\displaystyle \Delta {E}=E_{2}-E_{1}=h\nu \ ,}

  where h is Planck's constant. The frequency of the radiation emitted at an orbit of period T is as it would be in classical mechanics; it is the reciprocal of the classical orbit period:

  {\displaystyle \nu ={1 \over T}~~.}:

• {\displaystyle {1 \over 2}mv^{2}=n{h \over 2\pi }=n\hbar }

• Bohr's condition, that the angular momentum is an integer multiple of ? was later reinterpreted in 1924 by de Broglie as a standing wave condition: the electron is described by a wave and a whole number of wavelengths must fit along the circumference of the electron's orbit:

  {\displaystyle n\lambda =2\pi r~.}

• Classical mechanics

• The electron is held in a circular orbit by electrostatic attraction. The centripetal force is equal to the Coulomb force.

  {\displaystyle {m_{\mathrm {e} }v^{2} \over r}={Zk_{\mathrm {e} }e^{2} \over r^{2}}}

  where m_e is the electron's mass, e is the charge of the electron, k_e is Coulomb's constant and Z is the atom's atomic number. It is assumed here that the mass of the nucleus is much larger than the electron mass (which is a good assumption). This equation determines the electron's speed at any radius:

  {\displaystyle v={\sqrt {Zk_{\mathrm {e} }e^{2} \over m_{\mathrm {e} }r}}.}

  It also determines the electron's total energy at any radius:

  {\displaystyle E=-{1 \over 2}m_{\mathrm {e} }v^{2}}

  The total energy is negative and inversely proportional to r. This means that it takes energy to pull the orbiting electron away from the proton. For infinite values of r, the energy is zero, corresponding to a motionless electron infinitely far from the proton. The total energy is half the potential energy, which is also true for noncircular orbits by the virial theorem.

• A quantum rule

• The angular momentum L = m_evr is an integer multiple of ?:

  {\displaystyle m_{\mathrm {e} }vr=n\hbar }

• Substituting the expression for the velocity gives an equation for r in terms of n:

  {\displaystyle m_{\text{e}}{\sqrt {\dfrac {k_{\text{e}}Ze^{2}}{m_{\text{e}}r}}}r=n\hbar }

  so that the allowed orbit radius at any n is:

  {\displaystyle r_{n}={n^{2}\hbar ^{2} \over Zk_{\mathrm {e} }e^{2}m_{\mathrm {e} }}}

  The smallest possible value of r in the hydrogen atom (Z=1) is called the Bohr radius and is equal to:

  {\displaystyle r_{1}={\hbar ^{2} \over k_{\mathrm {e} }e^{2}m_{\mathrm {e} }}\approx 5.29\times 10^{-11}\mathrm {m} }

  The energy of the n-th level for any atom is determined by the radius and quantum number:

  {\displaystyle E=-{Zk_{\mathrm {e} }e^{2} \over 2r_{n}}=-{Z^{2}(k_{\mathrm {e} }e^{2})^{2}m_{\mathrm {e} } \over 2\hbar ^{2}n^{2}}\approx {-13.6Z^{2} \over n^{2}}\mathrm {eV} }

• 

  This expression is clarified by interpreting it in combinations that form more natural units:

  {\displaystyle \,m_{\mathrm {e} }c^{2}} is the rest mass energy of the electron (511 keV)

  {\displaystyle \,{k_{\mathrm {e} }e^{2} \over \hbar c}=\alpha \approx {1 \over 137}} is the fine structure constant

  {\displaystyle \,R_{\mathrm {E} }={1 \over 2}(m_{\mathrm {e} }c^{2})\alpha ^{2}}

  Since this derivation is with the assumption that the nucleus is orbited by one electron, we can generalize this result by letting the nucleus have a charge q = Z e where Z is the atomic number. This will now give us energy levels for hydrogenic atoms, which can serve as a rough order-of-magnitude approximation of the actual energy levels. So for nuclei with Z protons, the energy levels are (to a rough approximation):

  {\displaystyle E_{n}=-{Z^{2}R_{\mathrm {E} } \over n^{2}}}

• the formula uses the reduced mass also, but in this case, it is exactly the electron mass divided by 2. For any value of the radius, the electron and the positron are each moving at half the speed around their common center of mass, and each has only one fourth the kinetic energy. The total kinetic energy is half what it would be for a single electron moving around a heavy nucleus.

  {\displaystyle E_{n}={R_{\mathrm {E} } \over 2n^{2}}~~~~~~}