Atoms

Bohr Model of the Hydrogen Atom

Bohr Model of the hydrogen atom attempts to plug in certain gaps as suggested by Rutherford’s model by including ideas from the newly developing Quantum hypothesis. Bohr postulated that in an atom, electrons could revolve in stable orbits without emitting radiant energy.

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Bohr Model

Bohr model of the hydrogen atom attempts to plug in certain gaps as suggested by Rutherford’s model by including ideas from the newly developing Quantum hypothesis. According to Rutherford’s model, an atom has a central nucleus and electron/s revolve around it like the sun-planet system.

However, the fundamental difference between the two is that, while the planetary system is held in place by the gravitational force, the nucleus-electron system interacts by Coulomb’s Law of Force. This is because the nucleus and electrons are charged particles. Also, an object moving in a circle undergoes constant acceleration due to the centripetal force.

Further, electromagnetic theory teaches us that an accelerating charged particle emits radiation in the form of electromagnetic waves. Therefore, the energy of such an electron should constantly decrease and the electron should collapse into the nucleus. This would make the atom unstable.

The classical electromagnetic theory also states that the frequency of the electromagnetic waves emitted by an accelerating electron is equal to the frequency of revolution. This would mean that, as the electron spirals inwards, it would emit electromagnetic waves of changing frequencies. In other words, it would emit a continuous spectrum. However, actual observation tells us that the electron emits a line spectrum.

Watch Modern Atomic Theory –

Bohr Model Postulates

Bohr, in an attempt to understand the structure of an atom better, combined classical theory with the early quantum concepts and gave his theory in three postulates:

Postulate I

In a radical departure from the established principles of classical mechanics and electromagnetism, Bohr postulated that in an atom, electron/s could revolve in stable orbits without emitting radiant energy. Further, he stated that each atom can exist in certain stable states. Also, each state has a definite total energy. These are stationary states of the atom.

Postulate II

Bohr defined these stable orbits in his second postulate. According to this postulate:

  • An electron revolves around the nucleus in orbits
  • The angular momentum of revolution is an integral multiple of h/2p – where hàPlanck’s constant [h = 6.6 x 10-34 J-s].
  • Hence, the angular momentum (L) of the orbiting electron is: L = nh/2p

Postulate III

In this postulate, Bohr incorporated early quantum concepts into the atomic theory. According to this postulate, an electron can transition from a non-radiating orbit to another of a lower energy level. In doing so, a photon is emitted whose energy is equal to the energy difference between the two states. Hence, the frequency of the emitted photon is:

hv = Ei – Ef

(Ei is the energy of the initial state and Ef is the energy of the final state. Also, Ei > Ef).

Some important equations

Radii of Bohr’s stationary orbits

Bohr model

  • n – integer
  • rn – radius of the nth orbit
  • h – Planck’s constant
  • ε0 – Electric constant
  • m – Mass of the electron
  • Z – the Atomic number of the atom
  • e – Elementary charge

Since ε0, h, m, e, and p are constants and for a hydrogen atom, Z = 1, rn α n2

The velocity of Electron in Bohr’s Stationary Orbits

Bohr model

Since ε0, h, and e are constants and for a hydrogen atom, Z = 1, rn α (1/n)

Total Energy of Electron in Bohr’s Stationary Orbits

Bohr model

The negative sign means that the electron is bound to the nucleus.

Although these equations were derived under the assumption that electron orbits are circular, subsequent experiments conducted by Arnold Sommerfeld reaffirm the fact that the equations hold true even for elliptical orbits.

Energy Levels

When the electron is revolving in an orbit closest to the nucleus, the energy of the atom is the least or has the largest negative value. In other words, n = 1. For higher values of n, the energy is progressively larger.

The state of the atom wherein the electron is revolving in the orbit of smallest Bohr radius (a0) is the ‘Ground State’. In this state, the atom has the lowest energy. The energy in this state is:

E1 = -13.6 eV

Hence, the minimum energy required to free an electron from the ground state of an atom is 13.6 eV. This energy is the ‘Ionization Energy’ of the hydrogen atom. This value agrees with the experimental value of ionization energy too.

Now, a hydrogen atom is usually in ‘Ground State’ at room temperature. The atom might receive energy from processes like electron collision and acquire enough energy to raise the electron to higher energy states or orbits. This is an ‘excited’ state of the atom. Therefore, the energy required by the atom to excite an electron to the first excited state is:

E2 – E1 = -3.40 eV – (-13.6) eV = 10.2 eV

Similarly, to excite the electron to the second excited state, the energy needed is:

E3 – E1 = -1.51 eV – (-13.6) eV = 12.09 eV

Remember, that the electron can jump to a lower energy state by emitting a photon. Also, note that, as the excitation of the hydrogen atom increases, the minimum energy required to free the electron decreases.

Solved Examples for You

Question: How many postulates are present in the Bohr model of a hydrogen atom?
Solution: Bohr model of a hydrogen atom has three postulates. The postulate of the circular orbit, postulate of the selected orbit and postulate of the origin of spectral lines.

Question: According to the Bohr model, what is the energy of the atom in the ground state?
Solution: According to the Bohr model, the energy of the atom in the ground state is -13.6 eV.

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