Question

Question #431ec

Answer 1

Bohr's model is actually Rutherford-Bohr Model and it is an improvement over the Rutherford's model of atom. To understand set's start from here.

Explanation:

Following extract are required for this problem and therefore is reproduced below:

Rutherford model of atom proposed that all the atom's positive charge was concentrated in a very tiny volume at the centre of the atom. And that magnitude of this charge was proportional to the mass number of the atom.
He also proposed that electron cloud revolves around this central part of the atom much like planets rotate around the sun in our solar system, the size of the atom being about `10^5` times the size of this central part.

Bohr modified this and proposed that

The electrons can only orbit the nucleus stably, in fixed orbits at a fixed distances from the nucleus. These orbits are associated with definite energies and are also called energy shells or energy levels.

While in these orbits, the electron's acceleration does not produce any electromagnetic radiation and energy loss.

Bohr also proposed that any electron can gain or loose energy by jumping from one to another allowed energy levels by absorbing or radiating electromagnetic energy. The frequency `nu` of radiation was given as

`E_1-E_2=Delta E=h.nu`
where `h` is the Planck's constant.

Bohr's Model of Atom

On the basis of these postulates and detailed calculations, Bohr estimated radius of the simplest atom, Hydrogen, in the ground state. Limiting the calculations to first shell, `n=1`. This is called Bohr's radius `a_0` and is `approx 5.29 times 10^{-11}m`

Now Heisenberg's uncertainty principle or simply called Uncertainty Principle states that for any particle it is impossible to know simultaneously both position and momentum precisely.
Mathematically stated

`Deltax*Delta p approxh` .......(1)

where `Deltax` is uncertainty in position, `Delta p` is uncertainty in momentum, and `h` is Planck's Constant and is `6.626xx 10^(-34) m^2 kg // s`.

If an electron is confined within a circle of radius equal to Bohr's radius, uncertainty in position of electron will be of the size of the atom itself. The diameter of the hydrogen atom is `2xxa_0=2xx5.29 times 10^{-11}=1.058xx10^(−10)m`.

From equation (1), uncertainty in momentum is given by
`Delta p approxh/(Deltax)`
Assuming `Deltap=p`, also knowing that Kinetic energy `E=p^2/(2m_e)`, and inserting value of mass of an electron we obtain
`E=((6.626xx 10^(-34))/(1.058xx10^(−10)))^2/(2xx9.11×10^(-31))=2.15xx10^-15J`

This gives us the uncertainly in the energy of an electron when it circulates around the nucleus.

This is more than the ground state energy of Hydrogen which is about `2.2xx10^-18J`

Bohr's model postulates precise values of orbital radii and velocity of electrons. Where we can work out the energy differences between various orbits, giving the exact value of spectral transitions.

Thus the Heisenberg Uncertainty Principle does not support the Bohr atom model that an electron moves in a planar orbit of a fixed radius around the nucleus.