## Sunday, March 23, 2008

### Concept Review Ch.43 Bohr's Model and Physics of the Atom

Hydrogen spectra

When hydrogen gas in enclosed in a sealed tube and heated to a high temperature radiation is emitted. If this radiation is passed through a prism, components of different wavelengths in the radiation are deviated by different amounts and the picture that we capture on a screen is called the hydrogen spectrum.

The most striking feature of the hydrogen spectrum is that only some sharply defined discrete wavelengths exist in the radiation emitted by hydrogen.

For instance, light of wavelength 656.3 nm is observed and then light of wavelength 486.1 nm is observed. Hydrogen atoms do not emit any radiation between 656.3 nm and 486.1 nm.

Bohr's Model

In 1913, Niels Bohr proposed an atomic model that tried to explain the discrete sprectrum of radiation emitted by hydrogen gas.

The proposals of Bohr are termed as postulates

1. The electron revolves round the nucleus in circular orbits.

2. the circular orbits take only some special values of radius. In these orbits, the electron does not radiate energy, even though it is rotating around the nuclues. Radiation is expected from Maxwell's laws. But Bohr's conception is that electrons in the special radius orbits do not radiate energy.

3. The energy of the atom has a definite value when electrons of the atom are in specified stationary orbits. The electrons can jump from one stationary orbit another. If an electron jumps from an orbit of higher energy E2 to an orbit of lower energy E1, it emits a photon with energy equal to E2-E1.

The wavelength of the radiation of the photon (in wave nature theory) will be determined according to E2-E1 = hc/λ.

The electron can absorb energy from some source and jump from a lower energy orbit to a higher energy orbit also.

4. In the stationary orbits (orbits are stationary not the electrons) the angular momentum l of the electron around the nucleus is an integral multiple of the Planck constant h divided by 2π,

l = nh/2π

The last postulate is called Bohr's quantization rule.

Energy of a Hydrogen atom

The theory developed is applicable to hydrogen atoms and ions having just one electron. Thus it is valid for ions He+, Li++, Be+++ etc.

r = ε0h²n²/πmZe²

Kinetic energy of the electron in the nth orbit is

K = mv²/2 = mZ²e4/8ε0²h²n²

Potential energy of the atom when electron is in nth orbit

V = -mZ²e4/4ε0²h²n²

Total energy of the atom is E = K+V

= -mZ²e4/8ε0²h²n²

The formula of allowed radii derived from Bohr’s quantization rule is

r = ε0h²n²/πmZe²

So the radius of the smallest circle allowed can be found by putting n = 1 in that expression

r1 = ε0h²/πmZe²

For hydrogen, atomic number Z = 1. Substituting various values along with Z = 1 in the above expression we get r1 = 53 picometre or 0.053 nm.

1 picometre (pm) = 10^-12 m

53 pm or 0.053 nm is called the Bohr radium and it is used as a unit for measuring lengths in atomic physics. Bohr radius is denoted by the symbol a0.

For hydrogen atom, the second allowed radius for the electron will come out as 4 a0. (as n² term is in the radius formula).

The third radius allowed is 9 a0.
Extending it radius of the nth orbit allowed for the electron = n² a0

In the case of hydrogen like ions, Z will change and will be different from 1. Hence the radii formula in terms of Bohr radius a0 will be

rn = n² a0/Z

Ground and excited states

The expression for total energy of the atom is

Total energy of the atom is E = K+V

= -mZ²e4/8ε0²h²n²

For hydrogen atom Z = 1 and putting values for other constants and n = 1 we get energy when the electron is in the first allowed orbit as

E1 = -13.6 eV.

As we can see from the formula, energy is proportional to 1/ n² (All other constants will have the same value)

E2 = E1/4 = -13.6/4 = -3.4 eV

E3 = E1/9 = -13.6/9 = -1.5 eV

The lowest energy level is -13.6 eV (because it is negative, higher numerical values give lower energy levels)

The state of an atom with the lowest energy level is called its ground state.

If an electron makes a jump from the mth orbit(higher orbit) to nth orbit (lower orbit) (m>n), energy of the atom reduces to En from Em. The extra energy (Em-En) not any more there with the atom is emitted as a photon (electromagnetic radiation).

The wavelength of the emitted radiation will be according to the formula
according to Em-En = hc/λ.

It will come out as

1/ λ = RZ²(1/n² - 1/m²)

Where R = me4/8ε0²h³c.
R is called Rydberg constant and its value is equal to 1.0973*10^7 m^-1

In terms of the Rydberg constant energy formula becomes

E = -RhcZ²/n²

Rhc becomes 13.6 eV (For Z =1 and n = 1 we got that value earlier by using the more elaborate formula)

An energy of 1 rydberg means -13.6 eV ( energy of hydrogen atom in ground state.)