Question

6.a) In terms of wavelength, frequency and energy compare the two photons with wavelengths of 900 nm and 780 nm. b) What are the emission spectra of an element? c) How are the emission spectra of an element produced? d) How did Bohr (quantum theory) explain the emission (line) spectra of hydrogen?

Answer 1

(a) Given for Photon 1 : wavelength =900 nm = 9 X 10^-7 m, As frequency is the speed of light divided by wavelenth i.e v=c/λ, so frequncy = 3x10⁸ms^-1/9x10^-7m = 3.33 x 10¹⁴ s or 3.33 x 10¹⁴ Hertz. As energy = plank's constant x frequency =6.626 x10^-34 Js x 3.33 x 10¹⁴ s^-1 = 22.065 x10^-20J

Also, for Photon 2 : wavelength =780 nm = 7.8 X 10^-7 m, As frequency is the speed of light divided by wavelenth i.e v=c/λ, so frequncy = 3x10⁸ms^-1/7.8 x10^-7m = 3.84 x 10¹⁴ s or 3.84 x 10¹⁴ Hertz. As energy = plank's constant x frequency =6.626 x10^-34 Js x 3.84 x 10¹⁴ s^-1 = 25.443 x10^-20 J

(b) The emission spectrum of an element is the spectrum of frequencies of electromagnetic radiation emitted due to an atom or molecule making an atomic electron transition.

(c) As emission is the process by which a higher energy quantum mechanical state of a particle becomes converted to a lower one through the emission of a photon resulting in the production of light. The frequency of light emitted is a function of the energy of the transition. Since energy must be conserved, the energy difference between the two states equals the energy carried off by the photon. The energy states of the transitions can lead to emissions over a very large range of frequencies.resulting in production of an emission spectra.

(d) Bohr assumed that each electron with an associated energy orbit the nucleus at certain discrete, or quantized, radii,. He also assumed that when electrons "fall" from larger to smaller orbits, they release electromagnetic radiation obeying the Planck-Einstein relationship. Because the energies of the orbits are quantized, so are the wavelengths. Bohr's model calculated the energy of an electron in the shell, n as:

E(n)=-1/n².13.6 eV

Bohr explained the hydrogen spectrum in terms of electrons absorbing and emitting photons to change energy levels, where the photon energy is

hv =E = (1/n²_low - n_high²) .13.6 eV

In the Bohr’s model of the hydrogen atom, he suggested that the perhaps the electrons could only orbit the nucleus in specific orbits or shells with a fixed radius. Only shells with a radius given by the equation below would be allowed, and the electron could not exist in between these shells. Mathematically, the allowed values of the atomic radius can be written as r(n)=n². r(1), wheren n is a positive integer, and r(1) is the Bohr radius, the smallest allowed radius for hydrogen.

He found that Bohr, radius r(1) has the value 0.529×10​⁻¹⁰​​m

By keeping the electrons in circular, quantized orbits around the positively-charged nucleus, Bohr was able to calculate the energy of an electron in the nth energy level of hydrogen: E(n)=-1/n².13.6 eV, where the lowest possible energy or ground state energy of a hydrogen electron—E(1) is −13.6eV

Note that the energy is always going to be a negative number, and the ground state, n=1 has the most negative value. This is because the energy of an electron in orbit is relative to the energy of an electron that has been completely separated from its nucleus, n=∞, which is defined to have an energy 0eV. Since an electron in orbit around the nucleus is more stable than an electron that is infinitely far away from its nucleus, the energy of an electron in orbit is always negative.