 # Bohr Model of the Hydrogen Atom

### What Led to the Bohr Model of Hydrogen Atom?

In 1897, Sir J.J. Thomson discovered electrons as negatively charged particles present in every element's atom, but without any knowledge of the distribution of electrons, the positive charge, and the mass inside the atom. Subsequently, in 1904, Sir Thomson suggested a model for the atom, also known as the 'plum pudding model,' which stated that the electrons are embedded like plums in a distribution (or pudding) of positive charge within the atom. Thomson's model failed to explain emission spectra and alpha particle scattering. Rutherford came up with another model in which the electrons revolve around the nucleus in different orbits. The revolution is driven by the electrostatic force of attraction between the nucleus and the electrons. But Rutherford's model failed to account for the stability of atoms and the origin of line spectra. To address the shortcomings of these previous models, Prof. Neils Bohr, in 1913, applied Planck's quantum theory and proposed three postulates that came to be known as the Bohr Model of Atom. So, let us discuss the Bohr Model of Hydrogen Atom (class 12) in detail.

### Bohr Model of Atom

Applying the quantum theory of Planck, the hydrogen atom model put forward by Bohr is based on three postulates:

1. Electrons can only revolve in those orbits in which their angular momentum is an integral multiple of h/2π, where 'h' represents Planck's universal constant. Say, 'm' denotes the mass of an electron, and it is revolving with a velocity 'v' in an orbit of radius 'r', then its angular momentum will be 'mvr.' Therefore, according to Bohr's quantisation condition:

mvr = nh/2π,

where 'n' represents the principal quantum number of the orbit and is an integer (n = 1, 2, 3,…). Thus, Bohr's model of the atom postulates that electrons can only revolve in precise discrete orbits of specific radii, called 'stable orbits.'

1. While revolving in the stable orbits, the centripetally accelerated electrons do not radiate any energy. Hence, the atoms remain stable and exist in a 'stationary state.'

2. When the atom absorbs energy from outside, one or more of its outer electrons leaves its orbit and goes to some higher orbit of greater energy and is said to be in an 'excited state.' Within a time of 10-8 seconds, the electron returns to the lower orbit, and in the process, the electron radiates energy in the form of electromagnetic waves.

If the energy of the electron in the higher orbit is E2 and that in the lower orbit is E1, then the frequency 𝜈 of the radiated waves is given by Bohr's frequency condition:

𝜈 = (E2 - E1)/h

### Bohr's Theory of Hydrogen Atom and Hydrogen-like Atoms

A hydrogen-like atom consists of a tiny positively-charged nucleus and an electron revolving around the nucleus in a stable circular orbit.

If 'e,' 'm,' and 'v' be the charge, mass, and velocity of the electron respectively, 'r' be the radius of the orbit, and Z be the atomic number, the equation for the radii of the permitted orbits is given by r = n2 ｘr1, where 'n' is the principal quantum number, and r1 is the least allowed radius for a hydrogen atom, known as Bohr's radius having a value of 0.53 Å.

### The Energy Of Electrons:

For hydrogen and hydrogen-like atoms, the Bohr model of hydrogen gives the energy (E) of an electron present in the nth energy level (orbit) of hydrogen as:

E = (-1/n2) ｘ13.6 eV, where 'n' is the principal quantum number, and 13.6 eV is the least possible energy of an electron of hydrogen.

### Limitations of Bohr's Theory

• It fails to explain the spectra of atoms with multiple electrons.

• It does not explain the fine structure of the spectral lines of hydrogen.

• It gives no information about the relative intensities of an atom's spectral lines.

• It fails to account for the splitting of spectral lines in a magnetic (Zeeman effect) or electric field (Stark effect).

• The theory does not explain the distribution of electrons and their wave nature.

1. What is the Hydrogen Spectrum?

The Hydrogen spectrum was studied by Balmer. It consists of discrete bright lines (Hα, Hβ, H...) on a dark background. From one end to another, the brightness and the separation between these lines gradually decrease; these lines are members of a series called the Balmer series, and they are found in the visible part of the spectrum. Other series were found in the invisible parts of the spectrum - Lyman series in the ultraviolet region, and the Paschen, Brackett, and Pfund series in the infrared area. The wavelength of the lines in these series can be represented by the Rydberg equation:

1/λ = R [(1/n12) - (1/n22)], where R is the Rydberg constant, n1, and n2 are the lower and the higher energy levels, respectively, of the hydrogen atom. The transitions of electrons between the energy levels give rise to the line spectrum of hydrogen. For Balmer series, n1 = 2.