# Bohr's Model of An Atom

__DEVELOPMENTS LEADING TO THE BOHR’S MODEL OF ATOM__

__Wave Nature of Electromagnetic Radiation__

Electromagnetic Radiations (EMR) are energy radiations which do not need any medium for propagation, e.g. visible, ultraviolet, X-rays, etc. Following are the important characteristics of EMR:

(a) All electromagnetic radiations or waves travel with the velocity of light.

(b) These consist of electric and magnetic fields that oscillate in directions perpendicular to each other and perpendicular to the direction in which the wave is travelling.

(2) Characteristics

(i) All electromagnetic radiations travel with the velocity of light.

(ii) These consist of electric and magnetic fields components that oscillate in directions

perpendicular to each other and perpendicular to the direction in which the wave is travelling.

(3) A wave is always characterized by the following five characteristics,

Wavelength: The distance between two nearest crests or nearest troughs is called the wavelength. It is denoted by 𝝀 (lambda).

Frequency: It is defined as the number of waves which pass through a point in one second. It is denoted by the symbol ν (nu) and is expressed in terms of cycles (or waves) per second (cps) or hertz (Hz).

Velocity: It is defined as the distance covered in one second by the wave. It is denoted by the letter ‘c’. All electromagnetic waves travel with the same velocity,i.e

Wave number: This is the reciprocal of wavelength, i.e., the number of wavelengths per centimetre. It is expressed in

Amplitude: It is defined as the height of the crest or depth of the trough of a wave.

*Electromagnetic Spectrum*

The arrangement of various types of electromagnetic radiations in the order of their increasing or decreasing wavelengths or frequencies is known as electromagnetic spectrum.

*Electromagnetic Radiation Formula*

Frequency is defined as the number of waves that pass through a given point in one second. Mathematically it is equal to the reciprocal of the time period of electromagnetic radiation. A general equation relating the speed of light, frequency, and wavelength of electromagnetic radiation is given below:

c = ν 𝝀

Where,

c= speed of light,

ν= frequency of the electromagnetic wave and

𝝀 = wavelength of the electromagnetic wave.

__Particle Nature of Electromagnetic Radiation: Planck’s Quantum Theory__

Two points can explain this theory of electromagnetic radiation

The frequency of electromagnetic radiation is equal to the energy of absorbed or emitted radiation.

The atoms and molecules can emit and absorb a specific quantity of energy. The smallest amount of energy that can be absorbed or emitted is termed quantum.

According to Planck’s Quantum theory,

E= hv

Here, h is the Planck’s constant and its value is 6.626×10^{-34} J.s.

All electromagnetic radiation behaves according to wave theory or quantum theory. Electromagnetic waves consist of an electric field that varies in a direction perpendicular to the direction of propagation of the radiation and a magnetic field that is set at right angles to the electric field.

Both electric and magnetic fields travel at the speed of light. To understand remote sensing of electromagnetic radiation, you need to know the two main characteristics of electromagnetic radiation and their wavelength and frequency.

__Photoelectric Effect__

It was discovered by Hertz.

When a beam of light of certain frequency (threshold frequency) strikes the metal surface, electrons are emitted or ejected from the metal surface. this phenomenon is known as photoelectric effect.

Light of a particular frequency strikes a clean metal surface inside a vacuum chamber. Electrons are ejected from the metal and are counted by a detector that measures their kinetic energy.

*Observations in Photoelectric Effect*

(1)For each metal there is a characteristic minimum frequency below which photoelectric effect is not observed. This is called threshold frequency.

If frequency of light is less than the threshold frequency there is no ejection of electrons no matter how long it falls on surface or how high is its intensity

(2) The kinetic Energy of electrons emitted is directly proportional to frequency of striking photons & independent of their intensity

(3) The no. of electrons that are ejected per second from metal surface depends upon intensity of striking radiations and doesn’t depend upon their frequency.

*Explanation of Photoelectric Effect*

Einstein could explain photoelectric effect using Plank’s Quantum theory as follows: -

(a) Photoelectrons are ejected only when incident light has threshold frequency.

(b) If frequency of incident light is more than threshold frequency then the excess energy is imparted to electrons in the form of kinetic energy.

(c) Greater he frequency of incident light, greater the kinetic energy of e-.

(d) Greater the intensity of light more the no. of electrons ejected.

*Dual Behaviour of Electromagnetic Radiation*

The photoelectric effect could be explained considering that radiations consist of small packets of energy called quanta. These packets of energy can be treated as particles. On the other hand, radiations exhibit a phenomenon of interference and diffraction which indicated that they possess a wave nature. So it may be concluded that electromagnetic radiations possess dual nature.

- Particle nature
- Wave nature

*Wave Nature of Electromagnetic Radiation*

Due to a dispersion of white light, VIBGYOR appears. Vibgyor stands for – Violet, Indigo, Blue, Green, Yellow, Orange, and Red.

__Spectrum :-__

When a white light is passed through a prism, it splits into a series ofcoloured bands known as spectrum.

*Spectrum is of two types:*

**Continuous and line spectrum**

(a) The spectrum which consists of all the wavelengths is called **continuousspectrum.**

(b) **line spectrum **:A spectrum in which only specific wavelengths are present is known as a line spectrum. It has bright lines with dark spaces between them.

Electromagnetic spectrum is a continuous spectrum. It consists of a range of electromagnetic radiations arranged in the order of increasing wavelengths or decreasing frequencies. It extends from radio waves to gamma rays.

**Emission & Absorption Spectrum**

Spectra can be divided into two types based on absorption by gas or vapour & white light emission:

**Emission Spectrum:**

An spectrum due to the emission of white light by gas at high temperature is known as an emission spectrum. This kind of spectrum usually consists of bright lines on a dark background. The emission of energy by electrons generates an emission spectrum.

**Absorption Spectrum:**

The spectrum which occurs due to absorption of white light by gas and transmitted white light, is termed an absorption spectrum. Unlike the emission spectrum, it consists of dark lines on a bright background. it is due to the absorption of energy by electrons.

*Line Spectrum of Hydrogen*

When an electric discharge is passed through gaseous hydrogen, the H2 molecules dissociate and the energetically excited hydrogen atoms produced emit electromagnetic radiation of discrete frequencies. The hydrogen spectrum consists of several series of lines named after their discoverers. Balmer showed in 1885 on the basis of experimental observations that if spectral lines are expressed in terms of wavenumber (ν ), then the visible lines of the hydrogen spectrum obey the following formula :

where n is an integer equal to or greater than 3 (i.e., n = 3,4,5,....)

The series of lines described by this formula are called the Balmer series. The Balmer series of lines are the only lines in the hydrogen spectrum which appear in the visible region of the electromagnetic spectrum. The Swedish spectroscopist, Johannes Rydberg, noted that all series of lines in the hydrogen spectrum could be described by the following expression :

The value 109,677 cm–1 is called the Rydberg constant for hydrogen. The first five series of lines that correspond to n1 = 1, 2, 3, 4, 5 are known as Lyman, Balmer, Paschen, Bracket and Pfund series, respectively.

**The Spectral Lines for Atomic Hydrogen**

__Bohr Theory __

Neils Bohr (1913) was the first to explain quantitatively the general features of the structure of hydrogen atom and its spectrum.

** **

**Bohr’s model for hydrogen atom is based on the following postulates:**

i) The electron in the hydrogen atom can move around the nucleus in a circular path of fixed radius and energy. These paths are called orbits, stationary states or allowed energy states. These orbits are arranged concentrically around the nucleus.

ii) The energy of an electron in the orbit does not change with time. However, the electron will move from a lower stationary state to a higher stationary state when required amount of energy is absorbed by the electron or energy is emitted when electron moves from higher stationary state to lower stationary state. The energy change does not take place in a continuous manner.

** **

iii) The frequency of radiation absorbed or emitted when transition occurs between two stationary states that differ in energy by ∆E, is given by :

Where E_{1} and E_{2} are the energies of the lower and higher allowed energy states r espectively. This expression is commonly known as Bohr’s frequency rule.

iv) The angular momentum of an electron in a given stationary state can be expressed as

Thus an electron can move only in those orbits for which its angular momentum is integral multiple of h/2π that is why only certain fixed orbits are allowed.

*Bohr’s theory for hydrogen atom:*

- The stationary states for electron are numbered n = 1,2,3.......... These integral numbers are known as Principal quantum numbers.
- The radii of the stationary states are expressed as : $r_n=n^2a_0$
`r`_{n}=`n`^{2}`a`_{0}

where a_{0} = 52,9 pm. Thus the radius of the first stationary state, called the Bohr orbit, is 52.9 pm. Normally the electron in the hydrogen atom is found in this orbit (that is n=1). As n increases the value of r will increase. In other words the electron will be present away from the nucleus.

- The most important property associated with the electron, is the energy of its stationary state. It is given by the expression.

where R_{H} is called Rydberg constant and its value is 2.18×10^{–18} J.

- This figure depicts the energies of different stationary states or energy levels of hydrogen atom. This representation is called an energy level diagram.

When the electron is free from the influence of nucleus, the energy is taken as zero. The electron in this situation is associated with the stationary state of Principal Quantum number = n = ∞ and is called as ionized hydrogen atom. When the electron is attracted by the nucleus and is present in orbit n, the energy is emitted and its energy is lowered. That is the reason for the presence of negative sign in equation (2.13) and depicts its stability relative to the reference state of zero energy and n = ∞.

**Bohr’s theory can also be applied to the ions containing only one electron, similar to that present in hydrogen atom. For example, He ^{+} Li^{2+}, Be^{3+} and so on**

__Energy of the stationary states of hydrogen like species__

__Note – __

Total energy (E_{n}) = - Kinetic energy

Total energy (E_{n}) = Potential energy /2

__The radii of the stationary states are expressed as:__

where a_{0} = 52.9 pm. Thus the radius of the first stationary state, called the Bohr orbit, is 52.9 pm. where Z is the atomic number and has values 2,3 for the helium and lithium atoms respectively.

As n increases the value of r will increase. In other words the electron will be present away from the nucleus.

It is also possible to calculate the velocities of electrons moving in these orbits. The magnitude of velocity of electron increases with increase of positive charge on the nucleus and decreases with increase of principal quantum number

- The
**frequency**of radiation absorbed or emitted when transition occurs between two stationary states that differ in energy by ∆E, is given by:

Where E_{1} and E_{2} are the energies of the lower and higher allowed energy states respectively. This expression is commonly known **as Bohr’s frequency rule.**

and in terms of **wavenumber**

- The
**angular momentum**of an electron is quantized

Where m_{e} is the mass of electron, v is the velocity and r is the radius of the orbit in which electron is moving. Thus an electron can move only in those orbits for which its angular momentum is integral multiple of h/2π.

__Usefulness of Bohr theory – __

- It explains the stability of an atom .
- It explains the line spectra of hydrogen atom .

*Explanation of Line Spectrum of Hydrogen*

Line spectrum observed in case of hydrogen atom, as mentioned in section 2.3.3, can be explained quantitatively using Bohr’s model. According to assumption 2, radiation (energy) is absorbed if the electron moves from the orbit of smaller Principal quantum number to the orbit of higher Principal quantum number, whereas the radiation (energy) is emitted if the electron moves from higher orbit to lower orbit.

The energy gap between the two orbits is given by equation

The frequency (ν ) associated with the absorption and emission of the photon can be evaluated by using equation

and in terms of wavenumbers

In case of absorption spectrum, n_{f} > n_{i} and the term in the parenthesis is positive and energy is absorbed. On the other hand in case of emission spectrum n_{i} > n_{f} , ∆ E is negative and energy is released.

__Limitations of Bohr theory – __

- It could not explain the ability of atoms to form molecules by chemical bonds.
- It fails to account for the finer details (doublet, that is two closely spaced lines) of the hydrogen atom spectrum observed by using sophisticated spectroscopic techniques.
- This model is also unable to explain the spectrum of atoms other than hydrogen, for example, helium atom which possesses only two electrons.

Further, Bohr’s theory was also unable to explain the splitting of spectral lines in the presence of magnetic field (**Zeeman effect**) or an electric field (**Stark effect**).