# CCSU Bohr’s Theory Hydrogen Spectrum Physics Study Material

CCSU Bohr’s Theory Hydrogen Spectrum Physics Study Material

**CCSU Rutherford’s Model of Atom Study Material :**

In 1911, Rutherford, on the basis of α-particle scattering experiments, suggested a model of an atom. In this model, the entire positive charge and nearly whole of the mass of the atom are concentrated at the centre of the atom in a very small volume, known as the ‘nucleus’ of the atom (fig.).

The electrons revolve around the nucleus in planetary orbits at distances much larger than the size of the nucleus, the necessary centripetal force being provided by the electrostatic force of attraction between the electron and the nucleus. (Rutherford assumed orbital motion of electrons because without it the electrons would fall into the nucleus under the electrostatic attraction, and the atom would collapse.

The radius of the nucleus was estimated to be of the order of 10^{-15} m which is only 1/10,000 of the radius of the atom as a whole. Thus, Rutherford’s atom is largely hollow.

**Drawbacks of Rutherford’s model :** This model suffers from two drawbacks :

**Regarding Stability of Atom :**Electrons revolving around the nucleus have centripetal acceleration. According to electrodynamics, accelerated electrons would constantly radiate energy in the form of electromagnetic waves. Due to this continuous loss of energy of the electrons, the radii of their orbits would continuously decrease, and ultimately the electrons would fall into the nucleus. Thus, atom cannot remain stable.**Regarding Observation of Line Spectrum :**In Rutherford’s model, the electrons revolve in orbits of all possible radii and hence with all possible frequencies of revolution. As a result, electrons would radiate electromagnetic waves of all frequencies, that the spectrum of these waves will be ‘continuous’. But experimentally, atoms like hydrogen emit line spectra, each line corresponds to a particular frequency. So an atom should radiate waves of some definite frequencies only. Thus, Rutherford model was unable to explain the line spectrum.

**CCSU Bohr’s Model of Atom Study Material : **

**Quantisation of Angular Momentum :** In 1913, Bohr proposed a quantum model of atom in order to explain the stability of the atom and the emission of sharp spectral lines. For this, he proposed the following three postulates:

- An electron can revolve only in those orbits in which its angular momentum is an integral multiple of h/2π, where h is Planck’s universal constant. If the mass of the electron be m and it is revolving with velocity v in an orbit of radius r, then its angular momentum will be mvr. According to this Bohr’s postulate, we have

mvr = nh/2π

where n is an integer (n = 1,2,3,…………….) and is called the ‘principal quantum number’ of the orbit. This equation is called ‘Bohr’s quantization of angular momentum’.

Thus, according to Bohr’s atomic model, electrons can revolve only in certain discrete orbits of ‘definite’ radii, not in all. These are called ‘stable orbits’.

- While revolving in an stable orbit, the electron does not radiate energy inspite of its acceleration towards the centre of the orbit. Hence the atom remains stable and is said to exist in a stationary state.
- When the atom receives energy from outside, then one (or more) of its outer electrons shifts to some higher orbit. This state of the atom is called ‘excited state’. The electron in the higher orbit stays only for 10
^{-8 }second and returns back to a lower orbit. While returning back, the electron radiates energy in the form of electromagnetic waves (fig.)

If the energy of electron in the higher orbits be E_{2} and that in the lower orbit be E_{1} , then the frequency v of the radiated waves is given by

hv = E_{2} – E_{1}

_{ }**v = E _{2 }– E_{1 }/h**

**CCSU Hydrogen Spectrum Study Material :**

The spectrum of hydrogen atom consists of a number of lines. These lines have been grouped into a number of ‘series’. The lines in each series are such that **their separation and intensity decrease regularly towards shorter wavelengths,** converging to a limit called the ‘series limit’. The first spectral series was observed by Balmer in 1885, and is called the Balmer series of hydrogen (Fig.). The first line with the longest wavelength (6563 Å) is named H_{α, }the next H_{β }, and so on. The series limit lies at 3646 Å, beyond which is a faint continuous spectrum.

Balmer gave an empirical formula for the wavelengths of the series, which is

1/λ = R(1/2^{2} – 1/n^{2}), n = 3,4,5,…………..** (Balmer)**

Where R is a constant called ‘Rydberg’s constant’ and has the value

R = 1.097 x 10^{7 }meter^{-1 }.

The H_{α }line corresponds to n = 3, the H_{β }line to n = 4, and so on. The series limit corresponds to n = ∞.

The Balmer series contains only those spectral lines which fall in the visible part of the hydrogen spectrum. The lines falling in the ultraviolet and infrared parts form other series. The lines in the ultraviolet form the Lyman series whose wavelengths are given by

1/λ = R(1/1^{2} – 1/n^{2}), where n=2,3,4,…….**(Lyman)**

In the infra-red, three spectral series have been observed which are called Paschen, Brackett and Pfund series. The wavelengths of lines in these series can be expressed by the following formulae:

1/λ = R(1/3^{2} – 1/n^{2}), where n=4,5,6,…….**(Paschen)**

1/λ = R(1/4^{2} – 1/n^{2}), where n=5,6,7,…….**(Brackett)**

1/λ = R(1/5^{2} – 1/n^{2}), where n=6,7,8,…….**(Pfund)**

**(b)** **Bohr’s Theory of Hydrogen Spectrum :** Bohr explained the existence of sharp lines in the various series observed in hydrogen spectrum by applying Planck’s quantum hypothesis to the Rutherford’s atomic model. The Rutherford’s atom consists of a central massive nucleus in circular planetary orbits. The centripetal force required for the orbital motion is provided by electrostatic attraction between the positively-charged nucleus and the negatively-charged electron.

**Bohr proposed three postulates :**

- An electron can revolve only in those orbits in which its angular momentum is an integral multiple of h/2π, where h is Planck’s constant. These are called ‘stable orbits’.
- While revolving in an stable orbit, the electron does not radiate energy inspite of its acceleration towards the centre of the orbit. Hence the atom remains stable and is said to exist in a stationary state.
- When the atom receives energy from outside, then one (or more) of its outer electrons shifts to some higher orbit. This state of the atom is called ‘excited state’. The electron in the higher orbit stays only for 10
^{-8 }second and returns back to a lower orbit. While returning back, the electron radiates energy in the form of electromagnetic waves.

If the energy of electron in the higher orbit be E_{2 }and that in the lower orbit be E_{1 }, then the frequency v of the radiated waves is given by

hv = E_{2} – E_{1}

v = E_{2} – E_{1}/h

**CCSU Hydrogen Spectrum Study Material :**

The spectrum of hydrogen atom consists of a number of lines. These lines have been grouped into a number of ‘series’. The lines in each series are such that their separation and intensity decrease regularly towards shorter wavelengths, converging to a limit called the ‘series limit’. The first spectral series was observed be Balmer in 1885, and is called the Balmer series of hydrogen (fig.). The first line with the longest wavelength (6563 Å) is named H_{α }, the next H_{β, }and so on. The series limit lies at 3646 Å, beyond which is a faint continuous spectrum.

Balmer gave an empirical formula for the wavelengths of the series, which is

1/λ = R(1/2^{2 }– 1/n^{2}), n=3,4,5,…… ** (Balmer)**

Where R is a constant called ‘Rydherg’s constant’ and has the value

R=1.097 x 10^{7}meter^{-1 }

The H_{α }line corresponds to n = 3, the H_{β }line to n = 4, and so on. The series limit corresponds to n = ∞.

The Balmer series contains only those spectral lines which fall in the visible part of the hydrogen spectrum. The lines falling in the ultraviolet and infrared parts form other series. The lines in the ultraviolet form the Lyman series whose wavelengths are given by

1/λ = R(1/1^{2 }– 1/n^{2}), n=2,3,4,…… ** (Lyman)**

In the infra-red, three spectral series have been observed which are called Paschen, Brackett and Pfund series. The wavelengths of lines in these series can be expressed by the following formulae :

1/λ = R(1/3^{2 }– 1/n^{2}), n=4,5,6,…… **(Paschen)**

1/λ = R(1/4^{2 }– 1/n^{2}), n=5,6,7,…… ** (Brackett)**

1/λ = R(1/5^{2 }– 1/n^{2}), n=6,7,8,…… ** (Pfund)**

**CCSU Bohr’s Theory of Hydrogen Spectrum Study Material :**

Bohr explained the existence of sharp lines in the various series observed in hydrogen spectrum by applying Planck’s quantum hypothesis to the Rutherford’s atomic model. The Rutherford’s atom consist of a central massive nucleus containing the positive charge of the atom, and the electrons move round the nucleus in circular planetary orbits. The centripetal force required for the orbital motion is provided by electrostatic attraction between the positively-charged nucleus and the negatively-charged electron.

**Bohr proposed three postulates :**

An electron can revolve only in those orbits in which its angular momentum is an integral multiple of h/2π, where h is Planck’s constant. These are called ‘stable orbits’.

While revolving in an stable orbit, the electron does not radiate energy inspite of its acceleration towards the centre of the orbit. Hence the atom remains stable and is said to exist in a stationary state.

The emission (or absorption) of radiation by the atom takes place when an electron jumps from one stable orbit to another. The radiation is emitted (or absorbed) as a single quantum (photon) whose energy hv is equal to the difference in energies of the electron in the two orbits involved. Thus, if E_{i }that of the final orbit, then we have

hv = E_{i }– E_{f}

where v is the frequency of the emitted (or absorbed) radiation.

Let e, m and v be the charge, mass and velocity of the electron, and r the radius of the orbit. The positive charge on the nucleus is Ze, where Z is the atomic number (Fig.). In case of hydrogen atom, Z = 1. As the centripetal force is provided by the electrostatic force of attraction, we have

mv^{2}/r = 1/4πε_{0}.(Ze) x e/r^{2}

or mv^{2 }= Ze^{2}/4 πε_{0}r …………….(i)

From the first postulate, the angular momentum of the electron is

mvr = n h/2π, ……..(ii)

where n (=1,2,3,…………..) is ‘quantum number’. Squarring eq.(ii) and dividing by eq. (i), we get

**r = n ^{2} . h^{2} ε_{0}/πmZe^{2} **……………..(iii)

This is the equation for the radii of the permitted orbits.

**Energy of Electron in Stationary Orbits :** The energy E of an electron in an orbit is the sum of Kinetic and potential energies. The kinetic energy of the electron is

K = ½ mv^{2} =Ze^{2}/8πε_{0}r [From eq. (i)]

The potential energy of the electron at a distance r from the nucleus is equal to the work done in removing the electron from r to infinity against the electrostatic attraction of the nucleus (-1/4πε_{0}.(Ze) x e/r^{2}) and is given by

U = {^{∞}_{r}Ze^{2}/4 πε_{0}r^{2}dr = _{r}Ze^{2}/4 πε_{0}[1/r]∞_{r }= –_{ r}Ze^{2}/4 πε_{0}r

The total energy of the electron is therefore

E = K + U = Ze^{2}/8 πε_{0}r – Ze^{2}/4 πε_{0}r = -Ze^{2}/8 πε_{0}r

Substituting for r form eq. (iii), we get

**E = -mZ ^{2}e^{4}/8ε_{0}^{2}h^{2}.(1/n^{2}),** n = 1,2,3,………….(iv)

This is the expression for the energy of the electron in the n th orbit. We see that it is negative.

Suppose, in the ‘excited’ atom, an electron jumps from an initial (higher) orbit to a final (lower) orbit. The energy difference between these states (orbits) is

E_{i }– E_{f} = mZ^{2}e^{4}/8ε_{0}^{2}h^{2}.(1/n_{f}^{2 }– 1/n_{i}^{2})

Where n_{i} and n_{f} are the quantum numbers of the initial and the final states respectively.

According to Bohr’s third postulate, the frequency v of the emitted radiation (photon) is

v = E_{i }– E_{f}/h = mZ^{2}e^{4}/8ε_{0}^{2}h^{3}.(1/n_{f}^{2 }– 1/n_{i}^{2}).

The corresponding wavelength λ is given by

1/λ = v/c = mZ^{2}e^{4}/8ε_{0}^{2 }ch^{3}.(1/n_{f}^{2 }– 1/n_{i}^{2}).

Since n_{i} and n_{f} **can take only integral values, the radiation emitted by the excited atom would contain certain discrete wavelengths only.**

The value of the constant quantity me^{4}/8ε_{0}^{2}ch^{3} =R.

Thus ** 1/λ = Z ^{2}R(1/n_{f}^{2 }– 1/n_{i}^{2}).** …………..(v)

This is Bohr’s formula for hydrogen-like atoms (He^{+}, Li^{++},………).

For hydrogen, Z=1.

**1/λ = R(1/n _{f}^{2 }– 1/n_{i}^{2}).**

**Emission of Spectrum :** When the hydrogen atom gets sufficient energy from outside by some means, its electron moves from the lowest orbit to a higher orbit. This excited state of the atom lasts for about 10^{-8 }second after which the electron returns back to the lowest orbit either directly or through other lower orbits. While returning, the electron emits the difference in energy of the two orbits in the form of electromagnetic radiation.

If the electron jumps from an orbit n_{i} to an orbit n_{f}, the wavelength of the emitted radiation will be

1/λ = R(1/n_{f}^{2 }– 1/n_{i}^{2}).

It is found that for

n_{f} = 1, n_{i} = 2,3,4,……. We obtain **Lyman series**,

n_{f} = 2, n_{i} =3,4,5,……. We obtain **Balmer series**,

n_{f} = 3, n_{i} =4,5,6,……. We obtain **Paschen series,**

n_{f} = 4, n_{i} =5,6,7,……. We obtain** Brackett series,**

n_{f} = 5, n_{i} =6,7,8,……. We obtain Pfund series:

The corresponding energy level diagram is shown in Fig. The top horizontal line represents zero energy, that is, the energy of the electron outside the atom (n = ∞). The other horizontal lines represent energies of different orbits of hydrogen atom (Z=1) given by the formula

E = -me^{4}/8ε_{0}^{2}h^{2}.(1/n^{2}).

The arrows ending at the lines n= 1,2,3,4 and 5 represent the transitions responsible for the Lyman, Balmer, Paschen, Brackett and Pfund series respectively.

**Shortcomings of Bohr’s Theory Study Material :**

The Bohr theory of hydrogen and hydrogen-like atoms has a number of shortcomings :

- It explains the spectra of one-electron atoms only such as hydrogen, hydrogen isotopes, singly-ionised helium, doubly-ionised lithium, etc. It fails to explain the spectra of multi-electron atoms.
- When a spectral line is observed under a high resolving-power spectroscope, it is found to be made up of a number of closely-spaced lines. Bohr theory does not explain this ‘fine structure’ of spectral lines even in hydrogen atom.
- Bohr’s theory does not tell anything about the relative intensities of spectral lines of an atom.
- The theory cannot fully explain the splitting of a spectral line into a number of component-lines under the effect of a magnetic field (Zeeman effect) or an electric field (Stark effect).
- The theory does not explain the distribution of electrons in different orbits.
- It gives no information about the wave-nature of electron.

**No Balmer Lines in Absorption Spectrum of Hydrogen Study Material :**

The Bohr theory also explains the absorption line spectrum of hydrogen. When a beam of continuous light (containing all wavelengths) is passed through hydrogen and then sent into spectrograph, a set of dark lines is obtained. In terms of quantum theory the incident light is a beam of quanta (photons) of all sorts of energies. Now, according to Bohr theory, the hydrogen atoms absorb only those quanta whose energies correspond to transitions between its discrete energy levels. The resulting excited hydrogen atoms re-radiate the absorbed energy almost at once but these photons come off in random directions with only a few in the same direction as the original beam of continuous light. The dark lines in the absorption spectrum are therefore never completely black. Obviously, the absorption lines will have exactly the same frequencies as the emission lines.

Now, it is found that all the emission lines of hydrogen spectrum so not appear in the absorption spectrum. The reason is that normally the atom is always in the ground state, n = 1. Therefore, absorption transitions can only occur from n = 1 to n > 1. Hence lines of only the Lyman series can appear in absorption spectrum. To obtain Balmer series in absorption, the atom must initially be in the state n = 2, because Balmer lines require transitions from n = 2 to n > 2. Since atoms are usually in the ground state, Balmer lines are not obtained in absorption.

**CCSU Assumptions of Planck, Einstein and Bohr :**

Planck had assumed that the atoms of a hot body behave as oscillators and have discrete (quantized) energies. They do not emit radiant energy continuously, but only in ‘jumps’ or ‘quanta’. Planck, however, still maintained that radiation propagates continuously through space as electromagnetic waves.

Einstein, in order to explain the photoelectric effect, went a step ahead. He proposed that he radiation is not only emitted as quantum at a time, but also propagates as individual quanta (photons). He thus treated the propagation of radiation as particle propagation rather than wave propagation.

Bohr, in order to explain the hydrogen spectrum, adopted the Planck’s quantum hypothesis that the radiation is emitted discontinuously from the atom. He started with the quantization of the angular momentum of the electron in the orbit which ultimately results in the quantization of energy of the atom.

**CCSU Negative Energy of Hydrogen Orbits :**

An electron revolving in an hydrogen orbit has a negative potential energy by virtue of its attraction towards the nucleus, and also kinetic energy (which is positive) by virtue of its motion. The potential energy is greater in magnitude than the kinetic energy, so that the escape from the atom. A positive energy for a nucleus –electron combination would mean that electron is not bound to the nucleus. Such a combination cannot constitute an atom.

**Absorption of a Photon of energy greater that the Binding Energy Study Material :**

The binding energy of the hydrogen atom is the energy which must be supplied to it in order to remove its electron from the lowest state (level) to a state of zero energy. It is numerically equal to the energy of the lowest state (-13.6 eV).

The highest energy level of the atom corresponds to n = ∞ so that E = 0. Above this are the energy states of the system consisting of an unbound electron plus the ionized atom. The total energy of an unbound electron is not quantized, and is positive. Thus, any energy E>0 is possible for the electron, and the energy states form a continuum. When the hydrogen atom receives a photon of any energy which is greater than its binding energy of 13.6 eV, it absorbs the photon and the electron of the atom passes from its discrete ground state to a continuum state.

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