Write Bohr's postulates for the hydrogen atom model.

First Postulate: Electron revolves round the nucleus in discrete circular orbits called stationary orbits without emission of radiant energy. These orbits are called stable orbits or non-radiating orbits.Second Postulate: Electrons revolve around the nucleus only in orbits in which their angular momentum is an integral multiple of \\(h/2\pi\\).Third Postulate: When an electron makes a transition from one of its non-radiating orbits to another of lower energy, a photon is emitted having energy equal to the energy difference between the two states. The frequency of the emitted photon is then given by, \\(v=\dfrac{{E}_{i}-{E}_{f}}{h}\\)

By assuming Bohr's postulates derive an expression for radius of \\(n^{th}\\) orbit of electron, revolving round the nucleus of hydrogen atom.

Let the electron of mass \\(m\\) revolves around a nucleus (H-atom) in an orbit of radius \\(r_n\\) with linear velocity \\(v_n\\).As the electron revolves in an stationary orbit, thus centrifugal force acting on the electron is balanced by the coulombic force i.e. \\(F_{centrifugal} = F_{coulomb}\\)\\(\therefore\\) \\(\dfrac{mv_n^2}{r_n} = \dfrac{ke^2}{r_n^2}\\) where \\(k = \dfrac{1}{4\pi \epsilon_o} = 9\times 10^9\\)\\(\implies\\) \\(r_n = \dfrac{ke^2}{mv_n^2}\\)Also we use \\(mv_nr_n = \dfrac{nh}{2\pi}\\)Eliminating \\(v_n\\) from both equations, we get \\(r_n = \dfrac{ke^2}{m}.\dfrac{4\pi^2m^2 r_n^2}{n^2 h^2}\\)\\(\implies\\) \\(r_n = \dfrac{h^2 n^2}{4\pi^2 m ke^2} \\)Putting \\(m = 9.1\times 10^{-31} kg\\), \\(h = 6.626 \times 10^{-34} Js\\) and \\(e = 1.6\times 10^{-19} C\\)We get radius of \\(n^{th}\\) Bohr orbit \\(r_n = 0.529\\) \\( n^2\\) \\(A^o\\)

Hydrogen atom in ground state is excited by a monochromatic radiation of \\(\lambda= 975 \overset {o}{A}\\). Number of spectral lines in the resulting spectrum emitted will be

Energy of the photon, \\(E=\dfrac {hc}{\lambda}=\dfrac {1240}{97.5}=12.75eV\\)This energy is equal to energy gap between \\(n = 1 (-13.6)\\) and \\(n = 4(-0.85)\\). So by this energy, the electron will excite from \\(n = 1\\) to \\(n= 4\\). When the electron will fall back, numbers of spectral lines emmitted \\(=\dfrac {n(n-1)}{2}=\dfrac {(4)(4-1)}{2}=6\\)

What is meant by 'Quantization of charge'?

Quantization of charge implies that charge can assume only certain discrete values. That is to say the observed value of electric charge (q) of a particle will be integral multiples of (e) \\(1.6\times10^{-19}\\) coulombs. i.e.\\(q=ne\\) where \\(n=0,1,2,....\\) (both positive and negative integers)The charge cannot assume any value between the integers.

Write two limitations of Bohr's model.

The Limitations of Bohr's model are,1) The Bohr atomic model theory made correct predictions for smaller sized atoms like hydrogen, but poor spectral predictions are obtained when larger atoms are considered.2) It failed to explain the Zeeman effect when the spectral line is split into several components in the presence of a magnetic field.

An electron of stationary hydrogen atom passes from the fifth energy level to the ground level. The velocity that the atom of mass \\(m\\) acquired a result of photon emission will be : (\\(R\\) is Rydberg constant and \\(h\\) is Planck's constant)

We know that , \\(\dfrac{1}{\lambda}=R(1/n_1^2-1/n_2^2)\\)Here, \\(n_1=1, n_2=5\\) so \\(\dfrac{1}{\lambda}=R(1/1^2-1/5^2)=(24/25)R\\)Now, photon energy \\(E=\dfrac{hc}{\lambda}=(24/25)hcR\\)As atom of mass \\(m\\) acquired a velocity as a result of photo emission so using momentum conservation , the momentum photon = momentum of atom= \\(p\\)Thus, \\(p=E/c=(24/25)hR\\)Velocity of atom, \\(v=p/m=\dfrac{24hR}{25 m}\\)

The ratio of the energies of the hydrogen atom in its first to second excited state is

1st excited state corresponds to \\(n=2\\)2nd excited state corresponds to \\(n=3\\)\\(\therefore \displaystyle\frac{E_1}{E_2}=\frac{n^2_2}{n^2_1}=\frac{3^2}{2^2}=\frac{9}{4}\\)

In a Bohr's model of an atom, when an electron jumps from \\(n=1\\) to \\(n=3\\), how much energy will be emitted or absorbed?

\\(\Delta E=(-R_H/n_f^2)-(-R_H/n_i^2)\\)\\(n_i\\) & \\(n_f\\) stands for initial and final orbit respectively\\(R_H\\)=rydrberg's constant \\(=2.18\times 10^{-18}\\)\\(\Delta E=(-2.18\times 10^{-18}/3^2)-(-2.18\times 10^{-18}/1^2)\\)\\(\Delta E=2.18\times 10^{-18}(1/3^2-1)\\)\\(\Delta E=1.94\times 10^{-10} J\\)

The ground state energy of hydrogen atom is \\(-13.6\ eV\\). What are the kinetic and potential energies of the electron in this state?

Ground state energy of hydrogen atom is \\(E=-13.67eV\\)The kinetic energy of the electron is given as:\\(KE=-E\\)\\(=+13.67eV\\)The potential energy of the electron is given as:\\( PE= - 2\times KE\\)\\(= - 27.2eV\\)

Using Bohr's postulates of the atomic model, derive the expression for radius of n\\(^{th}\\) electron orbit. Hence obtain the expression for Bohr's radius.

Centripetal force is provided by the electrostatic force of attraction.\\(\dfrac{mv^2}{r} = \dfrac{e^2}{4 \pi \varepsilon_o r^2}\\)From bohr's second postulate.\\(L_n = m v_n r_n = \dfrac{nh}{2 \pi} \\)Eliminating \\(v_n\\) from above equations.\\(r_n = \dfrac{n^2 h^2\varepsilon_o}{\pi m e^2}\\)Bohr's radius is the radius of the innermost orbit. Hence, \\(n =1\\)\\(r_1 = \dfrac{h^2 \varepsilon_o}{\pi m e^2}\\)

Franck - Hertz Experiment | Definition, Examples, Diagrams

example

Wavelength of different spectral lines

The minimum wavelength of paschen series of hydrogen atom will be
For Paschen Series,

n = 3,

and mimimum wavelength occurs when transition occurs just from series limit,

\frac{h C}{λ} = - 13.6 [\frac{1}{n _{1}^{2}} - \frac{1}{n _{2}^{2}}]

\Rightarrow λ = \frac{- 4 . 1 4 \times 1 0 ^{- 15} \times 3 \times 1 0 ^{8}}{1 3 . 6 \times [ \frac{1}{9} - 1 ]}

\Rightarrow λ = \frac{4 . 1 4 \times 1 0 ^{- 15} \times 3 \times 1 0 ^{8} \times 9}{1 3 . 6 \times 8}

\Rightarrow λ = 1.878655 \times 1 0^{- 6} \times 9 \times 0.04861111

\Rightarrow λ = 8.21911 \times 1 0^{- 07}

\Rightarrow λ = 8219.11 \times 1 0^{- 10}

\Rightarrow λ = 8219 A^{\circ} \approx 8181 A^{\circ}

result

Franck and Herts Experiment

Franck and Hertz's original experiment used a heated vacuum tube containing a drop of mercury; they reported a tube temperature of 115

^{0}

C, at which the vapor pressure of mercury is about 100 pascals (and far below atmospheric pressure). It is fitted with three electrodes: an electron-emitting, hot cathode; a metal mesh grid; and an anode. The grid's voltage is positive relative to the cathode, so that electrons emitted from the hot cathode are drawn to it. The electric current measured in the experiment is due to electrons that pass through the grid and reach the anode. The anode's electric potential is slightly negative relative to the grid, so that electrons that reach the anode have at least a corresponding amount of kinetic energy after passing the grid.
The graphs published by Franck and Hertz show the dependence of the electric current flowing out of the anode upon the electric potential between the grid and the cathode.

definition

Conclusions from the plot of Franck and Hertz experiment

Some conclusions that can be drawn from the graph plotted in the Franck and Hertz experiment are as follows:
1. Rising curve in the current v/s accelerating plot corresponds to region where the electron gain kinetic energy due to excitation potential but not enough to ionize the medium (mercury).
2. Decaying curve corresponds to the region where the medium is ionized and hence energy is lost in ionisation.
3. Maxima corresponds to the point when the energy of the electrons is just enough to ionize the medium.
4. Minima corresponds to the point where electrons start to gain energy from the applied accelerating potential.
5. Ideally, the distance between two maxima is constant an equals the excitation potential of the medium. However, mercury has more than one excitation and ionization potential which makes the second and third peaks of the curve complicated.
6. The above observations suggest that the electrons give energy to the atoms in only discrete levels.

definition

Wavelength of a spectral line emitted by mercury and sodium lamps

The emission spectrum of atomic hydrogen is divided into a number of spectral series, with wavelengths given by the Rydberg formula. These observed spectral lines are due to the electron making transitions between two energy levels in the atom.
Rydberg formula:

\frac{1}{λ} = R Z^{2} (\frac{1}{n ^{'} ^{2}} - \frac{1}{n ^{2}})

where Z=1 for hydrogen atom.
Now for sodium and mercury put

Z = 11

and

Z = 80

respectively and calculate the wavelength of the light used.

Franck - Hertz Experiment

example

Wavelength of different spectral lines

result

Franck and Herts Experiment

definition

Conclusions from the plot of Franck and Hertz experiment

definition

Wavelength of a spectral line emitted by mercury and sodium lamps

REVISE WITH CONCEPTS

Bohr's Model

Energy Levels of Hydrogen Atom

Quick Summary With Stories

Franck and Hertz Experiment

Write Bohr's postulates for the hydrogen atom model.

By assuming Bohr's postulates derive an expression for radius of $n^{t h}$ orbit of electron, revolving round the nucleus of hydrogen atom.

Hydrogen atom in ground state is excited by a monochromatic radiation of $λ = 975 A o$ . Number of spectral lines in the resulting spectrum emitted will be

What is meant by 'Quantization of charge'?

Write two limitations of Bohr's model.

An electron of stationary hydrogen atom passes from the fifth energy level to the ground level. The velocity that the atom of mass $m$ acquired a result of photon emission will be :
( $R$ is Rydberg constant and $h$ is Planck's constant)

The ratio of the energies of the hydrogen atom in its first to second excited state is

In a Bohr's model of an atom, when an electron jumps from $n = 1$ to $n = 3$ , how much energy will be emitted or absorbed?

The ground state energy of hydrogen atom is $- 13.6 e V$ . What are the kinetic and potential energies of the electron in this state?

Using Bohr's postulates of the atomic model, derive the expression for radius of n $^{t h}$ electron orbit. Hence obtain the expression for Bohr's radius.

example

Wavelength of different spectral lines

result

Franck and Herts Experiment

definition

Conclusions from the plot of Franck and Hertz experiment

definition

Wavelength of a spectral line emitted by mercury and sodium lamps

REVISE WITH CONCEPTS

Bohr's Model

Energy Levels of Hydrogen Atom

Quick Summary With Stories

Franck and Hertz Experiment

Write Bohr's postulates for the hydrogen atom model.

By assuming Bohr's postulates derive an expression for radius of nth orbit of electron, revolving round the nucleus of hydrogen atom.

Hydrogen atom in ground state is excited by a monochromatic radiation of λ=975Ao. Number of spectral lines in the resulting spectrum emitted will be

What is meant by 'Quantization of charge'?

Write two limitations of Bohr's model.

An electron of stationary hydrogen atom passes from the fifth energy level to the ground level. The velocity that the atom of mass m acquired a result of photon emission will be : (R is Rydberg constant and h is Planck's constant)

The ratio of the energies of the hydrogen atom in its first to second excited state is

In a Bohr's model of an atom, when an electron jumps from n=1 to n=3, how much energy will be emitted or absorbed?

The ground state energy of hydrogen atom is −13.6 eV. What are the kinetic and potential energies of the electron in this state?

Using Bohr's postulates of the atomic model, derive the expression for radius of nth electron orbit. Hence obtain the expression for Bohr's radius.

By assuming Bohr's postulates derive an expression for radius of $n^{t h}$ orbit of electron, revolving round the nucleus of hydrogen atom.

Hydrogen atom in ground state is excited by a monochromatic radiation of $λ = 975 A o$ . Number of spectral lines in the resulting spectrum emitted will be

An electron of stationary hydrogen atom passes from the fifth energy level to the ground level. The velocity that the atom of mass $m$ acquired a result of photon emission will be :
( $R$ is Rydberg constant and $h$ is Planck's constant)

In a Bohr's model of an atom, when an electron jumps from $n = 1$ to $n = 3$ , how much energy will be emitted or absorbed?

The ground state energy of hydrogen atom is $- 13.6 e V$ . What are the kinetic and potential energies of the electron in this state?

Using Bohr's postulates of the atomic model, derive the expression for radius of n $^{t h}$ electron orbit. Hence obtain the expression for Bohr's radius.