Q: 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\$

Q: 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\$

Q: 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\$

Q: The electron in a hydrogen atom makes a transition \$n_1\rightarrow n_2\$, where \$n_1\$ and \$n_2\$ are the principal quantum numbers of the two energy states. Assume Bohr's model to be valid. The time period of the electron in the initial state is eight times that in the final state. What are the possible values of \$n_1\$ and \$n_2\$?

Here, \$Z= 1\$. We can derive the expression for the time period of electron in \$n^{th}\$ orbit as\$\quad T_n = \displaystyle\frac{1}{f_n} \propto n^3\$Hence \$\displaystyle\frac{T_1}{T_2} = \displaystyle\frac{n_1^3}{n_2^3}\$. As \$T_1 = 8T_2\$, therefore we get\$\quad 8 = \left(\displaystyle\frac{n_1}{n_2}\right)^3 \Rightarrow n_1 = 2n_2\$Thus, the possible values of \$n_1\$ and \$n_2\$ are \$n_1 = 2, n_2 = 1; n_1= 4, n_2=2;n_1=6, n_2 = 3\$ and so on.

Q: 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}\$

Question 1

Write Bohr's postulates for the hydrogen atom model.

Accepted Answer

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, $v=\dfrac{{E}_{i}-{E}_{f}}{h}$

Question 2

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

Accepted Answer

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$ $A^o$

Question 3

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

Accepted Answer

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 $=\dfrac {n(n-1)}{2}=\dfrac {(4)(4-1)}{2}=6$

Question 4

Write two limitations of Bohr's model.

Accepted Answer

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.

Question 5

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

Accepted Answer

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$

Question 6

What is meant by 'Quantization of charge'?

Accepted Answer

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.

Question 7

The electron in a hydrogen atom makes a transition $n_1\rightarrow n_2$, where $n_1$ and $n_2$ are the principal quantum numbers of the two energy states. Assume Bohr's model to be valid. The time period of the electron in the initial state is eight times that in the final state. What are the possible values of $n_1$ and $n_2$?

Accepted Answer

Here, $Z= 1$. We can derive the expression for the time period of electron in $n^{th}$ orbit as$\quad T_n = \displaystyle\frac{1}{f_n} \propto n^3$Hence $\displaystyle\frac{T_1}{T_2} = \displaystyle\frac{n_1^3}{n_2^3}$. As $T_1 = 8T_2$, therefore we get$\quad 8 = \left(\displaystyle\frac{n_1}{n_2}\right)^3 \Rightarrow n_1 = 2n_2$Thus, the possible values of $n_1$ and $n_2$ are $n_1 = 2, n_2 = 1; n_1= 4, n_2=2;n_1=6, n_2 = 3$ and so on.

Question 8

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)

Accepted Answer

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, $v=p/m=\dfrac{24hR}{25 m}$

Question 9

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.

Accepted Answer

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}$

Question 10

The ratio of kinetic energy to the total energy of an electron in a Bohr orbit of the hydrogen atom, is?

Accepted Answer

Since the relation between, K.E., P.E., and T.E. is given as $-T.E. = K.E. = -P.E./2$Ratio  $ K. E : T.E  ::  1 : -1$So, the correct option will be D.

Revise with Concepts

Bohr's Model

Energy Levels of Hydrogen Atom

Franck - Hertz Experiment

Learn with Videos

Quick Summary With Stories

Bohr's Model of Atom

Terms related to Bohr's Atomic Model - I

Terms related to Bohr's Atomic Model - II

Problems on Bohr Model.

Bohr's Theory and Quantum Number of Orbits

Introduction to Ionization Energy

Ionisation Energy

Hydrogen Like Atoms

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

Write two limitations of Bohr's model.

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?

What is meant by 'Quantization of charge'?

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)

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.

The ratio of kinetic energy to the total energy of an electron in a Bohr orbit of the hydrogen atom, is?

Revise with Concepts

Bohr's Model

Energy Levels of Hydrogen Atom

Franck - Hertz Experiment

Learn with Videos

Quick Summary With Stories

Bohr's Model of Atom

Terms related to Bohr's Atomic Model - I

Terms related to Bohr's Atomic Model - II

Problems on Bohr Model.

Bohr's Theory and Quantum Number of Orbits

Introduction to Ionization Energy

Ionisation Energy

Hydrogen Like Atoms

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

Write two limitations of Bohr's model.

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

What is meant by 'Quantization of charge'?

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)

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.

The ratio of kinetic energy to the total energy of an electron in a Bohr orbit of the hydrogen atom, is?

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

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?

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)

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.