Definitions [3]
Definition: Impact parameter
The impact parameter is the perpendicular distance of the initial velocity vector of the a-particle from the centre of the nucleus.
Definition: Emission Spectrum
When an excited gas emits radiation of specific discrete wavelengths, it produces bright lines on a dark background called an emission line spectrum.
Definition: Absorption Spectrum
When white light passes through a gas, some wavelengths are absorbed and appear as dark lines in the continuous spectrum, called the absorption spectrum
Formulae [9]
Formula: Coulomb Force between α-particle and Nucleus
\[F=\frac{1}{4\pi\varepsilon_0}\frac{(2e)(Ze)}{r^2}\]
Where:
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Z = atomic number
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r = distance between α-particle and nucleus
Formula: Distance of Closest Approach
\[d=\frac{1}{4\pi\varepsilon_0}\frac{2Ze^2}{K}\]
Formula: Electron in Circular Orbit
Centripetal Force = Electrostatic Force
\[\frac{1}{4\pi\varepsilon_0}\frac{e^2}{r^2}=\frac{mv^2}{r}\]
Formula: Relation between Radius and velocity
\[r=\frac{e^2}{4\pi\varepsilon_0mv^2}\]
Formula: Energies
Kinetic Energy:
\[K=\frac{1}{2}mv^2\]
Potential Energy:
\[U=-\frac{e^2}{4\pi\varepsilon_0r}\]
Total Energy:
\[E=-\frac{e^2}{8\pi\varepsilon_0r}\]
Formula: Radius of nth Orbit
\[r_n=\frac{\varepsilon_0n^2h^2}{\pi me^2}\]
Formula: Energy of nth Orbit
\[E_n=-\frac{13.6}{n^2}\mathrm{~eV}\]
Formula: Frequency of Emitted Radiation
\[h\nu=E_{n_i}-E_{n_f}\]
Since ni and nf are integers → Discrete line spectrum
Formula: De Broglie Theory and Bohr’s Quantisation
| Formula | Meaning |
|---|---|
| \[\lambda=\frac{h}{mv}\] | de Broglie wavelength |
| \[2\pi r_n=n\lambda\] | Standing wave condition |
| \[mvr_n=\frac{nh}{2\pi}\] | Bohr quantisation |
Key Points
Key Points: Bohr’s Model – Three Postulates
Postulate 1:
An atom could revolve in certain stable orbits without the emission of radiant energy.
Postulate 2:
The electron revolves around the nucleus only in those orbits for which the angular momentum is some integral multiple of h/2π, where h is Planck’s constant (= 6.6 × 10–34 J s). Thus, the angular momentum (L) of the orbiting electron is quantised. That is
\[L=\frac{nh}{2\pi}\]
Postulate 3:
An electron might make a transition from one of its specified non-radiating orbits to another of lower energy.
\[h\nu=E_i-E_f\]
Key Points: Limitations of Bohr Model
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Applicable only to hydrogenic atoms
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Cannot explain multi-electron atoms
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Cannot explain the relative intensity of spectral lines
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Does not include electron–electron interaction
Key Points: Limitations of Rutherford Model
1. An atom should be unstable
- Electron is accelerating
- Accelerating charge radiates energy
- Electron should spiral into the nucleus
2. Cannot explain the line spectrum of hydrogen
Key Points: Important Constants
| Quantity | Value |
|---|---|
| Planck’s constant (h) | \[6.6\times10^{-34}\mathrm{Js}\] |
| Electron charge (e) | \[1.6\times10^{-19}\mathrm{C}\] |
| 1 eV | \[1.6\times10^{-19}\mathrm{」}\] |
| Bohr radius | \[5.3\times10^{-11}\mathrm{m}\] |
| Ground state energy | –13.6 eV |
Key Points: Rutherford’s Nuclear Model
Based on the experiment, Rutherford proposed that:
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An atom has a small, dense, positively charged nucleus at its centre.
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Almost all the mass of the atom is concentrated in the nucleus.
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Electrons revolve around the nucleus.
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Most of the atom is empty space.
Important Questions [80]
- How is the size of a nucleus found experimentally? Write the relation between the radius and mass number of a nucleus.
- Define the distance of closest approach. An α-particle of kinetic energy 'K' is bombarded on a thin gold foil. The distance of the closest approach is 'r'. What will be the distance of closest approach for an α-particle of double the kinetic energy?
- In a Geiger-marsden Experiment, Calculate the Distance of Closest Approach to the Nucleus of Z = 75, When a α-particle of 5 Mev Energy Impinges on It before It Comes Momentarily to Rest and Reverses
- Using Rutherford'S Model of the Atom, Derive the Expression for the Total Energy of the Electron in Hydrogen Atom. What is the Significance of Total Negative Energy Possessed by the Electron?
- An Electron in an Atom Revolves Round the Nucleus in an Orbit of Radius R with Frequency V. Write the Expression for the Magnetic Moment of the Electron.
- Write two important limitations of Rutherford's nuclear model of the atom.
- Answer the Following Question. Explain Briefly How Rutherford Scattering of α-particle by a Target Nucleus Can Provide Information on the Size of the Nucleus.
- The electron in a hydrogen atom is typically found at a distance of about 5.3 × 10−11 m from the nucleus which has a diameter of about 1.0 × 10−15 m.
- The energy of hydrogen atom in an orbit is −1.51 eV. What are kinetic and potential energies of the electron in this orbit?
- Suppose you are given a chance to repeat the alpha-particle scattering experiment using a thin sheet of solid hydrogen in place of the gold foil. (Hydrogen is a solid at temperatures below 14 K.)
- Differentiate between the 'distance of the closest approach' and the 'impact parameter.'
- A Charged Particle Q is Moving in the Presence of a Magnetic Field B Which is Inclined to an Angle 30° with the Direction of the Motion of the Particle.
- Determine the distance of the closest approach when an alpha particle of kinetic energy 3.95 MeV approaches a nucleus of Z = 79, stops and reverses its directions.
- Draw a graph showing the variation of the number of particles scattered (N) with the scattering angle θ in the Geiger-Marsden experiment.
- In a Geiger-marsden Experiment, Calculate the Distance of Closest Approach to the Nucleus of Z = 80, When a α-particle of 8mev Energy Impinges on It before It Comes Momentarily to Rest and
- A narrow beam of protons, each having 4.1 MeV energy is approaching a sheet of lead (Z = 82). Calculate: the speed of a proton in the beam, and the distance of its closest approach
- In Both β− and β+ Decay Processes, the Mass Number of a Nucleus Remains the Same, Whereas the Atomic Number Z Increases by One in β− Decay and Decreases by One in β+ Decay. Explain Giving Reason.
- An electron jumps from fourth to first orbit in an atom. How many maximum number of spectral lines can be emitted by the atom? To which series these lines correspond?
- State Bohr’S Postulate of Hydrogen Atom Which Successfully Explains the Emission Lines in the Spectrum of Hydrogen Atom
- Using Bohr'S Postulates, Derive the Expression for the Total Energy of the Electron in the Stationary States of the Hydrogen Atom ?
- How will the energy of a hydrogen atom change if n increases from 1 to ∞?
- Using Bohr’S Postulates, Obtain the Expressions for (I) Kinetic Energy and (Ii) Potential Energy of the Electron in Stationary State of Hydrogen Atom.
- Using Bohr’S Postulates, Obtain the Expression for the Total Energy of the Electron in the Stationary States of the Hydrogen Atom.
- Using Bohr’S Postulates, Obtain the Expression for Total Energy of the Electron in the Nth Orbit of Hydrogen Atom.
- Use Bohr's postulate to prove that the radius of nth orbit in a hydrogen atom is proportional to n2.
- Using Bohr’S Postulates, Derive the Expression for the Frequency of Radiation Emitted When Electron in Hydrogen Atom Undergoes Transition from Higher Energy State (Quantum Number Ni) to the Lower
- The Electron in Hydrogen Atom is Initially in the Third Excited State. What is the Maximum Number of Spectral Lines Which Can Be Emitted When It Finally Moves to the Ground State?
- Calculate the de-Broglie wavelength associated with the electron revolving in the first excited state of the hydrogen atom. The ground state energy of the hydrogen atom is −13.6 eV.
- Using Bohr’S Postulates for Hydrogen Atom, Show that the Total Energy (E) of the Electron in the Stationary States Tan Be Expressed as the Sum of Kinetic Energy (K) and Potential Energy
- Use Bohr’s model of hydrogen atom to obtain the relationship between the angular momentum and the magnetic moment of the revolving electron.
- Write the Expression for Bohr’S Radius in Hydrogen Atom ?
- The radius of the nth orbit in the Bohr model of hydrogen is proportional to ______.
- State Bohr'S Quantization Condition for Defining Stationary Orbits.
- Obtain Bohr’S Quantisation Condition for Angular Momentum of Electron Orbiting in Nth Orbit in Hydrogen Atom on the Basis of the Wave Picture of an Electron Using De Broglie Hypothesis.
- When the electron orbiting in hydrogen atom in its ground state moves to the third excited state, show how the de Broglie wavelength associated with it would be affected.
- Answer the Following Question. Calculate the Orbital Period of the Electron in the First Excited State of the Hydrogen Atom.
- Specify the transition of an electron in the wavelength of the line in the Bohr model of the hydrogen atom which gives rise to the spectral line of the highest wavelength ______.
- Find the angular momentum of an electron revolving in the second orbit in Bohr's hydrogen atom.
- Using Bohr'S Postulates of the Atomic Model, Derive the Expression for Radius of Nth Electron Orbit
- State three postulates of Bohr's theory of hydrogen atom.
- Write the ionisation energy value for the hydrogen atom.
- What is meant by ionisation energy?
- State Bohr'S Postulate to Define Stable Orbits in Hydrogen Atom. How Does De Broglie'S Hypothesis Explain the Stability of These Orbits?
- Using Bohr'S Postulates, Derive the Expression for the Orbital Period of the Electron Moving in the Nth Orbit of Hydrogen Atom ?
- State Bohr Postulate of Hydrogen Atom that Gives the Relationship for the Frequency of Emitted Photon in a Transition.
- State Bohr's postulate to explain stable orbits in a hydrogen atom. Prove that the speed with which the electron revolves in nth orbit is proportional to n(1n).
- The wavelength of the second line of the Balmer series in the hydrogen spectrum is 4861 Å. Calculate the wavelength of the first line of the same series.
- Which Transition Corresponds to Emission of Radiation of Maximum Wavelength?
- Given the Ground State Energy E0 = - 13.6 eV and Bohr Radius a0 = 0.53 A. Find Out How the De Broglie Wavelength Associated with the Electron Orbiting in the Ground State Would Change When It Jumps into the First Excited State.
- A 12.5 eV Electron Beam is Used to Bombard Gaseous Hydrogen at Room Temperature. Upto Which Energy Level the Hydrogen Atoms Would Be Excited? Calculate the Wavelengths of the First Member of Lyman and First Member of Balmer Series.
- The ground state energy of a hydrogen atom is −13.6 eV. What are the kinetic and potential energies of the electron in this state?
- A Hydrogen Atom Initially in the Ground Level Absorbs a Photon, Which Excites It to the N = 4 Level. Determine the Wavelength and Frequency of the Photon.
- A 12.9 Ev Beam of Electronic is Used to Bombard Gaseous Hydrogen at Room Temperature. Upto Which Energy Level the Hydrogen Atoms Would Be Excited ?
- A 12.3 Ev Electron Beam is Used to Bombard Gaseous Hydrogen at Room Temperature. Upto Which Energy Level the Hydrogen Atoms Would Be Excited?
- Draw the Energy Level Diagram Showing How the Line Spectra Corresponding to Paschen Series Occur Due to Transition Between Energy Levels.
- The Energy Levels of an Atom Are as Shown Below. Which of Them Will Result in the Transition of a Photon of Wavelength 275 Nm?
- The diagram shows the four energy levels of an electron in the Bohr model of the hydrogen atom. Identify the transition in which the emitted photon will have the highest energy.
- A 12.5 eV electron beam is used to bombard gaseous hydrogen at room temperature. What series of wavelengths will be emitted?
- Find the Wavelength of the Electron Orbiting in the First Excited State in Hydrogen Atom.
- A hydrogen atom makes a transition from n = 5 to n = 1 orbit. The wavelength of photon emitted is λ. The wavelength of photon emitted when it makes a transition from n = 5 to n = 2 orbit is ______.
- Plot a Graph Showing the Variation of De Broglie Wavelength (X) Associated with a Charged a Particle of Mass M, Versus 1 √ V Where V
- Using Bohr’S Second Postulate of Quantization of Orbital Angular Momentum Show that the Circumference of the Electron in the Nth Orbital State in Hydrogen Atom is N Times the De Broglie Wavelength
- How Would the Ionization Energy Change When Electron in Hydrogen Atom is Replaced by a Particle of Mass 200 Times that of Electron but Having the Same Charge ?
- Show that the Radius of the Orbit in Hydrogen Atom Varies an N X N,Where N is the Principal Quantum Number of the Atom.
- How Does One Explain, Using De Broglie Hypothesis, Bohr'S Second Postulate of Quantization of Orbital Angular Momentum?
- Calculate the Shortest Wavelength of the Spectral Lines Emitted in Balmer Series.
- Define Ionization Energy.
- Answer the Following Question. State Bohr'S Quantization Condition of Angular Momentum. Calculate the Shortest Wavelength of the Bracket Series and State to Which Part of the Electromagnetic
- Calculate the De-broglie Wavelength of the Electron Orbitting in the N = 2 State of Hydrogen Atom.
- The Kinetic Energy of the Electron Orbiting in the First Excited State of Hydrogen Atom is 3.4 Ev. Determine the De Broglie Wavelength Associated with It.
- Calculate the Momentum of the Electron.
- Write the Expression for the De Broglie Wavelength Associated with a Charged Particle Having Charge ‘Q’ and Mass ‘M’, When It is Accelerated by a Potential V.
- When is Hα Line in the Emission Spectrum of Hydrogen Atom Obtained? Calculate the Frequency of the Photon Emitted During this Transition.
- The Ground State Energy of Hydrogen Atom is −13.6 Ev. If and Electron Make a Transition from the Energy Level −0.85 Ev to −3.4 Ev, Calculate Spectrum Does His Wavelength Belong?
- An electron in a hydrogen atom makes transitions from orbits of higher energies to orbits of lower energies. When will such transitions result in (a) Lyman (b) Balmer series?
- Using Rydberg Formula, Calculate the Longest Wavelength Belonging to Lyman and Balmer Series. in Which Region of Hydrogen Spectrum Do These Transitions Lie?
- Using Rydberg Formula, Calculate the Wavelengths of the Spectral Lines of the First Member of the Lyman Series and of the Balmer Series.
- Find Out the Wavelength of the Electron Orbiting in the Ground State of Hydrogen Atom.
- The ground state energy of hydrogen atom is – 13∙6 eV. If an electron makes a transition from an energy level – 1∙51 eV to – 3∙4 eV, calculate the wavelength of the spectral line emitted and name the series of hydrogen spectrum to which it belongs
- The Ground State Energy of Hydrogen Atom is −13.6 Ev. If an Electron Make a Transition from an Energy Level −0.85 Ev to −1.51 Ev, Calculate the Wavelength of the Spectral Line Emitted.
Concepts [12]
- Introduction of Atoms
- Alpha-particle Scattering and Rutherford’s Nuclear Model of Atom
- Atomic Spectra
- Bohr’s Model for Hydrogen Atom
- Energy Levels
- The Line Spectra of the Hydrogen Atom
- De Broglie’s Explanation of Bohr’s Second Postulate of Quantisation
- Heisenberg and De Broglie Hypothesis
- Thompson Model
- Dalton's Atomic Theory
- Hydrogen Spectrum
- Overview: Atoms
