49Atomic structure and Spectrum

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ATOMIC STRUCTURE AND SPECTRUM
▢ Thomson Atomic Model:
❖ Proposed By: J. J. Thomson
❖ Importance: First proposed atomic model
❖ Failed To Explain:
  • •
    Large angle scattering of alpha particles
  • •
    Origin of spectral lines in hydrogen spectrum
▢ Rutherford Atomic Model:
❖ Experiment: Gold foil experiment using alpha particles

Table 1: Rutherford Experiment Setup

Component
Description
Target
Thin gold foil
Projectile
\(\alpha\)-particles
Source
Radium
Shielding
Lead block to prevent unwanted radiation

Table 2: Observations and Conclusions

Observation
Conclusion
Most \(\alpha\)-particles pass straight without deviation
Most part of atom is empty
Few \(\alpha\)-particles deflected through small angles
Positive charge exists at centre
Very few \(\alpha\)-particles return back
Positive charge and mass concentrated in very small central region
Electrons revolve around nucleus
Circular orbits
❖ Limitations:
  • •
    According to Maxwell, revolving electron should radiate energy continuously
  • •
    Electron should spiral inward and fall into nucleus
  • •
    Cannot explain stability of atom
  • •
    Should give continuous spectrum, but atom gives line spectrum
  • •
    Cannot explain formation of spectral lines
❖ Important Point: Rutherford scattering experiment led to discovery of nucleus
▢ Bohr Atomic Model:
❖ Basis: Rutherford atomic model + quantum theory of radiation

Table 1: Bohr Postulates

Postulate
Statement / Formula
First postulate
Electron revolves around nucleus in circular path where Coulomb force provides centripetal force
Force balance
\(\frac{mv^2}{r}=\frac{1}{4\pi\epsilon_0}\frac{Ze^2}{r^2}\)
Second postulate
Only certain non-radiating stationary orbits are allowed
Angular momentum quantization
\(L=mvr=\frac{nh}{2\pi}\)
Third postulate
Radiation emitted/absorbed when electron jumps between energy levels
Energy quantum
\(E_2-E_1=h\nu\)
❖ Success:
  • •
    Explains stability of atom
  • •
    Explains line spectrum of hydrogen
  • •
    Calculates energy and radius of \(n^{th}\) orbit of one-electron systems like \(H, He^+, Li^{2+}\)
❖ Limitations:
  • •
    Cannot explain line spectra of multi-electron atoms
  • •
    Cannot explain Zeeman effect
  • •
    Cannot explain Stark effect
  • •
    Cannot explain three-dimensional model of atom
  • •
    Cannot explain shapes of molecules
  • •
    Not in accordance with de Broglie dual nature of matter
  • •
    Against Heisenberg uncertainty principle
▢ Radius of Orbit:

Table 1: Bohr Radius Formulae

Quantity
Formula / Relation
Radius of \(n^{th}\) orbit
\(r_n=r_0\frac{n^2}{Z}\)
First Bohr radius
\(r_0=0.53Å\)
Hydrogen radii ratio
\(r_1:r_2:r_3=1:4:9\)
When \(Z\) constant
\(r_n\propto n^2\)
When \(n\) constant
\(r_n\propto\frac{1}{Z}\)
❖ Important Point: Radius of orbit is inversely proportional to atomic number
▢ Velocity of Electron in Orbit:

Table 1: Velocity Formulae

Quantity
Formula / Value
Velocity in \(n^{th}\) orbit
\(v_n=\frac{Z}{n}\frac{c}{137}\)
Velocity in first Bohr orbit of H
\(v_1=2.19\times10^6\ m/s\)
General relation
\(v\propto\frac{Z}{n}\)
When \(Z\) constant
\(v\propto\frac{1}{n}\)
When \(n\) constant
\(v\propto Z\)
Radius relation
\(v\propto\frac{1}{\sqrt r}\)
▢ Energy of Electron in Orbit:

Table 1: Energy Formulae

Energy
Formula / Relation
Potential energy
\(U=-\frac{1}{4\pi\epsilon_0}\frac{Ze^2}{r}\)
Kinetic energy
\(K=\frac{1}{2}mv^2=\frac{1}{8\pi\epsilon_0}\frac{Ze^2}{r}\)
Total energy
\(E_n=K+U=-\frac{1}{8\pi\epsilon_0}\frac{Ze^2}{r}\)
Total energy
\(E_n=-Rhc\frac{Z^2}{n^2}\)
Energy in eV
\(E_n=-13.6\frac{Z^2}{n^2}\ eV\)
Rydberg constant
\(R=1.1\times10^7\ m^{-1}\)
Rydberg energy
\(Rhc=13.6\ eV\)
❖ Relations:
  • •
    \(P.E.=2\times Total\ energy\)
  • •
    \(P.E.=-2K.E.\)
  • •
    Total energy is always negative
  • •
    Negative energy means electron is bound to nucleus
  • •
    As \(n\) increases, total energy increases towards zero
  • •
    Energy of electron is maximum at infinity: \(E_\infty=0\)
❖ Proportionality:

Table 1: Energy Dependence

Condition
Relation
When \(Z\) constant
\(E_n\propto-\frac{1}{n^2}\)
When \(n\) constant
\(E_n\propto Z^2\)
▢ Important Relations:

Table 1: Bohr Orbit Relations

Quantity
Formula / Relation
Time period
\(T=\frac{2\pi r}{v}\)
Time period proportionality
\(T\propto\frac{n^3}{Z^2}\)
Kepler-like relation
\(T^2\propto r^3\)
Frequency of revolution
\(f=\frac{1}{T}\)
Frequency proportionality
\(f\propto\frac{Z^2}{n^3}\)
Angular momentum
\(L=mvr=\frac{nh}{2\pi}\)
Angular momentum proportionality
\(L\propto n\)
▢ Electronic Transition:
❖ Definition: Jump of electron from one energy level to another

Table 1: Transition Formulae

Quantity
Formula
Energy of radiation
\(\Delta E=E_2-E_1\)
Photon energy
\(h\nu=\Delta E\)
Wave number
\(\bar\nu=\frac{1}{\lambda}\)
Rydberg formula
\(\frac{1}{\lambda}=RZ^2\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)\)
Frequency
\(\nu=cRZ^2\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)\)
❖ Rules:
  • •
    Electron absorbs energy and jumps from lower to higher orbit
  • •
    Electron emits energy and jumps from higher to lower orbit
  • •
    As \(n\) increases, energy gap between consecutive levels decreases
  • •
    \(E_2-E_1>E_3-E_2>E_4-E_3>...\)
  • •
    Electron in \(n^{th}\) excited state means electron is in \((n+1)^{th}\) orbit
▢ Hydrogen Spectral Series:

Table 1: Spectral Series

Series
Final orbit \(n_1\)
Initial orbit \(n_2\)
Region
Series limit \(\lambda_{min}\)
First member \(\lambda_{max}\)
Lyman
1
2,3,4,...
Ultraviolet
\(\frac{1}{R}\)
\(\frac{4}{3R}\)
Balmer
2
3,4,5,...
Visible
\(\frac{4}{R}\)
\(\frac{36}{5R}\)
Paschen
3
4,5,6,...
Infrared
\(\frac{9}{R}\)
\(\frac{144}{7R}\)
Brackett
4
5,6,7,...
Infrared
\(\frac{16}{R}\)
\(\frac{400}{9R}\)
Pfund
5
6,7,8,...
Far-infrared
\(\frac{25}{R}\)
\(\frac{900}{11R}\)
❖ Wavelength Order: \(\lambda_{Pfund}>\lambda_{Brackett}>\lambda_{Paschen}>\lambda_{Balmer}>\lambda_{Lyman}\)
▢ General Spectral Formulae:

Table 1: Series Formulae

Quantity
Formula
Shortest wavelength / series limit
\(\lambda_{min}=\frac{n^2}{R}\)
First member / maximum wavelength
\(\lambda_{max}=\frac{n^2(n+1)^2}{(2n+1)R}\)
Number of possible emission lines from \(n^{th}\) orbit
\(\frac{n(n-1)}{2}\)
Number of possible absorption lines
\(n-1\)
Number of spectral lines between two levels
\(\frac{\Delta n(\Delta n+1)}{2}\)
❖ Series Index:
  • •
    \(n=1\) → Lyman
  • •
    \(n=2\) → Balmer
  • •
    \(n=3\) → Paschen
  • •
    \(n=4\) → Brackett
  • •
    \(n=5\) → Pfund
▢ Sommerfeld Atomic Model:
❖ Main Points:
  • •
    Electrons revolve around nucleus in elliptical orbits except first orbit which is circular
  • •
    Mass of electron changes relativistically with velocity: \(m=\frac{m_0}{\sqrt{1-v^2/c^2}}\)
  • •
    Total angular momentum includes orbital angular momentum and radial angular momentum
  • •
    For principal quantum number \(n\), there are \(n\) orbits
  • •
    Out of \(n\) orbits, \((n-1)\) are elliptical and 1 is circular
▢ Types of Spectra:

Table 1: Emission Spectrum

Type
Source
Nature
Line emission spectrum
Incandescent vapours/gases in atomic state
Individual atomic behaviour; characteristic of element
Band emission spectrum
Incandescent vapours/gases in molecular state
Molecular behaviour; characteristic of compound
Continuous emission spectrum
Incandescent solids and liquids
All wavelengths continuously distributed

Table 2: Absorption Spectrum

Type
Absorbing medium
Condition
Line absorption spectrum
Transparent vapours/gases in atomic state
White light passes through low-temperature gas
Band absorption spectrum
Transparent vapours/gases in molecular state
Molecular absorption
Continuous absorption spectrum
Transparent solids or liquids
Continuous absorption
❖ Important Points:
  • •
    No two different elements give identical line spectra
  • •
    No two different compounds give identical band spectra
  • •
    Fraunhofer spectrum is line absorption spectrum
  • •
    Fraunhofer lines are due to absorption of solar radiation by sun's atmosphere
  • •
    Origin of Fraunhofer lines was explained by Kirchhoff
▢ Excitation and Ionisation:

Table 1: Definitions

Term
Meaning
Excitation
Raising an electron from lower energy state to higher energy state
Ionisation
Complete removal of electron from atom
Ionisation energy
Energy required to remove electron completely from isolated atom
Ionisation potential
Potential difference through which electron is moved to gain ionisation energy
Ionisation energy of H atom
\(13.6eV\)
Ionisation potential of H atom
\(13.6V\)
❖ Nature: Ionisation is endothermic; energy is absorbed
▢ Pauli Exclusion Principle:
❖ Statement: No two electrons in an atom can have the same set of four quantum numbers
❖ Number of Elements by Shell:

Table 1: Total Possible Elements up to Shell n

Highest shell \(n\)
Total elements
2
\(2(1^2+2^2)=10\)
3
\(2(1^2+2^2+3^2)=28\)
4
\(2(1^2+2^2+3^2+4^2)=60\)
▢ Read and Digest:

Table 1: Important Points

Fact
Answer
Bohr angular momentum
\(L=\frac{nh}{2\pi}\)
Bohr angular momentum explained by
de Broglie
Main drawback of Bohr theory
Not consistent with de Broglie dual nature and Heisenberg uncertainty principle
Bohr postulate correctly measures
Angular momentum
First discovered hydrogen spectral series
Balmer series
Ionisation
Endothermic process
Infrared region
Between microwave and visible regions
Sodium vapour lamp
Line emission spectrum
Mercury vapour lamp
Line emission spectrum
Incandescent electric bulb
Continuous emission spectrum
Molecules like \(H_2,N_2,O_2,CO_2\)
Band spectrum
Black body
Continuous spectrum
\(\alpha\)-particles and gamma rays
Line spectra
\(\beta\)-particles
Continuous spectrum
Total energy of electron in atom
Always negative
When electron jumps lower to higher orbit
K.E. decreases, P.E. increases, total energy increases
Rydberg constant
Different for different elements
If electron mass becomes half
Rydberg constant becomes half
▢ High-Yield Recall:

Table 1: Atomic Structure One-Liners

Fact
Answer
Thomson model
First atomic model
Rutherford experiment
Gold foil alpha scattering
Most atom
Empty space
Nucleus
Small, massive, positively charged centre
Bohr angular momentum
\(mvr=\frac{nh}{2\pi}\)
Bohr radius
\(r_n=0.53\frac{n^2}{Z}Å\)
Velocity
\(v_n=\frac{Z}{n}\frac{c}{137}\)
First orbit H velocity
\(2.19\times10^6m/s\)
Energy
\(E_n=-13.6\frac{Z^2}{n^2}eV\)
Hydrogen ionisation energy
\(13.6eV\)
Hydrogen ionisation potential
\(13.6V\)
Rydberg formula
\(\frac{1}{\lambda}=RZ^2\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)\)
Lyman series
UV, \(n_1=1\)
Balmer series
Visible, \(n_1=2\)
Paschen series
IR, \(n_1=3\)
Brackett series
IR, \(n_1=4\)
Pfund series
Far IR, \(n_1=5\)
Number of emission lines
\(\frac{n(n-1)}{2}\)
Number of absorption lines
\(n-1\)
Line spectrum
Atomic gases/vapours
Band spectrum
Molecules
Continuous spectrum
Solids/liquids
Fraunhofer spectrum
Line absorption spectrum
Pauli principle
No two electrons have same four quantum numbers
Q1.
Electron transitions from first excited and second excited state to ground state. Ratio of emitted wavelengths is
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Q2.
Lyman series lies in:
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Q3.
The Ionization energy of hydrogen atom is 13.6 ev. Fo Bohr's theory, the energy corresponding to
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Q4.
If the radius of the first Bohr orbit is 0.53 Å, the radius of the third orbit is:
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Q5.
The velocity of electron in hydrogen atom is 7.29×10⁷ cm/sec, the potential energy of that electron is:
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Q6.
The ratio of energies of the hydrogen atom in its first excited state to second excited state is:
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Q7.
The ratio of minimum to the maximum wavelength of the Balmer series is:
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Q8.
In an electronic transition, atom cannot emit:
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Q9.
The angular momentum of the electron in the second orbit of hydrogen atom is
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Q10.
In an electronic transition, an atom cannot emit
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Q11.
The radius of hydrogen atom in ground state is 0.53 Å. After excitation, radius becomes 2.12 Å. The principal quantum number is
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Q12.
The ratio of energy of hydrogen atom in the first orbit to second orbit is
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Q13.
The radius of first orbit of electron in hydrogen atom is
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Q14.
The energy required to remove an electron from n = 2 state of hydrogen atom is
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Q15.
If an electron jumps from fourth excited state to second excited state, the number of possible emission transitions between these states is
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Q16.
The formation of closely spaced energy bands in solids is strongly associated with:
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Q17.
The energy required to excite hydrogen atom from n = 1 to n = 2 is 10.2 eV. The wavelength of radiation emitted when it returns to ground state is nearly
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Q18.
The ionization energy of hydrogen atom is 13.6 eV. The ionization energy when electron is already in first excited state is
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Q19.
Which series lies in visible region?
Q20.
The potential energy of an electron in the second Bohr orbit of hydrogen-like carbon ion \(\mathrm{C}^{5+}\) is \(E\). What is the total energy of an electron in the third orbit of \(\mathrm{He}^{+}\)?
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Q21.
The orbital angular momentum of electron in hydrogen atom varies as
📅MOE 2014
Q22.
If radius of first orbit of hydrogen atom is 0.53 Å, then radius of second Bohr orbit is
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Q23.
Radiations coming from Lyman series fall in
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Q24.
The ratio of shortest wavelength to longest wavelength among the first five hydrogen spectral series is
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Q25.
The ratio of wavelength of first line of Lyman series to first line of Balmer series is
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Q26.
Balmer series lies approximately between
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Q27.
Which physical constant was successfully calculated from the hydrogen energy levels in Bohr’s atomic model?
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Q28.
If energy of hydrogen atom in ground state is -13.6 eV, its energy in first excited state is
📅IE 2004
Q29.
Which transition in hydrogen atom gives an absorption line of higher frequency?
Q30.
Neglecting reduced-mass corrections, the ratio of the hydrogen-like spectral wavenumber coefficient for \(\mathrm{He}^{+}\) to that of hydrogen is:
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Q31.
Emission line spectrum is obtained from
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Q32.
The energy of lowest level of hydrogen atom is -13.6 eV. Energy of emitted photon in transition from n = 4 to n = 2 is
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Q33.
If r is the radius of a Bohr orbit of hydrogen atom, then radius of the same orbit of He+ is
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Q34.
The frequency of electron around first Bohr orbit is
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Q35.
If total energy of electron is E0, then its potential energy is
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Q36.
In which transition will the wavelength be minimum?
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Q37.
The radius of electron's second stationary orbit in Bohr atom is R. Radius of third orbit will be
Q38.
The radius of Bohr's first orbit is a0. Radius of electron in first orbit of singly ionised helium atom is
Q39.
The ratio of energies of hydrogen atom in its first excited state to second excited state is
Q40.
The ratio of total energy of an electron in hydrogen atom in n = 1 and n = 4 orbits is
Q41.
The ratio of kinetic energy of an electron in hydrogen atom in n = 1 and n = 4 orbits is
Q42.
The ionization energy of hydrogen atom is 13.6 eV. The ionization energy of helium ion would be
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Q43.
The speed of electron in fourth Bohr orbit of hydrogen atom is
Q44.
The first excitation potential of hydrogen atom is 10.2 V. The ionization potential is
Q45.
Hydrogen atoms in ground state are excited by photons of energy 12.1 eV. Number of spectral lines emitted according to Bohr theory is
Q46.
The Rydberg spectral coefficient for hydrogen is approximately \(R\). For hydrogen-like \(\mathrm{Na}^{10+}\), the corresponding coefficient (neglecting reduced mass) is:
Q47.
The wavelength of series limit of Lyman series is
Q48.
The wavelength of first line of Lyman series is 1216 Å. The wavelength of second member of Balmer series is
Q49.
The series limit of Balmer series is 3670 Å. The series limit for Paschen series will be
Q50.
Energy levels A, B, C have increasing energies EA < EB < EC. If λ1, λ2, λ3 correspond to C→B, B→A and C→A respectively, then
Q51.
The frequency of first line of Balmer series in hydrogen is ν0. Frequency of the same line emitted by doubly ionised lithium ion is
Q52.
Which transition in hydrogen atom emits photon of highest frequency?
Q53.
Minimum excitation potential of first orbit of hydrogen atom is
Q54.
If the shortest wavelength in Lyman series is 918 Å, the longest wavelength in the same series is
Q55.
If an electron has orbital angular momentum quantum number l = 7, its orbital angular momentum is
Q56.
The wavelength of first line of Lyman series of hydrogen is 1216 Å. The wavelength of second line of same series is
Q57.
The velocity of an electron in second orbit of ten-times ionised sodium atom is v. The velocity in fifth orbit will be
Q58.
The angular momentum of electron in hydrogen atom is proportional to
Q59.
Two electrons occupy circular Bohr orbits of radii \(R\) and \(4R\) in the same hydrogen-like atom. Their revolution periods are in the ratio:
Q60.
If an electron jumps from fifth excited state to second excited state, the number of possible emission transitions is
Q61.
In the simplified shell-capacity model, if no electron could have principal quantum number greater than \(n=4\), what would be the maximum atomic number possible?
Q62.
The process responsible for production of laser light is
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