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:
    ◈ _*type: bullet
  1. \(P.E.=2\times Total\ energy\)
  2. \(P.E.=-2K.E.\)
  3. Total energy is always negative
  4. Negative energy means electron is bound to nucleus
  5. As \(n\) increases, total energy increases towards zero
  6. 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:
    ▢ Electronic Transition:
    ❖ Definition: Jump of electron from one energy level to another
    ❖ **table:
      ❖ Rules:
        ◈ **type: bullet
      1. Electron absorbs energy and jumps from lower to higher orbit
      2. Electron emits energy and jumps from higher to lower orbit
      3. As \(n\) increases, energy gap between consecutive levels decreases
      4. \(E_2-E_1>E_3-E_2>E_4-E_3>...\)
      5. Electron in \(n^{th}\) excited state means electron is in \((n+1)^{th}\) orbit
      ▢ Hydrogen Spectral Series:
      ❖ **table:
        ❖ Wavelength Order: \(\lambda*{Pfund}>\lambda*{Brackett}>\lambda*{Paschen}>\lambda*{Balmer}>\lambda*{Lyman}\)
        ▢ General Spectral Formulae:
        ❖ **table:
          ❖ 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.
          The angular momentum of the electron in the second orbit of hydrogen atom is
          📅IOM 2011
          Q2.
          In an electronic transition, an atom cannot emit
          📅IOM 2011
          Q3.
          The radius of hydrogen atom in ground state is 0.53 Å. After excitation, radius becomes 2.12 Å. The principal quantum number is
          📅IOM 2010
          Q4.
          The ratio of energy of hydrogen atom in the first orbit to second orbit is
          📅MOE 2013
          Q5.
          The radius of first orbit of electron in hydrogen atom is
          📅MOE 2012
          Q6.
          The energy required to remove an electron from n = 2 state of hydrogen atom is
          📅MOE 2010
          Q7.
          If an electron jumps from fourth excited state to second excited state, the number of possible emission transitions between these states is
          📅MOE 2063
          Q8.
          Energy band in solids is explained by
          📅KU 2013
          Q9.
          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
          📅KU 2011
          Q10.
          The ionization energy of hydrogen atom is 13.6 eV. The ionization energy when electron is already in first excited state is
          📅BP 2010
          Q11.
          Which series lies in visible region?
          Q12.
          The potential energy of an electron in second orbit of carbon ion (Z = 6) is E. The total energy of an electron in third orbit of helium ion (Z = 2) is
          📅MOE 2014
          Q13.
          The orbital angular momentum of electron in hydrogen atom varies as
          📅MOE 2014
          Q14.
          If radius of first orbit of hydrogen atom is 0.53 Å, then radius of second Bohr orbit is
          📅MOE 2014
          Q15.
          Radiations coming from Lyman series fall in
          📅MOE 2009
          Q16.
          The ratio of shortest wavelength to longest wavelength among the first five hydrogen spectral series is
          📅BPKIHS 2000
          Q17.
          The ratio of wavelength of first line of Lyman series to first line of Balmer series is
          📅MOE Curriculum
          Q18.
          Balmer series lies approximately between
          📅MOE 2066
          Q19.
          Bohr's theory correctly predicts
          📅IE 2006
          Q20.
          If energy of hydrogen atom in ground state is -13.6 eV, its energy in first excited state is
          📅IE 2004
          Q21.
          Which transition in hydrogen atom gives an absorption line of higher frequency?
          Q22.
          The ratio of Rydberg constant for helium ion to Rydberg constant for hydrogen is
          📅BPKIHS 2008
          Q23.
          Emission line spectrum is obtained from
          📅BPKIHS 2004
          Q24.
          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
          📅BPKIHS 2005
          Q25.
          If r is the radius of a Bohr orbit of hydrogen atom, then radius of the same orbit of He+ is
          📅BPKIHS 2006
          Q26.
          The frequency of electron around first Bohr orbit is
          📅BPKIHS 2006
          Q27.
          If total energy of electron is E0, then its potential energy is
          📅BPKIHS 2006
          Q28.
          In which transition will the wavelength be minimum?
          📅BPKIHS 1995
          Q29.
          The radius of electron's second stationary orbit in Bohr atom is R. Radius of third orbit will be
          Q30.
          The radius of Bohr's first orbit is a0. Radius of electron in first orbit of singly ionised helium atom is
          Q31.
          The ratio of energies of hydrogen atom in its first excited state to second excited state is
          Q32.
          The ratio of total energy of an electron in hydrogen atom in n = 1 and n = 4 orbits is
          Q33.
          The ratio of kinetic energy of an electron in hydrogen atom in n = 1 and n = 4 orbits is
          Q34.
          The ionization energy of hydrogen atom is 13.6 eV. The ionization energy of helium ion would be
          📅IOM 2017
          Q35.
          The speed of electron in fourth Bohr orbit of hydrogen atom is
          Q36.
          The first excitation potential of hydrogen atom is 10.2 V. The ionization potential is
          Q37.
          Hydrogen atoms in ground state are excited by photons of energy 12.1 eV. Number of spectral lines emitted according to Bohr theory is
          Q38.
          The Rydberg constant for electron revolving around hydrogen atom is R. For electron revolving around 10-times ionised sodium atom, it will be
          Q39.
          The wavelength of series limit of Lyman series is
          Q40.
          The wavelength of first line of Lyman series is 1216 Å. The wavelength of second member of Balmer series is
          Q41.
          The series limit of Balmer series is 3670 Å. The series limit for Paschen series will be
          Q42.
          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
          Q43.
          The frequency of first line of Balmer series in hydrogen is ν0. Frequency of the same line emitted by doubly ionised lithium ion is
          Q44.
          Which transition in hydrogen atom emits photon of highest frequency?
          Q45.
          Minimum excitation potential of first orbit of hydrogen atom is
          Q46.
          If the shortest wavelength in Lyman series is 918 Å, the longest wavelength in the same series is
          Q47.
          If an electron has orbital angular momentum quantum number l = 7, its orbital angular momentum is
          Q48.
          The wavelength of first line of Lyman series of hydrogen is 1216 Å. The wavelength of second line of same series is
          Q49.
          The velocity of an electron in second orbit of ten-times ionised sodium atom is v. The velocity in fifth orbit will be
          Q50.
          The angular momentum of electron in hydrogen atom is proportional to
          Q51.
          In an atom, two electrons move in circular orbits of radii R and 4R. Ratio of times taken to complete one revolution is
          Q52.
          If an electron jumps from fifth excited state to second excited state, the number of possible emission transitions is
          Q53.
          If elements with principal quantum number n > 4 were not allowed in nature, the number of possible elements would be
          Q54.
          The process responsible for production of laser light is
          📅KU 2017