@aakashallen: Estimation of closest distance of approach (derivation) of α - particle
@aakashallen: Estimation of closest distance of approach (derivation) of α - particle
@aakashallen: Estimation of closest distance of approach (derivation) of α - particle
Summary
Estimation of closest distance of approach (derivation) of α - particle
4kZe 2
R=
mαVα 2
(closest distance of approach)
Size of the nucleus:
The radius of a nucleus
R = R0 (A)1/3 cm
where R0 can be 1.1 × 10-13 cm; to 1.44 × 10-13 cm; A = mass number; R = Radius of the nucleus.
Planck’s Quantum Theory
hc
Energy of one photon = hv =
λ
h = planck’s constant = 6.625 × 10-34 Js, v = frequency of light.
If a change ‘q’ is accelerated through a potential difference of ‘V’ volt then its kinetic energy will
be increased by q.V.
1 2
=K .E = mv qv
2
Photoelectric Effect:
1
= hv0 + me v 2
hv
2
Where me is the mass of the electron and v is the velocity associated with the ejected electron.
Bohr’s Model for Hydrogen like atoms:
h
1. mvr = n (Quantization of angular momentum)
2π
nh
v=
2πmr
2. E z2 z2
− 21 z 2 =
En = −2.178 ×10−18 2 J / atom =
−13.6 2 eV
n n n
−2π2 me 4 z 2
=E1 = ( z 1, for H − atom)
n2
n2 h2 0.529 × n 2
3. rn = × = Å
Z 4π 2 e 2 m Z
2πze 2 2.18 ×106 × z
4. v = = m/s
nh n
0.657 × Z 2 ×1016
5. Revolutions per sec = f = v / 2πr =
n3
1.52 ×10−16 × n3
6. Time taken for one revolution = 2πr/v =
Z2
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Atomic Structure
1
7. En = K .E + P.E =
− K .E = P.E
2
8. P.E = - 2K.E
9. T.E = - K.E
1 ze 2
10. K.E = 1/2 mv2, P.E = −
4πε 0 r
De – Broglie wavelength
h h
λ =
= (for photon)
mc p
If a charged particle Q is accelerated through potential difference V from rest then de – broglie
wavelength is
h h h
=λ = =
mv 2m( K .E ) 2mQV
The circumference of the nth orbit is equal to
n times the wavelength of the electron
2πrn =nλ
Energy of photon on the basis of Einstein’s mass energy relationship
h
E = mc 2 or λ =
mc
m0
m = dynamic mass =
2
v
1+
c
m0 = rest mass of particle, v = velocity of particle, C = speed of light.
If velocity of particle is zero then:
dynamic mass = rest mass
Rest mass of photon is zero that means photon is never at rest
h h
= λ =
2m( K .E.) 2m(q.V )
If an electron is accelerated through a potential difference of ‘V’ volt from rest then:
h 150 12.3
=⇒λ = ⇒λ Å= ⇒λ Å V in volt
2me (eV ) V V
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Atomic Structure
1 (13.6) z 2 1 1
= −
λ hc n12 n22
1 1 1
= v= RZ 2 2 − 2
λ n1 n2
Wave number
13.6eV
1.1×107 m −1 ; R = ; R ch =
R = Rydberg constant = 1.09678 × 107m-1; R = 13.6eV
hc
n12
The wavelength of marginal line = for all series.
RH
No. of photons emitted by a sample of H atom
If an electron is in any higher state n and makes a transition of ground state, then total no. of
n × (n − 1)
different photons emitted is equal to .
2
If an electron is in any higher state n and makes a transition to another excited state, then total no
∆n(∆n + 1)
of different photons emitted is equal to .
2
Note In case of single isolated atom if electron make transition from nth state to the ground state
then max. number of spectral lines observe = (n – 1)
Heisenberg’s uncertainty principle:
h h h
∆x.∆p > or m∆x.∆v ≥ or ∆x.∆v ≥
4π 4π 4πm
Where, Δx = uncertainly in position; Δp = uncertainty in momentum h = Planck’s constant ; m =
mass of the particle; Δv = uncertainty in velocity
Qunatum Numbers:
Principal quantum number (n) = 1, 2, 3, 4….to ∞
Number of subshell present in nth shell = n
Number of orbitals present in nth shell = n2.
The maximum number of electrons which can be present in a principal energy shell is
equal to 2n2.
nh
Orbital angular moment of electron in any orbit = .
2π
Azimuthal quantum number (ℓ) = 0, 1, ….to (n – 1).
Number of orbitals in a subshell = 2ℓ + 1.
Maximum number of electrons in particular subshell = 2 × (2ℓ +1)
h h
Orbital angular momentum = L ( +=
1) h ( + 1) h=
2π 2π
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Atomic Structure
m (magnetic quantum no.) can have values from - ℓ to + ℓ including zero. i.e., total
(2ℓ+1) values.
eh
Spin magnetic moment= µ3 s ( s + 1) or=µ n(n + 2) B.M. (n = no. of
2πmc
unpaired electrons)
h
It represents the value of spin angular momentum which is equal to s ( s + 1)
2π
1
Maximum spin of atom= × No. of unpaired electron.
2
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Atomic Structure
Practice Questions
1. Ejection of the photoelectron from metal in the photoelectric effect experiment can be stopped by
applying 0.5 V when the radiation of 250 nm is used. The work function of the metal is: (2018)
(a) 4.5 eV
(b) 3 eV
(c) 5.5 eV
(d) 5 eV
(b) Frequency of emitted radiation from a black body goes from a lower wavelength to higher
wavelength as the temperature increases
3. A stream of electrons from a heated filament was passed between two charged plates kept at a
potential difference V esu. If e and m are charge and mass of an electron, respectively, then the value of
h / λ (where λ, is wavelength associated with electron wave) is given by (2016)
(a) 2meV
(b) √meV
(c) √2meV
(d) meV
4. Rutherford's experiment, which established the nuclear model of the atom, used a beam of
(2002)
(c) helium atoms, which impinged on a metal foil and got scattered
(d) helium nuclei, which impinged on a metal foil and got scattered
5. Rutherford's alpha particle scattering experiment eventually led to the conclusion that (1986)
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Atomic Structure
(a) 10-10 cm
(b) 10-13 cm
(c) 10-15 cm
(d) I0-8 cm
8. The increasing order (lowest first) for the values of elm (charge/mass) for electron (e), proton (p),
neutron (n) and alpha particle (α) is (1984)
(a) e, p, n, α
(b) n, p, e, α
(c) n, P, α, e
(d) n, α, p, e
(a) nucleus
(b) atom
(c) electron
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Atomic Structure
(d) neutron
10. Rutherford's experiment on scattering of α-particles showed for the first time that the atom has
(1981)
(a) electrons
(b) protons
(c) nucleus
(d) neutrons
11. The radius of the second Bohr orbit for hydrogen atom is (Planck's constant (h )= 6.6262 x 10-30 Js;
mass of electron = 9.1091 x 10-31 kg ; charge of electron (e) = 1.60210 x 10-19 C; permittivity of vacuum
0
(∈0) = 8.854185 x 10-12kg-1 m-3 A2) (2017) (a) 1.65 A
0
(b) 4.76 A
0
(c) 0.529 A
0
(d) 2.12 A
12. P is the probability of finding the 1s electron of hydrogen atom in a spherical shell of infinitesimal
thickness, dr, at a distance r from the nucleus. The volume of this shell is 4πr2dr. The qualitative sketch
of the dependence of P on r is (2016)
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Atomic Structure
13. Which of the following is the energy of a possible excited state of hydrogen? (2015)
(a) + 13.6 eV
(b) -6.8 eV
(c) -3.4 eV
(d) + 6.8 eV
14. The correct set of four quantum numbers for the valence electrons of rubidium atom (Z= 37) is
(2013)
1
(a) 5, 0, 0, +
2
1
(b) 5, 1, 0, +
2
1
(c) 5, 1, 1, +
2
1
(d) 5, 0, 1, +
2
Z2
−2.178 ×10−18 J 2
E=
n
Wavelength of light required to excite all electron in an hydrogen atom from level n = 1 to n = 2 will be
(h = 6.62 x 10-34 Js and c = 3.0 x 108 ms-1) (2013)
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Atomic Structure
16. The kinetic energy of an electron in the second Bohr orbit of a hydrogen atom is [a0 is Bohr radius]
(2012)
h2
(a)
4π 2 ma02
h2
(b)
16π 2 ma02
h2
(c)
32π 2 ma02
h2
(d)
64π 2 ma02
(a) 2 and 0
(b) 0 and 2
(c) 1 and 2
(d) 2 and 1
18. Which hydrogen like species will have same radius as that of Bohr orbit of hydrogen atom? (2004)
(a) n = 2, Li2+
(b) n = 2, Be3+
(c) n = 2, He+
(d) n = 3, Li2+
19. If the nitrogen atom had electronic configuration 1s7, it would have energy lower than that of the
normal ground state configuration Is2 2s2 2p3, because the electrons would be closer to the nucleus, yet
1s7 is not observed, because it violates (2004)
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Atomic Structure
1 1
20. The quantum numbers + and − for the electron spin represent (2001) (a)
2 2
rotation of the electron in clockwise and anticlockwise direction respectively
(d) two quantum mechanical spin states which have no classical analogue
21. The wavelength associated with a golf ball weighing 200 g and moving at a speed of 5 m/hr is of the
order (2001)
(a) 10-10 m
(b) 10-20 m
(c) 10-30 m
(d) 10-14 m
(a) one
(b) two
(c) three
(d) zero
23. The electronic configuration of an element is 1s2, 2s2 2p6, 3s2 3p6 3d5, 4s1. This represents its (2000)
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Atomic Structure
25. The energy of an electron in the first Bohr orbit of H-atom is —13.6 eV. The possible energy value(s)
of the excited state(s) for electrons in Bohr orbits of hydrogen is (are) (1998)
(a) —3.4 eV
(b) — 4.2 eV
(c) —6.1 eV
(d) + 6.8 eV
h
(a) 6
2π
h
(b) 2
2π
h
(c)
2π
h
(d) 2
2π
27. The first use of quantum theory to explain the structure of atom was made by (1997)
(a) Heisenberg
(b) Bohr
(c) Planck
(d) Einstein
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Atomic Structure
28. Which of the following has the maximum number of unpaired electrons ? (1996)
(a) Mg2+
(b) Ti3+
(c) V3+
(d) Fe2+
1 h
(a) + .
2 2π
(b) zero
h
(c)
2π
h
(d) 2.
2π
30. Which of the following relates to photons both as wave motion and as a stream of particles? (1992)
(a) Interference
(b) E = mc2
(c) Diffraction
(d) E = hv
32. The correct set of quantum numbers for the unpaired electron of chlorine atom is (1989)
n l m n l m
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Atomic Structure
(a) 2 1 0 (b) 2 1 1
(c) 3 1 1 (d) 3 0 0
33. The correct ground state electronic configuration of chromium atom is (1989)
34. The outermost electronic configuration of the most electronegative element is (1988)
35. The orbital diagram in which the Aufbau’s principle is violated (1988)
(a)
(b)
(c)
(d)
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Atomic Structure
36. The wavelength of a spectral line for an electronic transition is inversely related to (1988)
(c) the difference in the energy of the energy levels involved in the transition
37. The ratio of the energy of a photon of 2000 A wavelength radiation to that of 4000 A radiation is
(1986)
1
(a)
4
(b) 4
1
(c)
2
(d) 2
38. Which one of the following sets of quantum numbers represents an impossible arrangement?
(1986)
(a) ultraviolet
(c) X-ray
(d) infrared
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Atomic Structure
40. Which electronic level would allow the hydrogen atom to absorb a photon but not to emit a photon?
(1984)
(a) 3s
(b) 2p
(c) 2s
(d) 1s
41. Correct set of four quantum numbers for the valence (outermost) electron of rubidium (Z= 37) is
(1984)
1
(a) 5, 0, 0, +
2
1
(b) 5, 1, 0 , +
2
1
(c) 5, 1, 1, + .
2
1
(d) 6, 0, 0, +
2
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Atomic Structure
44. Spin only magnetic moment of 25Mnx+ ion is √15 B.M. Then, the value of x is
(a) 1
(b) 2
(c) 3
(d) 4
45. The spin only magnetic moment of Cr3+ in aqueous solution would be :
(a) √13 B. M
46. ψ2 = 0 represent
(a) node
(b) orbital
ii. Nucleides having different atomic number(Z) and mass numbers (a) but same number of neutrons (n)
are called isotones.
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Atomic Structure
48. If n1 and n2 are the boundary value principal quantum numbers of a portion of spectrum of emission
spectrum of H atom, determine the wavelength (in metre) corresponding to last line (longest λ). Given :
n1 + n2 = 7, n2 - n1 = 3 and RH = 1.097 x 107 m-1. (Give your answer in multiple of 10-6)
(a) 8
(b) 4
(c) 10
(d) 15
49. What is the potential energy of an electron present in N-shell of the Be ion ?
(a) – 3.4 eV
(b) – 27.2 eV
(c) – 13.6 eV
(d) —6.8 eV
51. The ratio of kinetic energy and potential energy of an electron in a Bohr orbit of a hydrogen-like
species is:
(a) 1/2
(b) —1/2
(c) 1
(d) —1
52. Total Number of unpaired electrons in d-orbitals of an atom of element of atomic number 29 is: (a)
Ten
(b) One
(c) Zero
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Atomic Structure
(d) Five
53. Photons of minimum energy 496 k.J. mol-1 are needed to ionize sodium atoms. Calculate the lowest
frequency of light that will ionize a sodium atom.
54. According to Bohr's model, if the kinetic energy of an electron in 2nd orbit of He+ is x, then what
should be the ionisation energy of the electron revolving in 3rd orbit of M5+ ion
(a) x
(b) 4x
(c) x/4
(d) 2x
0
55. Last line of Lyman series for H—atom has wavelength λ1, A , 2nd line of Balmer series has
0
wavelength λ2, A then
16 9
(a) =
λ1 λ2
16 3
(b) =
λ2 λ1
4 1
(c) =
λ1 λ2
16 3
(d) =
λ1 λ2
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Atomic Structure
56. Wave function vs distance from nucleus graph of an orbital is given below:
(a) 1
(b) 2
(c) 3
(d) 4
57. Number of nodal planes (planes of zero electron density) in the dxy orbital is
(a) 1
(b) 2
(c) 0
(d) 4
58. The uncertainty in position and velocity of the particle are 0.2 nm and 10.54 x 10-27 ms-1 respectively
then the mass of the particle is (h = 6.625 x 10-34 Js)
(a) 48.86g
(b) 30.86 g
(c) 24.86 gm
(d) 100 g
59. A particle initially at rest having charge q coulomb. & mass m kg is accelerated by a potential
difference of V volts. What would be its K.E & de broglie wavelength respectively after acceleration
h
(a) qV,
2qVm
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Atomic Structure
h
(b) ,qV
2qVm
h
(c) qV,
mV
h
(d) ,qV
mV
60. The number of possible lines of Paschen series when electron jumps from 7th excited state to
ground state (in hydrogen like atom) is:
(a) 2
(b) 5
(c) 4
(d) 3
61. AIR service on Vividh Bharati is transmitted on 219 m band. What is its transmission frequency in
Hertz?
62. Calculate the maximum and minimum number of electrons. Which may have magnetic quantum
1
number m = + 1 and spin quantum number s = + in Chromium (Cr).
2
(a) 3, 2
(b) 6, 4
(c) 4, 2
(d) 2, 1
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Atomic Structure
63. In the emission spectrum of H-atom from energy level ‘n’ to ground state in one or more step, no
line belonging to the Brackett series is observed. The wave number of lines belonging to Balmer series
may be
8R 5R
(a) ,
9 36
3R 8R
(b) ,
16 9
5R 3R
(c) ,
36 16
3R 3R
(d) ,
4 16
64. The ionisation energy of H is 13.6 eV. Calculate the ionization energy of Li2+ ions.
(a) 54.4 eV
(b) 122.4 eV
(c) 244.8 eV
(d)108.8 eV
65. If Photon having wavelength 6.2 nm was allowed to strike a metal plate having work function 50 eV
then calculate wavelength associated with emitted electron:
(a) 1 x 10-10 m
(b) 2 x 10-10 m
(c) 3 x 10-18 m
(d) 4 x 10-14 m
(a) one
(b) two
(c) three
(d) four
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Atomic Structure
67. The wave number of electromagnetic radiation emitted during the transition of electron in between
two levels of Li2+ ion whose principal quantum numbers sum is 4 and difference is 2 is
(a) 3.5 R
(b) 4R
(c) 8R
8
(d) R
9
68. The correct order of wavelength of Hydrogen (1H1), Deuterium (1H2) and Tritium (1H3) moving with
same kinetic energy is
(b) λH = λD = λT
69. Spin angular momentum for unpaired electron in sodium (Atomic No. = 11) is
3
(a)
2
(b) 0.866 h/2π
3 h
(c) −
2 2π
(4) None of these
70. The distance between 4th and 3rd Bohr orbits of He+ is :
(d) None
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Atomic Structure
71. What is the frequency of revolution of electron present in 2nd Bohr's orbit of H-atom ?
Z2
(a) Potential energy of electron ∝
n2
(b) The product of velocity of electron and principle quantum number (n) ∝ Z2
Z2
(c) Frequency of revolution of electron in an orbit ∝
n3
Z2
(d) Coulombic force of attraction on the electron ∝
n2
73. If In Bohr's model, for uni-electronic atom, time period of revolution is represented as Tn, z where n
represents shell no. and Z represents atomic number then the value of T1, 2 : T2, 1 will be :
(a) 8 : 1
(b) 1 : 8
(c) 1 : 1
(d) 1 : 32
74. Be. and a proton are accelerated by the same potential, their de-Broglie wavelengths have the ratio
(assume mass of proton = mass of neutron) :
(a) 1 : 2
(b) 1 : 4
(c) 1 : 1
(d) 1: 3 3
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Atomic Structure
75. If the ionization energy of He+ is 19.6 x 10-18 J per atom then the energy of Be3+ ion in the second
stationary state is :
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Atomic Structure
Answer Key:
1. a 2. b 3. c 4. d 5. b 6. b 7. b 8. d 9. a 10. c
11. d 12. c 13. c 14. a 15. a 16. c 17. a 18. b 19. c 20. d
21. c 22. a 23. b 24. a 25. a 26. a 27. b 28. d 29. b 30. a
31. c 32. c 33. a 34. c 35. b 36. c 37. d 38. c 39. b 40. d
41. a 42. a 43. d 44. d 45. b 46. a 47. b 48. b 49. b 50. a
51. b 52. a 53. b 54. b 55. b 56. a 57. b 58. c 59. a 60. b
61. a 62. a 63. c 64. b 65. a 66. a 67. b 68. a 69. b 70. c
71. d 72. c 73. d 74. d 75. d
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