Bohrs model radius velocity of electron

Rutherford's planet-like model of the atom is contested by Bohr's Model in 1913 on two grounds, According to classical mechanics, whenever a charged particle is subjected to the acceleration it emits radiation and loses energy. An electron revolving around the nucleus would, therefore, be continually accelerated towards the center of the orbit and consequently emitting radiation.
    The result of this would be that the radius of curvature of its path would go on decreasing and due to spiral motion, the electrons will finally fall on the nucleus when all its rotational energy spent on the electromagnetic radiation and the atom would collapse.
    If the electrons lose energy continuously, the observed atomic spectra should be continuous, consisting of broad bands merging one into the other. The observed atomic, however, consists of well-defined lines of definite frequencies. To resolve the anomalous position Bohr's model proposed several novel postulates,

Postulates of Bohr's model

    • An atom possesses several stable circular orbits in which an electron can stay. So long as an electron can stay in a particular orbit there is no emission or absorption of energy. These orbits are called energy levels or main energy shells.
      These shells are numbered as 1, 2, 3, ........ starting from the nucleus and are designated as capital letters, K, L, M, ........ respectively.
      The energy associated with a certain energy level increases with the increase of its distance from the nucleus. Thus, if E₁, E₂, E₃ ...... denotes the energy levels numbered as 1 (K - Shell), 2 (L - Shell), 3 (M - Shell) ......., these are in the order,
      E1ㄑE2ㄑE3ㄑ......
    • An electron can jump from one orbit to another higher energy on the absorption of energy and one orbit to another lower energy orbit with the emission of energy.
    Emission or absorption spectrum in Bohrs model radius velocity of electron
    Emission and Absorption of Energy of an Electron
      The amount of energy (ΔE) emitted or absorption in this type of jump of the electron is given by Plank's Equation.
      ΔE = hν
      Where ν is the frequency of the energy (radiation) emitted or absorbed and h is the Plank Constant.
    • The angular momentum of an electron moving in an orbit is an integral multiple of h/2π. This is known as the principle of quantization of angular momentum according to which,
      mvr = n × (h/2π)
      Where m = mass of the electron, v = tangential velocity of the electron in its orbit, r = distance between the electron and nucleus and n = a whole number which has been given principle quantum number and atomic orbitals by Bohr.

    Radius of nth orbit in Bohr's model

      The nucleus has a mass m' and the electron has mass m. The radius of the circular orbit is r and the linear velocity of the electron is v. Evidently, on the revolving electron, two types of forces are acting,
    Postulates of Bohr's Theory and Bohrs model radius velocity of electron
    Bohrs model radius velocity of the electron
      The centrifugal force which is due to the motion of the electron and tends to take the electron away from the orbit.
      Centrifugal force = mv²/r, Acts outwards from the nucleus.
      The attractive force between the nucleus and the electron.
      Two attractive forces are in the operation, one being the electric force of attraction between the nucleus (Proton) and the electron, the other being the Gravitational force is comparatively weak and can be neglected.
      It is given by Coulomb's inverse squire low and is therefore equal to,
    e × (e/r²) = e²/r²
      It acts towards the nucleus. In order that the electron may keep on revolving in its orbit, these two forces, which acts in opposite direction must balance each other.
    That is, mv²/r = e²/r²
    ∴ mv² = e²/r
      Now Bohr made a remarkable suggestion that the angular momentum of the system, equal to mvr, can assume certain definite values or quanta. Thus all possible r, values only certain definite r values are permitted. Thus only certain, definite orbits are available to the revolving electron.
    Thus, mvr = nh/2π (where n have values 1,2,3, ......∞)
    or, v = n × (h/2π) × (1/mr)

    Then, e²/r = mv²

    or, e²/r = m × n² × (h/2π)² × (1/mr)²

    or, e²/r = n²h²/4π²mr²

    ∴ r = n²h²/4π²me²

    Relation between Bohr's nth and first orbit

    ∴ rn = n²h²/4π²me²
    or, rn = n² × (h²/4π²me²)
    Thus, rn = n² × r₁

    Radius of first Bohr orbit

      Taking n = 1, the radius of the first orbit is r₁.
    ∴ r₁ = 1 × h²/4π²me²
    = {1×(6.627×10⁻²⁷)²}/{4×(3.1416)²×(9.108×10⁻²⁸×(4.8 × 10⁻¹⁰)²}
    = 0.529 × 10⁻⁸ cm
    = 0.529 Å = a₀
      Thus the radius of first orbit r₁ = a₀, second orbit r₂ = 4 a₀ and third orbit r₃ = 9 a₀ and so on.

    Velocity of an electron in nth orbit in Bohr's model

      We have the following relations, mvr = nh/2π, and r = n²h²/4π²me²
    Then, v = (nh/2πm) × (1/r)
    = (nh/2πm) × (4π²me²/n²h²)


    ∴ v = 2πe²/nh
      This is the velocity of the electron in nth orbit in Bohr's model of the hydrogen atom.
      Putting the values of n (1, 2, 3, .....) we see the velocity in the second orbit will be one half of the first orbit and that in the third orbit will be one-third of that in the first orbit and so on.

    Energy of electron in nth orbit in Bohr's model

      The energy of an electron moving in one such Bohr orbit be calculated remembering that the total energy is the sum of the kinetic energy (T) and the Potential energy (V).
    Kinetic energy(T) = (1/2)mv²
      Potential energy(V) is the energy due to electric attraction and is given by,
    V = ∫(e²/r²)dr = - (e²/r)
    Thus the total energy, E = ½ mv² - (e²/r)
    = ½ mv² - mv² (where mv² = e²/r)
    = - ½ mv²
    = - ½ (e²/r)
      The energy associated with the permitted orbits is given by,
    E = - ½ (e²/r)
    = - (2π² me⁴/n²h²)
    = (En =1)/n²
    [where (En =1) = - (2π²me⁴/h²)]
      The energy being governed by the value of quantum number n. As n increases the energy becomes less negative and hence the system becomes less stable. Also note that with increasing n, r also increases. Thus increasing r also makes the orbit less stable.
      Thus if the energies associated with 1st, 2nd, 3rd,...., nth orbits are E1, E2, E3 ... En, these will be in the order,
    E₁ㄑE₂ㄑE₃ㄑ........ㄑEn
      Energy (E1) of the moving in the 1st Bohr orbit is obtained by putting n=1 in the energy expression of E.
    Thus, E1 = - {2 × (3.14)² × (9.109 × 10⁻²⁸)×(4.8 × 10⁻¹⁰)⁴}/{12 × (6.6256 × 10⁻²⁷)²}
    = - 21.79 × 10⁻¹² erg
    = - 13.6 eV
    = - 21.79 × 10⁻¹⁹ Joule
    = - 313.6 Kcal.

    Bohr's model questions answers

    • Question
      Calculate the velocity of the hydrogen electron in the first and third orbit. Also, calculate the number of rotation of an electron per second in the third orbit.
    • Answer
    The velocity of an electron in the Bohr's model, = 2πe²/nh
    where n = 1, 2, 3, ........
    Thus, the velocity of an electron in the first orbit,
    v₁= 2πe²/1 × h
    = 2πe²/h (when n = 1)
    ∴ v₁ = {2×(3.14)×(4.8×10⁻¹⁰)²}/(6.626×10⁻²⁷)
    = 2.188 × 10⁸ cm sec⁻¹

    Again the velocity of the third orbit,
    v₃= (1/3) × v₁ (when n = 3)
    Thus v₃ = (2.188 × 10⁸ cm sec⁻¹)/3
    = 7.30 × 10⁷ cm sec⁻¹
    ∴ The radius of the third orbit, = 32 × 0.529 × 10⁻⁸ cm

    Thus the circumference of the third orbit,
    2πr = 2 × 3.14 × 0.529 × 10⁻⁸ cm

    ∴ Rotation of an electron per second in the third orbit,
    = (7.30 × 10⁷)/(2 × 3.14 × 9 × 0.529 × 10⁻⁸ )
    = 2.44 × 10¹⁴ sec⁻¹
    • Question
      Calculate the kinetic energy of the electron in the first orbit of He+2. What will be the value if the electron is in the second orbit?
    • Answer
    Kinetic Energy = ½ mv²
    = 1/2 m (2πZe²/nh)²
    = 2π²mZ²e⁴/n²h²
    Where, e = 4.8 × 10⁻¹⁰ esu, Plank's Constant(h) = 6.626 × 10⁻²⁷ erg sec, and m = 9.1 × 10⁻²⁸ g
    ∴ Kinetic Energy = 871 × 10⁻¹³erg
    • Question
      The energy of an electron in the first Bohr orbit of the hydrogen atom is -13.6 eV. What is the energy value of the electron in the excited state of Li⁺²?
    • Answer
      The energy of an electron in the excited state of Li⁺² is ELi⁺² = - 30.6 eV
    • Question
      What is the radius of the second orbit of a hydrogen atom? Give radius of first orbit = 0.529 Å.
    • Answer
      Thus the radius of the second orbit of a hydrogen atom = 2.12 Å

    Bohrs model radius velocity of electron, postulates of bohr's theory and derivation of energy of electron in hydrogen atom related questions answers

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