### Rutherford model to Bohr's model of the hydrogen atom

Explanation of the hydrogen spectrum, Neils Bohr's in 1913 adopted the Rutherford model of the hydrogen atom in which an electron revolves around the single proton at the central nucleus.According to classical mechanics, when a charged particle is subjected to the acceleration it emits radiation and loses energy and the orbital radius must also be changed.

An electron revolving around the nucleus would, therefore, be continually accelerated towards the center of the orbit and consequently emitting radiation.

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 energy of the electron loses continuously, the observed atomic spectra should be continuous, consisting of broad bands merging one into the other. The observed emission spectrum consists of well-defined lines of definite frequencies. To resolve the anomalous position Niels Bohr's proposed a new atomic model of the hydrogen atom.

#### Bohr's postulates and hydrogen energy levels

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 of an atom.The energy levels are numbered as 1, 2, 3, ........ starting from the nucleus of an atom 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.

E₁, E₂, E₃ ...... denotes the number of energy levels of 1 (K - Shell), 2 (L - Shell), 3 (M - Shell) ......., these are in the order,

E1ã„‘E2ã„‘E3ã„‘......

An electron can jump from one energy level to another higher energy level on the absorption of energy and one energy level to another lower energy orbit with the emission of energy.

Hydrogen energy levels |

∴ mvr = n × (h/2Ï€)

where m = mass of an electron, v = tangential

**velocity of an electron**in an energy level, r = distance between the electron and nucleus of an atom and n = whole number which has been given the principal quantum number of an atom.

The amount of energy (Î”E) emitted or absorption in this type of jump of the electron is given by Plank's equation.

Î”E = hÎ½

where Î½ is the frequency of the radiation emitted or absorbed by an electron and h is the Plank constant.

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### Velocity and energy of an electron in a hydrogen atom

The nucleus has a mass m' and the electron has mass m. The radius of the circular orbit = r and the linear velocity of the electron = v. Evidently, on the revolving electron, two types of forces are acting, centrifugal force and electric force of attraction.The energy of an electron |

#### Centrifugal and electric force of an atom

Energy levels stable when the centrifugal force exerted by the moving electron must equal to the attractive force between the electron and nucleus of the hydrogen atom.Centrifugal force = mv²/r.

Two attractive forces are in the operation, one being the electric force of attraction between the nucleus and the electron, the other being the gravitational force. Gravitational force comparatively weak and can be neglected.

The electric force of attraction between two opposite charges is given by Coulomb's law.

Electric force = e × (e/r²) = e²/r².

Centrifugal force and electric force has acted in the opposite direction. The electron may keep on revolving in its orbit, these two forces, which acts in opposite direction must balance each other.

∴ mv²/r = e²/r²

or, mv² = e²/r.

or, mv² = e²/r.

#### Atomic size from Bohr's model

Bohr's remarkable suggestion for the angular momentum of the electron is an integral multiple of h/2Ï€. Angular momentum of an electron = mvr.∴ mvr = nh/2Ï€

Where n = integer called quantum number indicating

**hydrogen energy levels**having the values 1,2,3, ......∞.

v = n × (h/2Ï€) × (1/mr)

e²/r = mv²

Putting the value of v on this equation,

e²/r = m × n² × (h/2Ï€)² × (1/mr)².

or, e²/r = n²h²/4Ï€²mr²

∴ r = n²h²/4Ï€²me²

e²/r = mv²

Putting the value of v on this equation,

e²/r = m × n² × (h/2Ï€)² × (1/mr)².

or, e²/r = n²h²/4Ï€²mr²

∴ r = n²h²/4Ï€²me²

#### The radius of an orbit of the hydrogen atom

Solution for the radius of the permitted energy levels of the hydrogen atom in terms of the quantum number. When n = 1, the radius of the first stationary orbit of hydrogen.∴ r₁ = 1 × h²/4Ï€²me²

= {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 â„« = a₀

Thus the radius of first orbit r₁ = a₀, second orbit r₂ = 4 a₀ and third orbit r₃ = 9 a₀.= {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 â„« = a₀

r

or, r

∴ r

_{n}= n²h²/4Ï€²me²or, r

_{n}= n² × (h²/4Ï€²me²)∴ r

_{n}= n² × r₁ = r₁ × a₀Question

Calculate the radius of the second orbit of a hydrogen atom if the radius of the first orbit of hydrogen = 0.529 Ã….

Answer

The radius of the second orbit of a hydrogen atom

r₂ = n² × r₁ = 2.12 Ã…

#### The velocity of an electron of the hydrogen atom

mvr = nh/2Ï€

where v = velocity of an electron.

or, v = (nh/2Ï€m) × (1/r)

Putting the values of r = n²h²/4Ï€²me².

v = (nh/2Ï€m) × (4Ï€²me²/n²h²)

∴ v = 2Ï€e²/nh

where v = velocity of an electron.

or, v = (nh/2Ï€m) × (1/r)

Putting the values of r = n²h²/4Ï€²me².

v = (nh/2Ï€m) × (4Ï€²me²/n²h²)

∴ v = 2Ï€e²/nh

Thus the velocity of the second orbit will be one half of the first orbit and one-third of the first orbit and so on.

v₂ = v₁/2

v₃ = v₁/3.

∴ v

v₃ = v₁/3.

∴ v

_{n}= v₁/n = velocity of first orbit/principal quantum shell.Question

Calculate the velocity of the hydrogen electron in the first and third energy levels of an atom. How to calculate the number of rotation of an electron per second in the third energy level?

Answer

v = 2Ï€e²/nh

where n = 1, 2, 3, ........

∴ The velocity of an electron in the first energy level

v₁ = 2Ï€e²/1² × h

= 2Ï€e²/h

= {2×(3.14)×(4.8×10⁻¹⁰)²}/(6.626×10⁻²⁷)

= 2.188 × 10⁸ cm sec⁻¹.

where n = 1, 2, 3, ........

∴ The velocity of an electron in the first energy level

v₁ = 2Ï€e²/1² × h

= 2Ï€e²/h

= {2×(3.14)×(4.8×10⁻¹⁰)²}/(6.626×10⁻²⁷)

= 2.188 × 10⁸ cm sec⁻¹.

The velocity of the third energy level

v₃ = v₁/3.

∴ v₃ = (2.188 × 10⁸ cm sec⁻¹)/3

= 7.30 × 10⁷ cm sec⁻¹.

∴ The radius of the third energy level

= 3² × 0.529 × 10⁻⁸ cm.

v₃ = v₁/3.

∴ v₃ = (2.188 × 10⁸ cm sec⁻¹)/3

= 7.30 × 10⁷ cm sec⁻¹.

∴ The radius of the third energy level

= 3² × 0.529 × 10⁻⁸ cm.

Circumference of the third energy level

= 2Ï€r = 2 × 3.14 × 0.529 × 10⁻⁸ cm

= 2Ï€r = 2 × 3.14 × 0.529 × 10⁻⁸ cm

∴ Rotation of an electron per second in the third energy level of the hydrogen atom

= (7.30 × 10⁷)/(2 × 3.14 × 9 × 0.529 × 10⁻⁸ )

= 2.44 × 10¹⁴ sec⁻¹.

= 2.44 × 10¹⁴ sec⁻¹.

#### The kinetic and potential energy of an electron

The energy of an electron moving in one particular energy level can be calculated by the total energy or the sum of the kinetic and the potential energy of an electron.∴ The kinetic energy of a moving electron

= ½ mv².

V = ∫(e²/r²)dr = - (e²/r)

∴ Total energy = E = ½ mv² - e²/r.

Putting the value e²/r = mv².

∴ E = ½ mv² - mv²

= - ½ mv²

= - ½ e²/r

∴ Total energy = E = ½ mv² - e²/r.

Putting the value e²/r = mv².

∴ E = ½ mv² - mv²

= - ½ mv²

= - ½ e²/r

#### Energy of an electron in hydrogen energy levels

Energy of an electron, E = - ½ e²/r

where r = n²h²/4Ï€²me².

∴ E = - 2Ï€² me⁴/n²h².

When n = 1, that is the energy of the first orbit of the hydrogen atom

E₁ = - 2Ï€²me⁴/h²

∴ En = E₁/n².

where r = n²h²/4Ï€²me².

∴ E = - 2Ï€² me⁴/n²h².

When n = 1, that is the energy of the first orbit of the hydrogen atom

E₁ = - 2Ï€²me⁴/h²

∴ En = E₁/n².

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.

If energies associated with 1st, 2nd, 3rd,...., nth orbits are E1, E2, E3 ... En, these will be in the order,

E₁ã„‘E₂ã„‘E₃ã„‘........ã„‘En.

The energy of the moving electron in the first energy level obtained by putting n=1 in the energy expression of the hydrogen atom.

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

= - 21.79 × 10⁻¹² erg

= - 13.6 eV

= - 21.79 × 10⁻¹⁹ Joule

= - 313.6 Kcal

Question

H, H⁺, He⁺ and Li⁺² - for which of the species

**Bohr's model**is not applicable?

Answer

From the above species H, He⁺ and Li⁺² contain one electron but H⁺-ion has no electron. Bohr's model is applicable for one electronic system thus for H⁺-ion Bohr's model is not applicable.

#### The kinetic energy of moving electron of an atom

The kinetic energy of a moving electron

T = ½ mv²

= ½ m × (2Ï€Ze²/nh)²

= (2Ï€²mZ²e⁴)/n²h².

where the charge of an electron (e) = 4.8 × 10⁻¹⁰ esu

Plank's constant (h) = 6.626 × 10⁻²⁷ erg sec

mass of an electron (m) = 9.1 × 10⁻²⁸ gm.

∴ The kinetic energy of hydrogen atom in first energy level

= 13.6 eV

QuestionT = ½ mv²

= ½ m × (2Ï€Ze²/nh)²

= (2Ï€²mZ²e⁴)/n²h².

where the charge of an electron (e) = 4.8 × 10⁻¹⁰ esu

Plank's constant (h) = 6.626 × 10⁻²⁷ erg sec

mass of an electron (m) = 9.1 × 10⁻²⁸ gm.

∴ The kinetic energy of hydrogen atom in first energy level

= 13.6 eV

The energy of an electron in the first energy level of the hydrogen atom = -13.6 eV. What is the energy value of the electron in the excited level of lithium-ion?

Answer

The energy of an electron in the excited level of lithium-ion

= - 30.6 eV

= - 30.6 eV