```Radius and Energy Levels of Hydrogen Atom:
Consider an electron of mass ‘m’ and charge ‘e’ revolving around a nucleus of charge Ze (where, Z = atomic number and e is the charge of the proton) with a tangential velocity v. r is the radius of the orbit in which electron is revolving.
By Coulomb’s Law, the electrostatic force of attraction between the moving electron and nucleus is Coulombic force = KZe2/r2
K = 1/4π∈o  (where ∈o is permittivity of free space)
K = 9 x109 Nm2 C–2
In C.G.S. Units, value of K = 1 dyne cm2 (esu)–2
The centrifugal force acting on the electron is mv2/r
Since the electrostatic force balances the centrifugal force, for the stable electron orbit.
mv2/r= KZe2/r2                                         … (1)
or     v2 =   KZe2/mr                                          … (2)
According to Bohr’s postulate of angular momentum quantization, we have
mvr = nh/2π
v = nh/2πmr
v2 = n2h2/4π2m2r2                                         … (3)
Equating (2) and (3)
KZe2/mr = n2h2/4π2m2r2
Solving for r we getr = n2h2/4π2mKZe2
Where n = 1, 2, 3 - - - - - ∞
Hence only certain orbits whose radii are given by the above equation are available for the electron. The greater the value of n, i.e., farther the energy level from the nucleus the greater is the radius.
The radius of the smallest orbit (n=1) for hydrogen atom (Z=1) is ro.
ro = n2h2/4π2me2K = 12 x (6.626 x 10-34)2 / 4x(3.14)2 x9x10-31 x (1.6x10-19)2x9x109 =5.29 x 10–11 m=0.529 Å
Radius of nth orbit for an atom with atomic number Z is simply written as
rn = 0.529 x n2/z Å```
Related Resources
Problems with Solutions

Problems with Solutions True and False Problem:...

Calculation of velocity of Electron

Calculation of Velocity: We know that mvr =...

Quantum Numbers

QUANTUM NUMBERS An atom contains large number of...

Atomic Models

Atomic Models We know the fundamental particles of...

Quantum Mechanical Model of Atom

Quantum Mechanical Model of Atom...

Discovery of Electron Proton and Neutron

Discovery of Electron, Proton & Neutron Lets...

Fundamental Particles

Relation-Kinetic Energy and Wavelength

Relation between Kinetic Energy and Wavelength:...

Atomic Spectrum

ATOMIC SPECTRUM If the atom gains energy the...

Discovery of Proton

DISCOVERY OF PROTON: POSITIVE RAYS OR CANAL RAYS...

Atomic Structure Terms

ATOMIC TERM Nuclide: Various species of atoms in...

Hunds rule of Maximum Multiplicity

Hund’s Rule of Maximum Multiplicity: This...

Solved Questions of Atomic Structure Part I

Solved Questions of Atomic Structure Part I...

Dual Nature of matter and Photoelectric Effect

Dual Nature of Matter and Photoelectric Effcet...

Characteristics of Wave

Characteristics of Wave SOME IMPORTANT...

A-Scattering Experiment

A-Scattering Experiment Conclusions of...

Heisenbergs Uncertainty Principle

HEISENBERG’S UNCERTAINTY PRINCIPLE All...

Bohrs Atomic Model

BOHR’S ATOMIC MODEL Bohr developed a model...

Calculation of Energy of Electron

Calculation of Energy of an Electron: The total...

Hydrogen Atom

HYDROGEN ATOM If an electric discharge is passed...

Objective Questions of Atomic Structure

Objective Questions of Atomic Structure Problem:...

Derivation of Angular Momentum from de Broglie Equation

Derivation of Angular Momentum from de Broglie...

Plancks Quantum Theory

PLANCK’S QUANTUM THEORY When a black body is...

Electronic Configuration

Electronic Configuration Rules for filling of...

Discovery of Neutron

DISCOVERY OF NEUTRON After the discovery of...

Shapes and Size of Orbitals

Pauli’s Exclusion Principle: According to...

Limitations of Bohrs Theory

Merits of Bohr’s Theory: * The experimental...

Photoelectric Effect

PHOTOELECTRIC EFFECT Sir J.J. Thomson, observed...