Determine the radius, momentum, and energy of the nth orbital of hydrogen or any other atom
Derivation of Radius, Velocity, and Energy Equations for the n-th Orbit of Hydrogen and Hydrogen-like Atoms
1. Determination of the Radius of the n-th Orbit:
Let the total positive charge in the nucleus of a hydrogen or any hydrogen-like atom be = Ze (where Z = atomic number and e = charge of a proton or an electron). In a hydrogen atom, a single electron revolves around the nucleus in a circular orbit of radius r with a velocity v.
According to Coulomb’s law, the electrostatic force of attraction exerted by the nucleus on the electron (i.e., the inward centripetal force) is—
Again, for the electron revolving with a velocity v, the outward centrifugal force is—
[Here, m = mass of the electron and r = radius of the orbit]
The essential condition for an electron to remain in a stable circular orbit is that the centripetal force and centrifugal force must be perfectly balanced and equal to each other. Therefore—
or,
or, mv2r = Ze2
or, v2 =
Furthermore, according to Bohr’s postulate, the angular momentum of an electron in a stable stationary orbit is given by—
or, v =
or, v2 =
Equating expressions (iii) and (iv), we get—
or, Ze2 =
or, Ze2 · 4π2mr = n2h2
or, r =
Equation (v) is used to determine the radius of any specific stationary orbit of a revolving electron.
2. Determination of the Velocity of an Electron Revolving in a Bohr Orbit:
According to the principles of Bohr’s atomic model—
or, r =
Again, we know that the radius of the orbit is given by—
From equations (i) and (ii), we get—
or,
or, v =
For a hydrogen atom (Z = 1), the orbital velocity of the electron in different orbits is—
3. Determination of the Total Energy Expression of an Electron in the n-th Orbit:
The total energy (En) of an electron in the n-th orbit of an atom is equal to the sum of its Potential Energy and Kinetic Energy.
or, En =
[From equation (iii) of the radius derivation, we know: mv2 =
or, En =
or, En =
or, En =
or, En =
Now, substituting the expression for radius r from equation (v), we get—
For a hydrogen atom (Z = 1), the total energy in the n-th orbit is given by—
Values of Essential Constants for Determining Radius, Velocity, and Energy:
| Sl. No. | Constant / Parameter | Symbol | C.G.S Unit System | M.K.S / S.I Unit System |
|---|---|---|---|---|
| 1 | Velocity of Light | c | 3 × 1010 cm s-1 | 3 × 108 m s-1 |
| 2 | Planck’s Constant | h | 6.626 × 10-27 erg s | 6.626 × 10-34 J s |
| 3 | Mass of Electron | m | 9.108 × 10-28 g | 9.108 × 10-31 kg |
| 4 | Charge of Electron | e | 4.8 × 10-10 e.s.u | 1.602 × 10-19 C |
| 5 | Rydberg Constant | RH | 109678 cm-1 | 10967800 m-1 |
| 6 | Energy of 1 eV Electron | E | 1.6 × 10-12 erg | 1.6 × 10-19 J |
| 7 | Radius (1st Orbit of H) | r | 5.292 × 10-9 cm | 5.292 × 10-11 m |
| 8 | Velocity (1st Orbit of H) | v | 2.18 × 108 cm s-1 | 2.18 × 106 m s-1 |
| 9 | Total Energy (1st Orbit of H) | En | -2.18 × 10 Granger-11 erg | -2.18 × 10-18 J |
1 erg = 1 g cm2 s-2 | 1 e.s.u = 1 g1/2 cm3/2 s-1
(i) 1 m = 102 cm (or, 1 cm = 10-2 m) (ii) 1 cm s-1 = 10-2 m s-1 (iii) 1 erg = 10-7 J (or, 1 J = 107 erg)
Task: Determine the radius of the first orbit, and the velocity and total energy of the electron in that orbit, for H, He+, Li2+, and Be3+.
Derivation of the Rydberg Equation (From Bohr’s Corollaries):
When an electron transitions (de-excites) from a higher orbit (n2) to a lower orbit (n1), the frequency of the emitted electromagnetic radiation is ν. According to Bohr’s theory, it can be written as:
or, h ·
or,
or,
or,
or,
The Rydberg Equation
1. For Hydrogen atom (Z = 1), the Rydberg equation is expressed as:
2. For Hydrogen-like ions (e.g., He+, Li2+, Be3+, etc.), the equation becomes:
Theoretical Determination of the Rydberg Constant (RH)
According to Bohr’s theory, the formula for the Rydberg constant is:
The experimentally determined actual value of the Rydberg constant is 109,678 cm-1. Since the value calculated theoretically using Bohr’s postulates is exceptionally close to the experimental value, it conclusively demonstrates that Bohr’s atomic model is highly realistic, valid, and accurate.
Key Formulas of the Chapter (At a Glance)
