Bohr's
radius for the nth Orbit: (n = principal quantum number)

Let

R = Bohr' radius for an atom of atomic number Z,

n = orbit number = principal quantum number = 2

h = Planck's constant = 6.626 * 10⁻³⁴ units

K = 1/(4πε₀) = 9
* 10⁹ N-m²/C² = Coulomb's constant

Z = 2 for Helium

m = mass of an electron = 9.1 * 10⁻³¹ kg

e = charge on the electron = 1.602 * 10⁻¹⁹ C

1) centripetal force = electrostatic attraction between an electron and protons.

m v² / R = K (Z*e) * e / R²

=>** v² = K Z e² / (m R) --- (1)**

2) Angular momentum = m v R = n h / 2π (integral multiple of h/2π)

=> ** v = n h / (2 π m R) ---
(2)**

3) from (1) and (2):

n² h² / (4π² m² R²) = K Z e² / (m R)

=>** R = n² h² / (4π² m K e²
Z) ** --- (3)

4) So speed of electron (linear along the circular orbit) by substituting value of R,

=> **v = (2 π K e² Z) / (n h)**

5) **Potential energy of the electron:**

We ignore gravitational potential energy here.

PE = - K * Z * e * e / R = - K Z e² / R --- (4)

= **- [4 π² m K² Z² e⁴ ] / (n² h²)**

6) **Kinetic energy of electron:**

=> 1/2 * m * v² = (π m * R e² Z ) / (n h)

=** [ 2 π² K² Z² e⁴ m ] / (n² h²) ** = - P.E / 2

=========================================

7) The total energy of the electron : (a simple formula)

KE + PE = P.E / 2

**Total energy = - 13.6 Z² / n² eV **=** - 13.6 e V **

So **K.E. = 13.6 e V ** and ** P.E. = - 27.2 e V**

Bohr's Radius of Hydrogen atom R for n = 1 is 0.529 °A

So for** Helium in n =2, R = n² * 0.529 / Z °A = 1.058 °A**

**Speed of electron in Hydrogen **(n = 1) is v = 2,185 km/s (**≈ speed of light / 137) **

so for **Helium** in n = 2, v = z * 2,185 / n km/s

So* v = 2, 185 km/s *