Formula sheet: Atoms

1.1 How to use this sheet

This page collects every relation the chapter uses, in the order the chapter develops them. Each group has the formula, the meaning of every symbol and the unit to quote. Learn the three numbers first, because most numerical questions in this chapter fall out of them: the Bohr radius goes as n squared, the speed of the electron goes as 1 over n, and the total energy goes as 1 over n squared with a negative sign.

1.2 Rutherford scattering

Coulomb force between an alpha particle of charge 2e and a nucleus of charge Ze at separation r:

F = (1/4πε0) (2e)(Ze)/r squared

Distance of closest approach in a head-on collision, where all the kinetic energy of the alpha particle becomes electrical potential energy:

d = (1/4πε0) (2Ze squared)/K with K the initial kinetic energy. For a natural alpha particle K is at most 7.7 MeV.

1.3 Bohr model

Quantisation of angular momentum, the second postulate:

L = m v r = n h/(2π), n = 1, 2, 3, ...

Radius of the nth orbit:

r n = (4πε0) n squared h squared/(4π squared m e squared) = n squared a0, a0 = 0.53 Å

Speed of the electron:

v n = e squared/(2 ε0 n h) = 2.18 × 10 to the 6/n m/s

Total energy, kinetic energy and potential energy:

E n = -(m e to the fourth)/(8 ε0 squared n squared h squared) = -13.6/n squared eV

K n = 13.6/n squared eV, U n = -27.2/n squared eV, so K = -E and U = 2E.

1.4 Transitions and spectra

Energy of a photon emitted or absorbed:

hν = E i - E f

Rydberg relation for the wavelength of a spectral line:

1/λ = R (1/n f squared - 1/n i squared), R = 1.097 × 10 to the 7 per metre

Photon energy from wavelength in nanometres, using hc = 1240 eV nm:

E in eV = 1240/λ in nm

The series are fixed by the lower level: Lyman has n f = 1 and lies in the ultraviolet, Balmer has n f = 2 and its first lines are visible, Paschen has n f = 3 and lies in the infrared, Brackett has n f = 4, Pfund has n f = 5.

1.5 de Broglie link

Matter wavelength:

λ = h/(m v)

Standing-wave condition on the nth orbit:

2π r n = n λ

1.6 Constants and conversions

e = 1.6 × 10 to the -19 C, m = 9.1 × 10 to the -31 kg, h = 6.6 × 10 to the -34 J s, 1/4πε0 = 9.0 × 10 to the 9 N m squared/C squared, 1 eV = 1.6 × 10 to the -19 J, 1 fm = 10 to the -15 m, 1 Å = 10 to the -10 m. Also remember 13.6 eV = 2.2 × 10 to the -18 J, the magnitude of the ground-state energy of hydrogen.