Answer:
a) E = - (2π² k m e²/h²) 1 / n²
b) r = n² h² / (4π² m k e²)
c) f = - (2π² k m e² / h³) 1 / n²
Step-by-step explanation:
a) It is asked to find the energy of the Hydrogen atom, in this case the electron is subjected to an electrical force by the nucleus that has a positively charged proton, let's write Newton's second law
F = m a
the electric force is
F = k qq / r²
the acceleration is centripetal
a = v² / r
we substitute
k q² / r² = m v² / r
v² = k q² / r m (1)
now we can use the energy of the electron
E = K + U = ½ m v² + k q² / r
E = ½ m (k q² / r m) - k q² / r
E = - ½ q² / r
the charge q is the charge of the electron q = e
as the energy is negative it indicates that the system is stably linked
now let's use the angular momentum
L = n h / 2π
the angular momentum of an electron is
L = p r = m v r
we substitute
m v r = n h / 2π
v = n h / 2π 1/mr (2)
we substitute the equation 2 in 1
n² h² / (4π² m² r²) = k e² / (m r)
n² h² / (4π² m r) = k e²
r = n² h² / (4π² m k e²)
we substitute in the energy equation
E = - ½ e² (4π² m k e²) / h² n²
E = - (2π² k m e²/h²) 1 / n²
b) r = n² h² / (4π² m k e²)
c) to find the frequency we use the Planck equation
E = h f
f = E / h
f = - (2π² k m e² / h³) 1 / n²