Question

Express the speed of the electron in the Bohr model in terms of the fundamental constants (me, e, h, e0), the nuclear charge Z, and the quantum number n. Evaluate the speed of an electron in the ground states of He1 ion and U911. Compare these speeds with the speed of light c. As the speed of an object approaches the speed of light, relativistic effects become important. In which kinds of atoms do you expect relativistic effects to be greatest

Answer 1

a) v = 4.37 10⁶ m / s, speed is much less than c

b) v = 2.01 10⁸ m / s, this value is 67% of the speed of light, , for which relativistic corrections should be used

Step-by-step explanation:

The bohr model for the hydrogen atom and dendroids is a classical model with a quantization of the angular momentum

let's start by using Newton's second law with the electric force

F = m a

Coulomb's law electric force

F =
`k (q_1q_2)/(r^2)`

in this case in an atom the number of protons is equal to the atomic number and there is only one electron

q₁ = Ze

q₂ = e

acceleration is centripetal

a = v² / r

we substitute

`k (Z e^2)/(r^2) = m (v^2)/(r)`

v² =
`k (Ze^2)/(m r)`

quantization is imposed without justification in this model,

L = p x r = n
`\hbar`

\hbar= h /2π

if we consider circular orbits, the speed and position are perpendicular

m v r = n \hbar

r =
`(n \hbar)/(m v)`

we substitute

v² =
`k (Z e^2)/(m) (m v)/(n \hbar)`

v =
`k (Z e^2 )/( n \hbar)`

let's apply this equation

\hbar= h / 2π

\hbar= 6.626 10-34 / 2π

\hbar= 1.05456 10⁻³⁴ J s

a) He1 ion, the atomic number of helium is 2

v =
`(9 \ 10^9 \ 2 ( 1.6 \ 10^(-19))^2 )/(n \ 1.0546 \ 10^(-34))`

v =4.3695 10⁶ / n m / s

the ground state occurs for N = 1

v = 4.37 10⁶ m / s

the relationship of this value to the speed of light is

v / c = 4.37 10⁶/3 10⁸

v / c = 1.46 10⁻²

speed is much less than c

b) the uranium ion with atomic number Z = 92

v =
`(9 \ 10^9 \ 92 ( 1.6 \ 10^(-19))^2 )/(n \ 1.054 \ 10^(-34) )`

v = 2.01 10⁸ m / s

v/c =
`(2.01 \ 10^8 )/(3 \ 10^8)`

v/c = 0.67

this value is 67% of the speed of light, for atoms with a higher atomic number the effects are increasingly important, for which relativistic corrections should be used