Question

(a) For one-electron ions with nuclear charge Z, what is the speed of the electron in a Bohr-model orbit labeled with n? Give your answer in terms of v1, the orbital speed for the n = 1 Bohr orbit in hydrogen. (b) What is the largest value of Z for which the n = 1 orbital speed is less than 10% of the speed of light in vacuum?

Short answer

(a) the equation of nth orbital speed for 1-electron ion is${\mathrm{v}}_{\mathrm{n},\mathrm{ion}}=\mathrm{Z}\frac{{\mathrm{v}}_1}{\mathrm{n}}$

(B) the largest Z value for which orbital speed is less than 10% of speed of light is 13.

Step-by-step solution

Given Data

According to de Broglie, Light has dual nature as in some situations it behaves like waves and in others like particles. Electrons, which sometimes exhibit particle nature may also behave like wave in some situations.

When a particle acts like a wave, it carries a wavelength which is known as de Broglie wavelength.

Orbital speed of nth orbit in Bohr model for hydrogen atom is:

${\mathbf{v}}_{\mathbf{n}}\mathbf{=}\frac{\mathbf{1}}{{\mathbf{\epsilon }}_{\mathbf{0}}}\mathbf{}\frac{{\mathbf{e}}^{\mathbf{2}}}{\mathbf{2}\mathbf{nh}}$

Speed of electron

For n = 1 the orbital speed of electron is given by:

${\mathrm{v}}_1=\frac{1}{{\mathrm{\epsilon }}_0}\frac{{\mathrm{e}}^2}{2\mathrm{h}}$

On comparing with nth orbital speed of electron

${\mathrm{v}}_{\mathrm{n}}=\frac{{\mathrm{v}}_1}{\mathrm{n}}$

The equation of orbital speed for ions is given by:

${\mathrm{v}}_{\mathrm{n},\mathrm{ion}}=\frac{1}{{\mathrm{\epsilon }}_0}\frac{{\mathrm{Ze}}^2}{2\mathrm{nh}}={\mathrm{Zv}}_{\mathrm{n}}=\mathrm{Z}\frac{{\mathrm{v}}_1}{\mathrm{n}}$

Therefore, the equation of nth orbital speed for 1-electron ion is ${\mathrm{v}}_{\mathrm{n},\mathrm{ion}}=\mathrm{Z}\frac{{\mathrm{v}}_1}{\mathrm{n}}$.

Largest Z value

Orbital velocity for n = 1 in hydrogen atom is given by

${\mathrm{v}}_1=\frac{1}{{\mathrm{\epsilon }}_0}\frac{{\mathrm{e}}^2}{2\mathrm{h}}=\frac{1}{8.85*{10}^{-12}}\frac{(1.6*{10}^{-19}\mathrm{C}{)}^2}{\left(2\right(6.626*{10}^{-34}\mathrm{Js})}=2.18*{10}^6\mathrm{m}/\mathrm{s}$

The orbital speed of ion should be

${\mathrm{v}}_{\mathrm{n},\mathrm{ion}}=0.10\mathrm{c}=3.0*{10}^7\mathrm{m}/\mathrm{s}$

So, Z value will be given by

$\mathrm{Z}=\frac{{\mathrm{nv}}_{\mathrm{n},\mathrm{ion}}}{{\mathrm{v}}_1}=\frac{\left(1\right)(3.0*{10}^7\mathrm{m}/\mathrm{s})}{(2.18*{10}^6\mathrm{m}/\mathrm{s})}=13.8$

Therefore, the largest Z value for which orbital speed is less than 10% of speed of light is 13.