Question

If the de-Broglie wavelength of the electron in $n_{th}$ Bohr orbit in a hydrogenic atom is equal to $1.5a_0$ ($a_0$ is Bohr radius), then the value of $\left(\frac{n}{Z}\right)$ is:

• A. $1.0$
• B. $0.75$
• C. $0.40$
• D. $1.0$

Answer 1

The correct option is B $0.75$

According to Bohr’s model of H-atom
$Numberofwaves=\frac{Circumference}{Wavelength}$
⇒ $n=2\pi r/\lambda$
∴ $2\pi r=n\lambda$
$n$= principal quantum number
$\lambda$ = wavelength of electron ...(i)
Also, we know that radius ($r$) of an atom is given by
$r=\frac{a_0{n}^2}{Z}$
Thus, Eq. (i) becomes $\left(\frac{2\pi a_0{n}^2}{Z}\right)=n\lambda$ ...(ii)
$\left(\frac{2\pi a_0{n}^2}{Z}\right)$ = $n\times 1.5\pi a_0$ [Given, $\lambda =n\times 1.5\pi a_0$]
⇒ $\frac{n}{Z}$ = $\frac{1.5\pi a_0}{2\pi a_0}$
= $\frac{1.5}{2}$
= $0.75$