91Ó°ÊÓ

An electron, in a certain state n, moves with a speed equals to $7.3×{10}^5\mathrm{m}/\mathrm{s}$around the hydrogen's nucleus

Our aim is to find the radius of the hydrogen atom. An electron, moving in a circular orbit $r$around a certain nucleus, makes a standing waves where

$2\mathrm{π}{r}_n=n\mathrm{λ}$

Every particle, which has a certain momentum, has an associated wavelength where

$\mathrm{λ}=\frac{h}{p}=\frac{h}{mv}$

By substituting (2) into (1), we get

$2\mathrm{π}{r}_n=n\frac{h}{m_c{v}_n}⇒r_n=\frac{nh}{2\mathrm{π}{m}_c{v}_n}$

From Fig.(38.27) in the book, we know that.

role="math" localid="1650742362891" $r_n=n^2{a}_B\text{(where}{a}_D\text{is the bohr radius)}$

By equating (3) and (4), we get

role="math" localid="1650742483141" $$\begin{gathered} n^2{a}_B=\frac{yh}{2\mathrm{π}{m}_e{v}_n}→n=\frac{h}{2\mathrm{π}{m}_e{v}_n{a}_{Hs}} \\ ≃3 \end{gathered}$$

Hence the radius of the hydrogen atom, in which an electron moves with a speed equals to $\mathbf{7}.\mathbf{3}×\mathbf{105}\mathrm{m}/\mathrm{s}$in the third state, is