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  • Using Bohr’s atomic model, derive the expression for the radius of orbit of the revolving electron in a hydrogen atom.   <div style="float:lef

Using Bohr’s atomic model, derive the expression for the radius of n^{th} orbit of the revolving electron in a hydrogen atom.

 

 
 
 
 
 

Answers (1)

According to Bohr's postulates, 

L_n=mv_nr_n=\frac{nh}{2\pi }

For a dynamically stable orbit in a hydrogen atom (Z=1)

F_e=F_c

 \frac{mv_n^2}{r_n}=\frac{1}{4\pi \epsilon _0}\frac{e^2}{r_n^2}

mv_n^2=\frac{1}{4\pi \epsilon _0}\frac{e^2}{r_n}

v_n^2=\frac{e^2}{4\pi \epsilon _0mr_n}

v_n=\frac{e}{\sqrt{4\pi \epsilon _0mr_n}}\: \: \rightarrow (1)

Also, v_n=\frac{nh}{2\pi mr_n}\: \: \rightarrow (2)

Equating (1) and (2),

\frac{e}{\sqrt{4\pi \epsilon _0 mr_n}}=\frac{nh}{2\pi mr_n}

\frac{e^2}{4\pi \epsilon _0 mr_n}=\frac{n^2h^2}{4\pi^2 m^2r_n^2}

\frac{e^2}{\epsilon _0}=\frac{n^2h^2}{\pi mr_n}

r_n=\frac{n^2h^2\epsilon _0}{\pi me^2}

Posted by

Safeer PP

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