In the Bohr model of the hydrogen atom,let R, | vedclass

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(D) According to the Bohr model for a hydrogen atom:$1$. The radius of the $n^{th}$ orbit is given by $R_n = \frac{\varepsilon_0 n^2 h^2}{\pi m Z e^2}

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$vR$

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(D) According to the Bohr model for a hydrogen atom:$1$. The radius of the $n^{th}$ orbit is given by $R_n = \frac{\varepsilon_0 n^2 h^2}{\pi m Z e^2}$,which implies $R \propto n^2$.$2$. The speed of the electron in the $n^{th}$ orbit is given by $v_n = \frac{Z e^2}{2 \varepsilon_0 n h}$,which implies $v \propto \frac{1}{n}$.$3$. The total energy of the electron in the $n^{th}$ orbit is given by $E_n = -\frac{m Z^2 e^4}{8 \varepsilon_0^2 n^2 h^2}$,which implies $E \propto \frac{1}{n^2}$.Now,let us check the proportionality for the given options:- For $vR$: $v \cdot R \propto (\frac{1}{n}) \cdot (n^2) = n$.Therefore,the quantity $vR$ is proportional to the quantum number $n$.

In the Bohr model of the hydrogen atom,let $R, v$,and $E$ represent the radius of the orbit,the speed of the electron,and the total energy of the electron,respectively. Which of the following quantities is proportional to the quantum number $n$?

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