Which of the following statements are true re | vedclass

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(A) According to Bohr's model for a hydrogen atom:$(I)$ The orbital speed $v_n$ is given by $v_n \propto \frac{1}{n}$. As $n$ increases (moving away f

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$I$ and $III$

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(A) According to Bohr's model for a hydrogen atom:$(I)$ The orbital speed $v_n$ is given by $v_n \propto \frac{1}{n}$. As $n$ increases (moving away from the nucleus),$v_n$ decreases. Thus,statement $(I)$ is true.$(II)$ The radius of the $n^{th}$ orbit is $r_n \propto n^2$. Statement $(II)$ is false because it is proportional to the square of the principal quantum number.$(III)$ The frequency of revolution $f$ is given by $f = \frac{v}{2\pi r}$. Since $v \propto \frac{1}{n}$ and $r \propto n^2$,we have $f \propto \frac{1/n}{n^2} = \frac{1}{n^3}$. Thus,statement $(III)$ is true.$(IV)$ The electrostatic binding force is $F = \frac{ke^2}{r^2}$. Since $r$ increases as $n$ increases,the force $F$ decreases. Thus,statement $(IV)$ is false.Therefore,statements $(I)$ and $(III)$ are correct.

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