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Validating the Bohr's Model

Suppose an el...

Question

Suppose an electron is attracted towards the originby a force $\frac{k}{r}$ where 'k' is a constant and 'r' is the distance of the electron from the origin. By applying Bohr model to this system, the radius of the $n^{t h}$ orbital of the electrons is found to be ' $r_{n}$ ' and the kinetic energy of the electron to be ' $T_{n}$ '. Then which of the following is true?

T_{n} \propto \frac{1}{n^{2}}, r_{n} \propto n^{2}

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T_{n}

independent of n,

r_{n} \propto n

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T_{n} \propto \frac{1}{n}, r_{n} \propto n

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T_{n} \propto \frac{1}{n}, r_{n} \propto n^{2}

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Solution

The correct option is B $T_{n}$ independent of n, $r_{n} \propto n$
$\frac{k}{r} = \frac{m v^{2}}{r}$
$\Rightarrow m v^{2} = k$ (Independent of n)
Hence, $T = \frac{1}{2} m v^{2}$ is also independent of n.
Now, $m v r = \frac{n h}{2 π}$
So, $r \propto n$

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