Suppose an electron is attracted towards the origin by a force  k / r where k is constant r is the distance of the electron from the origin. By applying Bohr model to this system, the radius of the  nth orbital of the electron is found to be rn and the kinetic energy of the electron to be Tn Then which of the following is true?

  • Option 1)

    T_{n}\alpha \frac{1}{n},r_{n}\alpha\: n^{2}

  • Option 2)

    T_{n}\alpha \frac{1}{n^{2}},r_{n}\alpha\: n^{2}

  • Option 3)

    T_{n}\: Independent \: of \: \: n,\: \: \: r_{n}\alpha\: n

  • Option 4)

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

 

Answers (1)

As we learnt in

Bohr quantisation principle -

mvr=\frac{nh}{2\pi } \\2\pi r= n\lambda

- wherein

Angular momentum of an electron in  stationary orbit is quantised.

 

 

Supposing that the force of attraction in Bohr atom does not follow inverse square law but inversely proportional to r ,

\frac{1}{4\pi \varepsilon _{0}}\frac{e^{2}}{r} would have been     = \frac{m\nu ^{2}}{r}

\therefore m\nu ^{2}= \frac{e^{2}}{4\pi \varepsilon _{0}}= k\Rightarrow \frac{1}{2}m\nu ^{2}= \frac{1}{2}k

This is independent of n 

From m\nu r_{n}= \frac{nh}{2\pi }

as\: m\nu \: \: is \: independent \: of\: r,r_{n}\alpha n

Correct option is 3.


Option 1)

T_{n}\alpha \frac{1}{n},r_{n}\alpha\: n^{2}

This is an incorrect option.

Option 2)

T_{n}\alpha \frac{1}{n^{2}},r_{n}\alpha\: n^{2}

This is an incorrect option.

Option 3)

T_{n}\: Independent \: of \: \: n,\: \: \: r_{n}\alpha\: n

This is the correct option.

Option 4)

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

This is an incorrect option.

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