(a) Using Bohr’s second postulate of quantization of orbital angular momentum show that the circumference of the electron in the n^th orbital state in hydrogen atom is n times the de Broglie wavelength associated with it.

(b) The electron in hydrogen atom is initially in the third excited state.
What is the maximum number of spectral lines which can be emitted when it finally moves to the ground state?

Solution

i) According to Bohr’s second postulate, we have
$space space space space space mvr subscript straight n straight space equals straight space straight n fraction numerator straight h over denominator 2 straight pi end fraction rightwards double arrow straight space 2 πr subscript straight n straight space equals straight space nh over mv$

But, as per De- broglie hypothesis

$straight h over mv equals straight space straight h over straight p equals straight lambda$

Therefore, $2 πr subscript straight n straight space equals straight space nλ$ ; where is the de- broglie wavelength.
ii) Given, electron in the hydrogen atom is in the third excited state.

For third excited state, n = 4

For ground state n= 1

Now, total number of possible spectral lines is given by,
$N space equals space fraction numerator n left parenthesis n minus 1 right parenthesis over denominator 2 end fraction N space equals space fraction numerator italic 4 italic left parenthesis italic 4 italic minus italic 1 italic right parenthesis over denominator italic 2 end fraction N equals space fraction numerator italic 4 italic cross times italic 3 over denominator italic 2 end fraction equals space 6$