At what energy level does the Bohr hydrogen atom have diameter 8.57 nm ?
Radius of the atom is r = n2 * h2 * Є0/π * m * e2 ….( 1 )
it is given that diameter D = 8.57 nm = 8.57 * 10-9 m
So radius is r = D/2 = 8.57 * 10-9 m/2 = 4.285 * 10-9 m
Here we can find the energy level simply by substituting value of r in the equation ( 1 ) which gives n = 9
But we can also calculate it by calculating the energy of the energy level
Energy of hydrogen atom in an energy level n is given by
E = - m * e4/8 * n2 * h2 * Є02 ………. ( 2)
From ( 1 ) and ( 2 ) we can observe that
E = ( - e2/8 π * Є0 ) * ( π * m * e2/n2 * h2 * Є0 )
= ( - e2/8 π Є0 * r) ……… ( 3 )
E = - ( 1.6 * 10-19 )2 / ( 8 * 3.14 * 8.85 * 10-12 * 4.285 * 10-9 )
= - ( 2.56 * 10-38/952.60 * 10-21 )
= - 0.00269 * 10-17 J
= - ( 0.00269 * 10-17/ 1.6 * 10-19 ) eV
= - 0.168 eV
Now energy of an hydrogen is given by E = - 13.6 eV/n2
So - 0.168 = - 13.6 /n2
Then n2 = 13.6/0.168 = 80.9 = 81 ( appr )
Finally n = 9
So the hydrogen atom is in the energy level n = 9
At what energy level does the Bohr hydrogen atom have diameter 8.57 nm ?
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