The orbital electron of a hydrogen atom moves with a speed of 5.456 ｘ10⁵ m s⁻¹ 

(a) find the value of the quantum number 'n' associated with this electron,

(b) calculate the radius of this orbit and (c) the energy of the electron in this orbit? 

(Ans: n = 4. r₄ = 0.846 nm: E₄ = -0.85 eV)



Data Given:

Hydrogen Atom 

Speed of the electron = v = 5.456 ｘ10⁵ m s⁻¹ 

∴ Plank's Constant = h =  6.62 ｘ 10⁻³⁴ m² kg s⁻¹

∴ Coulomb's Constant = K = 9 ｘ 10⁹ N m² C⁻²

∴ Charge on the electron = e = 1.6 ｘ 10⁻¹⁹ C

∴ Mass of the electron = m = 9.11 ｘ 10⁻³¹ kg

∴ Radius of 1st Bohr Orbit = r = 0.53 ｘ 10⁻¹⁰ m


To Find:

(a) Number of Quantum No. = n =  ?

(a) Radius of the energy level = `\r_n` =  ?

(b) Energy of the energy level = `\E_n` =  ?




Solution:

(a) Number of Quantum No. = n =  ?

The  general formula for finding the velocity of an electron in any orbit is 

 `\v_n` = `\frac{2πke²}{nh}`

or

 n = `\frac{2πke²}{hv_n}`

by putting the corresponding values 

n = `\frac{2 ｘ 3.1416 ｘ 9 ｘ 10⁹ N m² C⁻² ｘ (1.6 ｘ 10⁻¹⁹ C)²}{6.62 ｘ 10⁻³⁴ m² kg s⁻¹ ｘ 5.456 ｘ10⁵ m s⁻¹ }`

n = `\frac{56.5488ｘ 10⁹ kg m s⁻² m² C⁻² ｘ 2.56 ｘ 10⁻³⁸ C²}{ 36.11872 ｘ 10⁻²⁹ m³ kg s⁻²}`

n = `\frac{144.764928 ｘ 10⁻²⁹ kg s⁻² m³ }{  36.11872 ｘ 10⁻²⁹ m³ kg s⁻²}`

n = 4.008

or

n = 4 ---------------Ans.1


(b) Radius of the energy level = `\r_n` =  ?

The general formula for the radius of quantized orbit for a Hydrogen atom is 

`\r_n` = 0.053 nm ｘn²

for n = 4 (calculation Ans.1)

`\r_4` = 0.053 nm ｘ(4)²

`\r_5` = 0.053 nm ｘ16

`\r_5` = 0.848 nm  ------------------Ans.2




(c) Energy of the energy level = `\E_n` =  ?

The general formula for the radius of quantized orbit for a Hydrogen atom is 

`\E_n` = -`\frac {E_0}{n^2}`

Where `\E_0` = constant = 2.179 ｘ10⁻¹⁸ J  = 13.6 eV and n = 4 (calculation Ans.1) So,


`\E_n` = - `\frac {2.179 ｘ10⁻¹⁸ J}{4^2}`

`\E_n` = - `\frac {2.179 ｘ10⁻¹⁸ J}{16}`

`\E_n` = - 0.1361875 ｘ10⁻¹⁸ J


By expressing in eV  ( 1 eV = 1.602 ｘ10⁻¹⁹ J)

`\E_n` = - `\frac {0.1361875 ｘ10⁻¹⁸ J }{1.6 ｘ10⁻¹⁹}` eV

`\E_n` = - 0.0851171875 ｘ10¹ eV

or

`\E_n` = - 0.851  eV  ----------------Ans. 3


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