Find the wavelength associated with an electron in the state n=4 of hydrogen. (Answer: 1.33 nm) 



Given:

Atom: Hydrogen

Energy level n = 4

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

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

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

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


To Find:


The wavelength associated with electron = Î» =  ?


Solution:

The  formula for wavelength is 

 Î» = `\frac{h}{sqrt {2mE_n}}`

Where

`\E_n` = `\frac{2Ï€²k²me⁴}{n²h²}`  So,

 Î» = `\frac{h}{sqrt {frac{2m2Ï€²k²me⁴}{n²h²}}}`

 Î» = `\frac{h}{frac{2mÏ€ke²}{nh}}`

or

 Î» = `\frac{nh²}{2mÏ€ke²}`

by putting the corresponding values 



 Î» = `\frac{4 ｘ (6.62 ｘ 10⁻³⁴ m² kg s⁻¹)²}{2ｘ9.11 ｘ 10⁻³¹ kg ｘ 3.1416 ｘ 9 ｘ 10⁹ N m² C⁻² ｘ(1.6 ｘ 10⁻¹⁹ C)²}`


 Î» = `\frac{4 ｘ 43.8244 ｘ 10⁻⁶⁸ m⁴ kg² s⁻²}{515.159568 ｘ 10⁻²² kg N m² C⁻² ｘ2.56 ｘ 10⁻³⁸ C²}`

[N = kg m s⁻²]

 Î» = `\frac{175.2976ｘ 10⁻⁶⁸ m⁴ kg² s⁻²}{1318.80849408 ｘ 10⁻⁶⁰ kg m s⁻² ｘ kg m² }`

or

 Î» = 0.1329ｘ 10⁻⁸ m

or

 Î» = 1.329ｘ 10⁻⁹ m

Or

 Î» = 1.329 nm  ---------------Ans.

Thus the wavelength associated with an electron in the state n=4 of hydrogen is 121.9  nm  



************************************

Shortcut Links For 


1. Website for School and College Level Physics   
2. Website for School and College Level Mathematics  
3. Website for Single National Curriculum Pakistan - All Subjects Notes 

© 2022-Onwards by Academic Skills and Knowledge (ASK)     

Note:  Write me in the comment box below for any query and also Share this information with your class-fellows and friends.