What are the energies in eV of quanta of wavelength? λ = 400, 500 and 700 nm? (Ans: 3.10 eV. 2.49 eV. 1.77 eV)



Data Given:

Let 

(1) `\λ_a` = 400 nm = 400 ｘ 10⁻⁹ m 

(2) `\λ_b` = 500 nm = 500 ｘ 10⁻⁹ m 

(3) `\λ_c` = 700 nm = 700 ｘ 10⁻⁹ m 


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

∴ Speed of light = 3 ｘ 10⁸ m s⁻¹


To Find:


(1) `\E_a` =  ? (in eV)   

(2) `\E_b` =  ? (in eV) 

(3) `\E_c` =  ? (in eV) 


Solution:


The general formula for finding energy for a given wavelength λ is

E = `\frac {hc}{λ}` 

where c is speed of light = 3 ｘ 10⁸ m s⁻¹


(1) `\E_a` =  ? (in eV)  

`\E_a` = `\frac {hc}{λ_a}`

by putting values

`\E_a` = `\frac { 6.62 ｘ 10⁻³⁴ m² kg s⁻¹ ｘ 3 ｘ 10⁸ m s⁻¹
c}{400 ｘ 10⁻⁹ m}` 

`\E_a` = `\frac { 19.86 ｘ 10⁻²⁶ m³ kg s⁻²}{400 ｘ 10⁻⁹ m}`

`\E_a` = 0.04965ｘ 10⁻¹⁷ J

`\E_a` = 4.965ｘ 10⁻¹⁹ J

Expressing in eV we have

`\E_a` = `\frac {4.965ｘ 10⁻¹⁹ }{1.6 ｘ 10⁻¹⁹ }` eV

`\E_a` = 3.103  eV --------------Ans.1




(2) `\E_b` =  ? (in eV)  

`\E_b` = `\frac {hc}{λ_b}` 

by putting values

`\E_b` = `\frac { 6.62 ｘ 10⁻³⁴ m² kg s⁻¹ ｘ 3 ｘ 10⁸ m s⁻¹
c}{500 ｘ 10⁻⁹ m}`

`\E_b` = `\frac { 19.86 ｘ 10⁻²⁶ m³ kg s⁻²}{500 ｘ 10⁻⁹ m}`

`\E_b` = 0.03972ｘ 10⁻¹⁷ J

`\E_b` = 3.972 10⁻¹⁹ J

Expressing in eV we have

`\E_b` = `\frac {3.972 10⁻¹⁹ｘ 10⁻¹⁹ }{1.6 ｘ 10⁻¹⁹ }` eV

`\E_b` = 3.483  eV --------------Ans.2





(3) `\E_c` =  ? (in eV)  

`\E_c` = `\frac {hc}{λ_b}`

by putting values

`\E_c` = `\frac { 6.62 ｘ 10⁻³⁴ m² kg s⁻¹ ｘ 3 ｘ 10⁸ m s⁻¹
c}{500 ｘ 10⁻⁹ m}`

`\E_c` = `\frac { 19.86 ｘ 10⁻²⁶ m³ kg s⁻²}{700 ｘ 10⁻⁹ m}`

`\E_c` = 0.02837ｘ 10⁻¹⁷ J

`\E_c` = 2.837 ｘ 10⁻¹⁹ J

Expressing in eV we have

`\E_c` = `\frac {2.837 ｘ 10⁻¹⁹ｘ 10⁻¹⁹ }{1.6 ｘ 10⁻¹⁹ }` eV

`\E_c` = 1.773  eV --------------Ans.3

 


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