Question

In: Physics

Absorpstion spectrum of hydrogen shows line in 404.7 nm ; 435.8 nm and 486.3 nm. Determine:...

Absorpstion spectrum of hydrogen shows line in 404.7 nm ; 435.8 nm and 486.3 nm.

Determine:
a. Transtition for each line above
b. Ionization energy of Hydrogen

[Question doesn't state about spectral series (Lyman, Balmer.....). I am confused about this part. How I determine n1?]

Solutions

Expert Solution

a) Given wavelengths fall in visible region. So the transition is related to Balmer series.

Relation between wavelength and corresponding states of transition is given by rydberg formula ,

1 / = R * [ 1 /n12 - 1/ n22 ] ............... ( 1 )

where R = 1.09677 * 107 / m is Rydberg constant

and n1 , n2 are initial and final states of electron transition

( i ) Here n1 = 2 , as the transition corresponds to balmer series

wavelength given is = 404.7 nm = 404.7 × 10-9 m

Then using Rydberg relation ,

1 / ( 404.7 × 10-9 ) = 1.09677 × 107 ×  [ 1/22 - 1 / n22 ]

0.00247 × 109 = 1.09677 × 107 × ( 0.25 - 1/n22 )

0.25 - 1/ n22 = ( 0.00247 × 109 / 1.09677 × 107 )

0.25 - 1/n22 = 0.225

1/n22 = 0.25 - 0.225

1/n22 = 0.025

n22 = 1 ÷ 0.025

n2 = 40

n2 = 6.32 = 6

So electron jumps from n1 = 2 to n2 = 6 to produce a line of wavelength 404.7 nm

( ii ) Wavelength = 435.8 nm

So we can write ,

1/ ( 435.8 × 10-9 ) = 1.09677 × 107 [ 1/4 - 1/n22 ]

( 0.00229 × 109 ) / ( 1.09677 × 107? ) = ( 0.25 - 1/n22 )

0.2092 = 0.25 - 1/n22

1 /n22 = 0.25 - 0.2092

n22 = 1 / 0.041

n2 = 24.3

n2 = 5 ( approximately )

So a transition from n1 = 2 to n2 = 5 gives the wavelength 435.8 nm

( iii ) Wavelength = 486.3 nm = 486.3 × 10-9 m

Using Rydberg formula again , we can write

1 / ( 486.3 × 10-9 ) = 1.09677 × 107 × [ 1/4 - 1/n22 ]

0.00205 × 109 / ( 1.09677 × 107 ) = ( 0.25 - 1/n22 )

0.1874 = 0.25 - 1/ n22

1/n22 = 0.25 - 0.1874

n22 = 1 / 0.0626

n2 = 15.97

n2 = 4

( b ) When hydrogen atom is ionized , electron is said to move to a quantum state n2 = infinity from ground state n1 = 1

So we can write Rydberg equation as follows in this case

1/ = 1.09677 × 107 × ( 1/ 12 - 1 / 2 )

1/ = 1.09677 × 107

   = 1 / 1.09677 × 107 = 0.911 × 10-7 m

Corresponding energy E = h c /

= ( 6.626 × 10-34 × 3 × 108 ) / ( 0.911 × 10-7? )

= 2.18 × 10-19 J

= ( 2.18 × 10-19 / 1.6 × 10-19 ) eV

= 13.6 eV

So ionization energy of hydrogen is E = 13.6 eV


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