In: Physics
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?]
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