What is the wavelength and frequency of electromagnetic radiation
having an energy of I rydberg (a rydberg is equal to 109 680 cm-1). Convert l
rydberg to electron volts and to kilojoules/mole (kJ ·mol-1).
Answers
Answer:
The idea here is to use Rydberg's equation to find the wavelength of the emitted electromagnetic radiation first, then convert this wavelength to energy using the Einstein-Planck equation.
So, the Rydberg equation looks like this
1
λ
=
R
⋅
(
1
n
2
1
−
1
n
2
2
)
, where
λ
- the wavelength of the emitted photon;
R
- the Rydberg constant, equal to
1.097
⋅
10
−
2
nm
−
1
;
n
1
- the principal quantum number of the orbital from which the transition is taking place;
n
2
- the principal quantum number of the orbital to which the transition is taking place.
Now, the first ionization energy is the energy needed to completely remove one mole of electrons from one mole of atoms in the gaseous state.
Since you didn't specify if you're looking for ionization energy per mole, I'll show you the value you'd have per atom.
So, completely removing an electron from an atom is equivalent to having
n
2
=
+
∞
.
Removing an electron from
n
1
=
1
to
n
2
=
+
∞
will require electromagnetic radiation of the wavelength
1
λ
=
R
⋅
(
1
1
2
−
1
∞
)
=
R
⋅
(
1
−
0
)
=
R
This means that you have
λ
=
1
R
=
1
1.097
⋅
10
−
2
nm
−
1
=
91.16 nm
The Einstein-Planck equation, which establishes a relationship between photon energy and its frequency, looks like this
E
=
h
⋅
f
, where
R
- the energy of the photon;
h
- Planck's constant, equal to
6.626
⋅
10
−
34
J s
;
f
- the frequency of the photon.
SInce frequency and wavelength have the following relationship
c
=
λ
⋅
f
, where
c
- the speed of light in vaccuum,
≈
3.0
⋅
10
8
ms
−
1
it follows that you can write
E
=
h
⋅
c
λ
Before doing any calculation, convert the wavelength from nanometers to meters
91.16
nm
⋅
1 m
10
9
nm
=
9.116
⋅
10
−
8
m
This means that you have
E
=
6.626
⋅
10
−
34
J
s
⋅
3.0
⋅
10
8
m
s
−
1
9.116
⋅
10
−
8
m
=
2.181
⋅
10
−
18
J
Expressed in kilojoules, this is equaivalent to
2.181
⋅
10
−
18
J
⋅
1 kJ
10
3
J
=
2.181
⋅
10
−
21
kJ
This means that if you supply this much energy to a hydrogen atom in its ground state, you will remove its electron completely.
SIDE NOTE To get the value in kJ per mole, the way it's usually reported, you need to first convert Joules to kilojoules, then use Avogadro's number.
2.181
⋅
10
−
21
kJ
atom
⋅
6.022
⋅
10
23
atoms
mole
=
1313 kJ/mol
similar question only you have to follow q rule and formula