SME
Inorganic Chemistry · Problem type
Absolute hardness and electronegativity
Use this for “Cu⁺ has I = 20.29 eV and EA = 7.73 eV. Find its absolute hardness η and its Mulliken electronegativity χ.”
Diagram
I: filled, −I → free electron, +47.29 eV
EA: empty, −EA → free electron, +5.14 eV
χ: −χ → free electron, +26.22 eV
2η: filled, −I → empty, −EA, +42.15 eV
This is an example. Type your own values to start a new problem.
η
=
(
I (eV)
−
EA (eV)
)
÷
2
=
η (eV)
χ
=
(
I (eV)
+
EA (eV)
)
÷
2
=
χ (eV)
ΔE
Gap between the levels
calculated · eV
σ
Softness
calculated · eV⁻¹
Formulas
ΔE
=
I
−
EA
42.15 = 47.29 − 5.14
η
=
ΔE
2
21.08
=
42.15
2
χ
=
I
+ EA
2
26.22
=
47.29 + 5.14
2
σ
=
1
η
0.04745
=
1
21.08
Assumptions
•
I is the energy to remove an electron from the species and EA the energy given out when it takes one on, both in eV.
•
For an ion Mⁿ⁺, I is the next ionization energy of the metal and EA the one before it (Na⁺: the second and the first).
•
Absolute hardness η is half the gap I − EA, and Mulliken’s χ is their mean; softness σ is 1 ÷ η.
•
The levels −I and −EA stand for the highest filled and lowest empty orbitals, measured from 0 eV, an electron at rest far away.
Step-by-step
WE KNOW
I = 47.29 eV, EA = 5.14 eV
FIND
gap between the levels (ΔE), absolute hardness (η), Mulliken electronegativity (χ), softness (σ)
Step 1 · Find gap between the levels (ΔE)
Use ΔE = I − EA
The gap from the filled level up to the empty one is the ionization energy less the electron affinity.
ΔE = 47.29 − 5.14
ΔE = 42.15 eV
Step 2 · Find absolute hardness (η)
Use η = ΔE ÷ 2
The hardness is half the gap.
η
=
42.15
2
η
=
21.08
eV
→
For
a
metal
ion,
η
above
about
10
eV
usually
means
a
hard
acid
(Na⁺
21.1,
Al³⁺
45.8).
Step 3 · Find Mulliken electronegativity (χ)
Use χ = (I + EA) ÷ 2
Mulliken’s electronegativity is the mean of the ionization energy and the electron affinity.
χ
=
47.29 + 5.14
2
χ
=
52.43
2
χ
=
26.22
eV
Step 4 · Find softness (σ)
Use σ = 1 ÷ η
Softness is the reciprocal of hardness.
σ
=
1
21.075
σ
=
0.04745
eV⁻¹
Check
Put the numbers back in. Both sides should match.
42.15 = 47.29 − 5.14 ✓
21.08
=
42.15
2
✓
26.22
=
47.29 + 5.14
2
✓
0.04745
=
1
21.075
✓
Topic check
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