HSAB Theory - Chemical Hardness

Chemical Hardness

Chemical hardness in electron volt
Acids Bases
Hydrogen H+ infinite Fluoride F- 7
Aluminum Al3+ 45.8 Ammonia NH3 6.8
Lithium Li+ 35.1 hydride H- 6.8
Scandium Sc3+ 24.6 carbon monoxide CO 6.0
Sodium Na+ 21.1 hydroxyl OH- 5.6
Lanthanum La3+ 15.4 cyanide CN- 5.3
Zinc Zn2+ 10.8 phosphane PH3 5.0
Carbon dioxide CO2 10.8 nitrite NO2- 4.5
Sulfur dioxide SO2 5.6 Hydrosulfide SH- 4.1
Iodine I2 3.4 Methane CH3- 4.0
Table 2. Chemical hardness data

In 1983 Pearson together with Robert Parr extended the qualitative HSAB theory with a quantitative definition of the chemical hardness (η) as being proportional to the second derivative of the total energy of a chemical system with respect to changes in the number of electrons at a fixed nuclear environment:

The factor of one-half is arbitrary and often dropped as Pearson has noted.

An operational definition for the chemical hardness is obtained by applying a three-point finite difference approximation to the second derivative:

\begin{align} \eta &\approx \frac{E(N+1)-2E(N)+E(N-1)}{2},\\ &=\frac{(E(N-1)-E(N)) - (E(N)-E(N+1))}{2},\\ &=\frac{1}{2}(I-A), \end{align}

where I is the ionization potential and A the electron affinity. This expression implies that the chemical hardness is proportional to the band gap of a chemical system, when a gap exists.

The first derivative of the energy with respect to the number of electrons is equal to the chemical potential, μ, of the system,

from which an operational definition for the chemical potential is obtained from a finite difference approximation to the first order derivative as

\begin{align} \mu &\approx \frac{E(N+1)-E(N-1)}{2},\\ &=\frac{-(E(N-1)-E(N))-(E(N)-E(N+1))}{2},\\ &=-\frac{1}{2}(I+A), \end{align}

which is equal to the negative of the electronegativity (χ) definition on the Mulliken scale: μ = −χ.

The hardness and Mulliken electronegativity are related as

and in this sense hardness is a measure for resistance to deformation or change. Likewise a value of zero denotes maximum softness, where softness is defined as the reciprocal of hardness.

In a compilation of hardness values only that of the hydride anion deviates. Another discrepancy noted in the original 1983 article are the apparent higher hardness of Tl3+ compared to Tl+.

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