ELECTROSTATIC THEORY :-
In this theory, the charged ions are considered to be conducting sphere and the solvent having continuous dielectric having a fixed dielectric constant (ε)
According to electrostatic theory, Double sphere, Scientist Mathew was proposed we calculate the effect of 'k' on ε
Initiating the ions are at on infinite distance from one another and the activated state, the distance is dAB.
When the ions are separated by a distance x, the force active between them is according to coolum's law.
f ∝ ZA e. ZB e
x²
f = 1 ZA.ZB . e² .x →...(1)
4πε₀ε x²
Here,
ε = dielectric constant
ε₀ = permittivity of vacuum
[8.854 × 10 ⁻¹² c² N⁻¹ m⁻²]
The workdone on the system, in moving them together a distance dx is,
W = - ZA ZB e² dx → (2)
4π ε₀ εx²
The negative sign indicates decrease of x by dx.
The work done in moving, the ions from initial state to dAB is,
W = ZA ZB e² [ 1 − 1 ]
4π ε₀ ε dAB ∝
W = ZA ZB e² .......(3) [ 1 = 0 ]
4π ε₀ ε dAB ∝
The work is positive, if the ionic charges to the Gibb's energy of activation when two ions forms the activated complex.
∆Ges = ZA ZB e²
4π ε₀ ε dAB
Considering molar reaction,
∆Ges ≠ = L. ZA ZB e²
4π ε₀ ε dAB
Here,
L = avagadro number
The total molar Gibb's energy of activation is,
∆G ≠ = ∆G≠ nes + ∆Ges ≠
According to Eyring,
k = kBT e −∆G≠
h RT
lnk = ln kBT − ∆Gnes − L. ZA ZB e²
h RT 4π ε₀ε dAB.kT
kB = R
L
lnk = ln kBT - ∆Ges − ZA. ZB e²
h RT 4π ε₀ ε dAB kBT
lnk = lnk₀ − ZA B e²
y c mx
According to the above equation,
According to the above equation the natural logarithm of rate constant of a between ions should vary linearly with reciprocal of dielectric constant
( 1 )
ε
SINGLE SPHERE MODEL :-
Born derive another equation reaction the rate constant k with the dielectric constant (ε) based on single sphere model. The equation is,
lnk = lnk₀ − e²
8π ε₀ ε kBT
[ ( ZA + ZB )² − ZA² − ZB ]
ʋ≠ ʋA ʋB
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