46
Electrochemical Supercapacitors for Energy Storage and Delivery
2
2
ψ
d x 1 d ⎛ dψ ⎞
x
,
dx
2
=
⎜
⎟
2 dψ x ⎝ dx ⎠
Equation (2.10) can be rewritten as:
⎛
⎞
2
dψ
2
⎛ z F ⎞
d ⎜
x ⎟ = −
∑ z FC
o exp ⎜ −
i ψ x
i
⎟ d dψ
⎝ dx
x
(2.11)
⎠
ε ε
r o
⎝ RT ⎠
dψ
When integrating the conditions x → ∞, ψ
x
x = 0 and
= 0
dx
into Equation (2.10), Equation (2.12) is obtained:
⎛ dψ ⎞
2
x
2RT
∑
o
⎡ ⎡
⎛ z F
i ψ ⎞ ⎤ 2RTC
o ⎡
⎛ z Fψ ⎞
⎛ z Fψ ⎞ ⎤
⎜
⎟ = −
C ⎢exp⎜−
x ⎟ − 1⎥ =
⎢exp⎜
i
x ⎟ − exp ⎜
i
x ⎟⎥
⎝ dx ⎠
ε ε
r o
⎣
⎝ RT ⎠ ⎦
ε ε
r o
⎣
⎝ RT ⎠
⎝ RT T ⎠⎦
(2.12)
At the point of x = d, ψ x = ψ 1 , Equation (2.12) can be alternatively expressed as:
⎛ dψ ⎞
2
2RTC
o ⎡
⎛ z Fψ ⎞
⎛ z Fψ ⎞ ⎤
⎜
x ⎟ =
⎢exp⎜
i
1 ⎟ − exp x ⎜ −
i
1 ⎟⎥
(2.13)
⎝ dx ⎠
ε ε
x d
=
r o
⎣
⎝ RT ⎠
⎝ RT ⎠⎦
In order to obtain the total net charge within the diffuse layer, Equation (2.9)
can be integrated from x = d to x = ∞, resulting in:
x=∞
⎛ dψ
⎜
⎞
x
1
q
⎟ =
ρdx = −
(2.14)
⎝ dx ⎠ x d
=
ε ε
r o
∫
ε ε
r o
x d
=
Combining Equations (2.13) and (2.14), the total net charge within the diffuse
layer becomes:
⎡
⎛ z F ⎞
⎛ z F ⎞⎤
q
i ψ
ψ
= 2ε ε RTC
o ⎢ p ⎜
1
r o
ex
⎟ − exp ⎜ −
i
1 ⎟⎥
(2.15)
⎣
⎝ RT ⎠
⎝ RT ⎠ ⎠⎦
Because ρ is expressed as μC.m –3 , q in Equation (2.15) should be in μC.m –2
units.