Derivation of the Nernst
Equation:
[𝑪]𝒓
𝒍𝒏 =(𝑽𝒍
[𝑪]𝒍
−𝑽𝒓 )
Why else do we care?
What else?
Other health conditions besides atrial fibrillation
may result from problems with membrane
potential:
1)Cystic fibrosis—poor chloride movement
across the membranes
2)Epilepsy may be due to poorly working voltage
gated channels
Intuitive picture for Flux
We start with diffusive flux:
Concentration per volume=mol/cm^3* 1/cm
Putting them together:
𝜕𝑉
Electric Drift: 𝐽𝐷𝑟𝑖𝑓𝑡= − 𝜇𝑧[𝐶]
𝜕𝑥
𝑐𝑚2 𝑚𝑜𝑙
𝑠
𝑐𝑚4
𝑐𝑚2 𝑚𝑜𝑙 𝑉
𝑉𝑠 𝑐𝑚3 𝑐𝑚
Combining the Drift and the Diffusion:
𝜕[𝐶]
𝐽𝐷𝑟𝑖𝑓𝑡 + 𝐽𝐷𝑖𝑓𝑓= -D
𝜕𝑥
-
𝜕𝑉
𝜇𝑧[𝐶] :
𝜕𝑥
Getting everything in terms of mobility:
Replace D with the Boltzmann constant : D= 𝑘𝑇𝑞 𝜇
𝑘𝑇 𝜕[𝐶]
𝐽𝑇𝑜𝑡𝑎𝑙= - 𝜇
𝑞
𝜕𝑥
-
𝜕𝑉
𝜇𝑧[𝐶]
𝜕𝑥
More on the Boltzmann constant from Wikipedia:
The Boltzmann constant (k or kB) is a physical
constant relating energy at the individual particle
level with temperature. It is the gas constant R
divided by the Avogadro constant NA.
k=R/ NA
(See thermally agitated molecule)
Looking more like it:
𝑅𝑇 𝜕[𝐶]
𝐽𝑇𝑜𝑡𝑎𝑙= - 𝜇
𝐹
𝜕𝑥
-
Replace
𝜕𝑉
𝜇𝑧[𝐶]
𝜕𝑥
R is the ideal gas constant and
F is the Faraday constant
𝑘𝑇
𝜇
𝑞
with
𝑅𝑇
𝜇
𝐹
More on the Faraday constant from Wikipedia:
(one mole of electrons)
In physics and chemistry, the Faraday constant (named
after Michael Faraday) is the magnitude of electric charge
per mole of electrons.[1] It has the currently accepted value
F = 96,485.3365(21) C/mol.[2] The constant F has a
simple relation to two other physical constants:
where: F=eNA
e ≈ 1.6021766×10−-19 C;[3] NA ≈ 6.022141×1023 mol−1.[4]
NA is the Avogadro constant (the ratio of the number of
particles 'N' to the amount of substance 'n' - a unit mole),
and e is the elementary charge or the magnitude of the
charge of an electron.
One Mole of Particles:
Multiply both sides by F and z:
F*z*(𝐽𝑇𝑜𝑡𝑎𝑙 )=F*z* (-
𝑅𝑇 𝜕[𝐶]
𝜇
𝐹 𝜕𝑥
-
𝜕𝑉
𝜇𝑧[𝐶] )
𝜕𝑥
Cross out F in the diffusive flux; add the factor z in the drift expression
Current Flux:
𝐼𝑐𝑢𝑟𝑟𝑒𝑛𝑡 𝑓𝑙𝑢𝑥 = 𝐶𝑜𝑢𝑙.∗ 𝑣𝑎𝑙𝑒𝑛𝑐𝑒
𝑐𝑚2 𝑠
𝜕[𝐶]
𝜇𝑧𝑅𝑇
𝜕𝑥
- 𝜇𝐹𝑧
2
𝜕𝑉
[𝐶]
𝜕𝑥
Set equation=0 to get Nernst equation (no current)
0=-
𝜕[𝐶]
𝜇𝑧𝑅𝑇
𝜕𝑥
- 𝜇𝐹𝑧
2
𝜕𝑉
[𝐶]
𝜕𝑥
Factor out - 𝜇𝑧
0= -
𝜕[𝐶]
𝜇𝑧(𝑅𝑇
𝜕𝑥
+𝐹𝑧
𝜕𝑉
[𝐶] )
𝜕𝑥
(We see that -𝜇𝑧 is a solution, so we can get rid of it.)
Now we have the variables we want:
0=𝑅𝑇
𝜕[𝐶]
𝜕𝑥
+ 𝐹𝑧 [𝐶]
𝜕𝑉
𝜕𝑥
Move the diffusive flux term over to the LHS
𝜕[𝐶]
-𝑅𝑇
𝜕𝑥
𝜕𝑉
=𝐹𝑧[𝐶]
𝜕𝑥
Divide by -RT/Fz:
𝜕[𝐶]
𝜕𝑥
𝐹𝑧
𝜕𝑉
=− [𝐶]
𝑅𝑇
𝜕𝑥
Separation of Variables:
𝜕[𝐶]
𝜕𝑥
𝐹𝑧
𝜕𝑉
=− [𝐶]
𝑅𝑇
𝜕𝑥
Becomes
𝜕[𝐶]
[𝐶]𝜕𝑥
𝐹𝑧 𝜕𝑉
=−
𝑅𝑇 𝜕𝑥
𝜕[𝐶] 𝐹𝑧
=−
[𝐶]
𝑅𝑇
𝜕𝑉
Integration:
Goldman-Hodgkin
A little applet
• http://www.nernstgoldman.physiology.arizon
a.edu/#download