for the current density equation (I) , solve for dV/dx when I = 0
I = - [ uz2F[C] dV/dx + uzRT d[C]/dx]
According to your equation.
dV/dx= uzRT/uz2[C] d[C]/dx
I am uncertain what your symbols mean.
It's supposed to be the current density equation I think??
My teacher also gave the form:
I = -(ƒÊ(z^2)F(C))(dv/dt)+ (UZRT d(C)/dx))
C is concentration.
z valence
F Faraday constant
U I am not sure of
R is gas constant
T is temp
and he wanted us to set I to 0 and solve.
also, he wanted us to show how
RT/zF ln (C)out/(C)in ends up to be
58millivolts log10 (C)out/(C)in
with constants:
deg K = absolute T = deg C + 273.16
1 cal = 4.2 Joules
1V = 1Joule/Coulomb
F = Faradays�f Constant (96,480 Coulombs/mol)
R = Gas Constant (1.987 cal / mol degK)
e = elementary electrical charge = 1.602x10-19 C
Avogadro�fs Number 6.02x1023 molecules/mole
I am really confused about the ln and log10 and how to get 58
dV/dx= uzRT/uz2[C] d[C]/dx
integrate with respect to x to get
v= uzRT/uz2 ln(C)
μ is mobility of ions in solution in units of cm2v-1s-1
To solve for dV/dx when I = 0 in the current density equation, let's start by rearranging the equation:
I = -[uz^2F[C] (dV/dx) + uzRT (d[C]/dx)]
Since we want to solve for dV/dx when I = 0, we can set I to 0:
0 = -[uz^2F[C] (dV/dx) + uzRT (d[C]/dx)]
Now, let's isolate dV/dx:
uz^2F[C] (dV/dx) = -uzRT (d[C]/dx)
dV/dx = -(uzRT/uz^2F[C]) * (d[C]/dx)
Simplifying further:
dV/dx = -(RT/zF) * (d[C]/dx)
Now, let's move on to the expression RT/zF ln (C)out/(C)in and how it becomes 58 millivolts log10 (C)out/(C)in.
First, let's rewrite the expression RT/zF ln (C)out/(C)in:
RT/zF ln (C)out/(C)in
Now, let's substitute the given constants:
R = 1.987 cal/mol degK (Gas constant)
T = absolute temperature in Kelvin (T = deg C + 273.16)
z = valence
F = 96,480 Coulombs/mol (Faraday's constant)
We also need to convert ln to log10:
ln (C)out/(C)in = log10 (C)out / log10 (C)in
Now, let's substitute the given conversion factors:
1 cal = 4.2 Joules
1V = 1 Joule/Coulomb
1 Coulomb = 6.02x10^23 / (1.602x10^-19) elementary charges (Avogadro's number / elementary electrical charge)
Let's combine all these constants and units:
RT/zF ln (C)out/(C)in = (1.987 cal/mol degK) * (temperature in Kelvin) / (valence * 96,480 Coulombs/mol) * ln (C)out / ln (C)in
And then convert the units:
RT/zF ln (C)out/(C)in = (1.987 * 4.2 Joules/mol degK) * (temperature in Kelvin) / (valence * 96,480 Joules/Coulomb) * ln (C)out / ln (C)in
Now let's substitute the values given:
RT/zF ln (C)out/(C)in = (1.987 * 4.2 * (temperature in Kelvin)) / (valence * 96,480) * ln (C)out / ln (C)in
Finally, let's simplify further and convert the result to millivolts:
RT/zF ln (C)out/(C)in = (1.987 * 4.2 * (temperature in Kelvin)) / (valence * 96,480) * 0.0592 * log10 (C)out / log10 (C)in
This expression simplifies to 0.0592 log10 (C)out / log10 (C)in, which represents 58 millivolts log10 (C)out / log10 (C)in.