Gibbs equation, potentials and flows

(1)

„Innowacyjna dydaktyka bez ograniczeń - zintegrowany

rozwój Politechniki Łódzkiej - zarządzanie Uczelnią,

nowoczesna oferta edukacyjna i wzmacniania zdolności

do zatrudniania osób niepełnosprawnych”

Prezentacja multimedialna współfinansowana przez

Unię Europejską w ramach

(2)

Bogdan Walkowiak

Department of Biophysics

Institute of Materials Science and Engineering

Technical University of Lodz

POTENTIALS (5)

(3)

Projekt współfinansowany przez Unię Europejską

Gibbs’ formula in the differential form

dU = TdS + dW

(dU = dQ + dW; dS = dQ/T)

various kind of work:

- volumetric work

dW

v

= - p dV

- mechanical work

dW

m

= F dl

- electric work

dW

e

= dq

- chemical work

dW

c

= dn

chemical potential

generally the work is given by the formula:

dW = d W

v

+ dW

m

+ dW

e

+ dW

c

= - p dV + F dl + dq + dn

Gibbs’ formula can be rewritten to:

dU = TdS - p dV + F dl + dq + dn

(4)

Let assume: dS = 0; dV = 0; dl = 0; dq = 0

then:

dU =

dn

m

or generally:

dU =

_i

dn

_i

i=1

Chemical Potential

(5)

When system is doing the work (-dW) a chemical potential is diminishing (-d

-d

pdV

The equation of state for ideal gass is given by the formula:

pV = nRT

R=8.31 J/mol deg – the universal constant

for isothermal process (dT = 0) valid is the equality:

p dV = - V dp

then:

-d

V dp;

and V = nRT/p

d

nRT dp/p

d

nRT dp/p

2 nRT ln p + W

1 Chemical Potential

(6)

nRT (ln p

2 - ln p

1 ) +

+

+ nRT ln p

2 for p

1 = 1

since p = nRT/V

and n/V = c

thus

+

+ nRT ln c

Chemical Potential

(7)

Electric

charge of n mols of ions is described by the equation:

q = z n F

and change in the charge amount:

dq = z F dn

or for several types of ions:

m

dq = F z

i

dn

i

i=1

the Gibbs’ formula will be expressed as:

m

dU = TdS - p dV + F dl + F z

i

dn

i

+

i

dn

i

i=1 i=1

Electrochemical Potential

(8)

Elements of the sum can be ordered:

m

F z

i

dn

i

+

i

dn

i

= dn

i

(

i

- z

i

F )

i=1 i=1

i=1

i

e

=

_i

+ z

_i

F

i

e

=

_i

eo

+ RT ln c

_i

+ z

_i

F

Electrochemical Potential

(9)

Osmotic Pressure

dp

1

2 subsystem 1:

+ RT ln c

₁

subsystem 2:

+ RT ln c

₂

+ RT ln c

₁

=

+ RT ln c

₂

RT ln c

₁

/c

₂

=

V dp

dp = (RT/V) ln c

₁

/c

₂

= (RT/V) ln c

₁

/c

₂

(10)

If the subsystem 2 contains pure solvent then c

₂

= 1, and:

= (RT/V) ln c

₁

for low concentrations of solutions true is dependence:

= RTc

₁

Van’t Hoff’ formula

(11)

Osmotic Pressure

(12)

+ -

Ag

+

C

₁

NO

₃

-

C

₂

dV

J

_d

= L

_d

d + L

_de

dV

J

_e

= L

_ed

d + L

_e

dV

For equilibrium exchange J

e

= 0, and then:

dV = - d L

ed

/L

e

and assuming that d = 0, one can write:

L

ed

/L

e

= J

d

/J

e

J

d

= c (v

- v

+

) ; J

e

= z F c (v

+ v

+

)

(13)

v

+

- v

-

1 dV = --- ---- d

u = v/E - mobility

v

+

+ v

-

z F

u

+

- u

-

RT c

1 dV = - ln

----u

+

+ u

-

z F c

2

(14)

-

+

B

-

c

₁

K

+

c

₂

B

-

c

₁

K

+

(c

₁

+ x)

K

+

(c

₂

- x)

K

+

c

₁

Cl

-

c

₂

Cl

-

x

Cl

-

(c

₂

- x)

Starting state

Equilibrium state

d

e

= 0

(15)

x (c

1 +x) = (c

2 -x) (c

2 -x)

x /( c

2 -x) = (c

2 -x) / (c

1 +x)

[Cl

1 -] / [Cl

2 -] = [K

2 +

] / [K

1 +

]

dV = (RT/F) ln ([Cl

1 -] / [Cl

2 -]) = (RT/F) ln ([K

2 +

] / [K

1 +

])

Donnan’s Equilibrium

(16)

INTERIOR

EXTERIOR c

I

/ c

E

E (mV)

Na

+

9.2 mM

Na

+

120 mM

13:1

+ 67

K

+

140 mM

K

+

2.5 mM

1:56

- 102

Cl

-

4.0 mM

Cl

-

120 mM

30:1

- 86

Nerst’s equation (membrane potential)

RT

c

z

dV = --- ln ----

z F

c

w

Membrane Potential

(17)

Goldman’s equation (with permeability coefficients)

RT

P

_K

[K

_I

+

] + P

_Na

[Na

_I

+

] + P

_Cl

[Cl

_E

-

]

dV = --- ln ---

z F

P

_K

[K

_E

+

] + P

_Na

[Na

_E

+

] + P

_Cl

[Cl

_I

-

]

For premeability coefficients P

_K

: P

_Na

: P

_Cl

= 1 : 0.04 : 0.05,

and concentrations as on previous transparency, the calculated

membrane potential is about - 90 mV

(18)

(19)

Action Potential

(20)

(21)

(22)

(23)

Transmembrane Signalling

(24)

(25)

Intracellular Signalling

(26)

(27)

(28)

(29)

„Innowacyjna dydaktyka bez ograniczeń - zintegrowany

rozwój Politechniki Łódzkiej - zarządzanie Uczelnią,

nowoczesna oferta edukacyjna i wzmacniania zdolności

do zatrudniania osób niepełnosprawnych”

Prezentacja multimedialna współfinansowana przez

Unię Europejską w ramach