010 Ionic Equilibria

8/8/2019 010 Ionic Equilibria

1/16

University of Balamand

Faculty of Medicine and Medical Sciences

Dr. M. Khachab

GPHY 402

September 13, 20

Ionic Equilibria Pag

Ionic Equilibria

Principles ofEquilibria

Learning Objectives

1. Define the electrochemical potential () of an ion

2. Write the equation for the difference between theelectrochemical potential of an ion on one side of amembrane and that on the other side of the membrane ()

What does it mean?

3. Use the Nernst Equation to determine whether an ion is inequilibrium

If the ion is not in equilibrium, determine in whichdirection the ion will tend to flow

4. Compute the electrical potential difference across amembrane that is permeable to only one ionic species

5. Identify the Gibbs-Donnan Equilibrium and explain itsimplications for distribution of ions and water between thecytosol and extracellular fluid


2/16



Dr. M. Khachab

GPHY 402

September 13, 20


Principles of Equilibria

I. The ElectrochemicalPotential of an Ion

In Which Way Will an Ion Flow

Spontaneously?In situation 1: Na+ will tend to flow from A

to B because of both theforce of diffusion(concentration gradient) andthe electrical force (electricalpotential difference or voltagegradient)

In situation 2: Na+ will tend to flow

from A to B because ofthe force of diffusion(concentration gradient)

and from B to A because of

the electrical force(electrical potentialdifference or voltagegradient)


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Dr. M. Khachab

GPHY 402

September 13, 20


In Which Way Will Na+ Net Flow Occur inSituation 2 ?

The force of diffusion andthe electrical force areopposing forces

In order to know whether anet flow of Na+ will occur,and if so, in whichdirection, we need toquantitatively comparethe strength of theopposing forces

Electrochemicalpotential () of an ion

The Electrochemical Potential () of an

Ion Defined as:

= o+RTlnC +zFE

o is the electrochemical potential in areference state (1M concentration at 20oC)

R is the ideal gas constant

T is the absolute temperature (0K)

C is the concentration of the ion (mole/L)

z is the valence of the ion

F is Faradays Number ( 96,500coulomb/mole)

E is the electrical potential (in volts)

has units of energy/mole


4/16



Dr. M. Khachab

GPHY 402

September 13, 20


The Electrochemical Potential () of an Ion

= o + RTlnC + zFE

RTlnC is the concentration term

zFE is the electrical term

RTlnC is the energy that 1 mole of ions possesses dueto concentration

zFE is the energy that 1 mole of ions possesses due tothe electrical potential

The Electrochemical Potential of an Ion

Across a Membrane ()

A = o + RTlnCA + zFEA B = o + RTlnCB + zFEB

= (A) - (B)

= RT ln (C(A) / C(B))+ zF (EA EB)


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Dr. M. Khachab

GPHY 402

September 13, 20


Meaning of

=RT ln (C(A) / C(B)) + zF(EA EB)

RT ln (C(A) / C(B)) is the energy difference between amole of ions on side A and a mole of ions on side B asa result of the concentration difference

zF(EA EB) is the energy difference between a mole ofions on side A and a mole of ions on side B as a resultof the electrical potential difference

A positive value of indicates a greater

electrochemical potential on side A than on side B A negative value of indicates a greaterelectrochemical potential on side B than on side A

Meaning of =RT ln (C(A) / C(B)) + zF(EA EB)

The net flow of an ion will spontaneously occur fromwhere its is higher to where its is lower

RTln(C(A)/C(B)) is the tendency of the ion to flow becauseof the concentration gradient (Force of diffusion)

zF(EA - EB) is the tendency of the ion to flow because ofthe electrical potential difference (Electrical force)

A positive value of indicates a tendency for the ion toflow from A to B

A negative value of of indicates a tendency for the ionto flow from B to A


6/16



Dr. M. Khachab

GPHY 402

September 13, 20


Principles of Equilibria

II. ElectrochemicalEquilibrium

An Ion in Electrochemical Equilibrium

= RT ln (C(A) / C(B)) + zF(EA EB)

At equilibrium, =0

Equilibrium occurs when the force of diffusion andthe electrical force are equal in magnitude butopposite in direction

When an ion is in electrochemical equilibrium across amembrane between side A and side B:

Its electrochemical potential on side A is equal tothat on side B

Net force on the ion = 0

The ion has no net tendency to flow in eitherdirection


7/16



Dr. M. Khachab

GPHY 402

September 13, 20


Electrochemical Equilibrium and theNernst Equation

= RT ln (C(A) / C(B)) + zF (EA EB) = 0EA EB = (-RT/zF) ln (C(A) / C(B)) = (RT/zF) ln (C(B) / C(A))

This is the Nernst Equation

The Nernst Equation allows to calculate the electricalpotential difference (EA EB) that just balances aparticular concentration ratio provided the force ofdiffusion and the electrical force are in opposite directions

The Nernst Equation holds ONLY for an ion that is inelectrochemical equilibrium

Any ion that is in electrochemical equilibrium willsatisfy the Nernst Equation

The Nernst Equation

EA EB = (-RT/zF) ln (C(A)/ C(B)) = (RT/zF) ln (C(B) /C(A))

ln(x) = 2.303 log(x)

2.303RT/F ~= 60 mV

A useful form of the Nernst Equation:

EA EB = (-60mV/z) log (C(A) / C(B))or

EA EB = (60mV/z) log (C(B)/C(A))


8/16



Dr. M. Khachab

GPHY 402

September 13, 20


Examples of Uses of the Nernst Equation

EA EB = (-60mV/z) log(C(A)/ C(B)) = (-60mV/+1) log (0.1/0.01)

EA EB = (-60mV/+1) log(10) = -(60mV)(1) = -60 mV

The Nernst equation tells us that at equilibrium, side A must be60 mV negative relative to side B

Question: If K+ is in equilibriumacross the membrane,calculateEA EB?

0.01 M KCl0.1 M KCl

A B


EA EB = (-60mv/z) log (C(A)/C(B) )= (-60mV/-1)log (1/0.1)

EA EB = (-60mv/-1) log (10)= (60mV)(1) = +60mV

If EA EB were+60 mV, the electrical force and the

diffusion forces would be equal in magnitude butopposite in direction

However, the measured electrical potential difference EA EB across the membrane is +100mv

Therefore, Cl- is not in electrochemical equilibriumacross the membrane

Question: is Cl- inelectrochemical equilibriumacross the membrane in thesituation shown in the figure?

-

-

- 0.1 M KCl

-

-

+

+

1 M KCl +

+

+

100 mV

A B


9/16



Dr. M. Khachab

GPHY 402

September 13, 20



Cl- is not in electrochemical equilibrium across themembrane

Question: How could we determine then the direction ofnet Cl- flow?

When ions are not in equilibrium, the NernstEquation can be used to predict the direction inwhich ions will flow

Cl- net flow will be from B to A

--

- 0.1 M KCl

-

-

++

1 M KCl +

+

+

100 mV

A B

Using the Nernst Equation

The Nernst Equation indicates what the membraneelectrical potential would have to bein order for an ionto be in electrochemical equilibrium

The Nernst Equation gives a theoreticalnumber

However, we can measureby a voltmeter themembrane electrical potential

The measurement gives a realnumber

Compare the potential calculated using the Nernst

Equation with the actual (measured) membraneelectrical potential to predict the direction in whichions tend to flow


10/16


11/16



Dr. M. Khachab

GPHY 402

September 13, 20


The Gibbs-Donnan Equilibrium

The cytosol contains some ionic species to which theplasma membrane is permeable, and others to which theplasma membrane is impermeable

The cytosol has a significant concentration of charged,impermeant macromolecules, such as proteins andnucleic acids

These are predominantly anionic (negatively charged)

The presence of impermeant ionic species influences theequilibrium distribution of permeant ionic species

A Gibbs-Donnan Model Situation The membrane is permeable to

K+ and Cl-, but is impermeable toX-

There is no electrical potentialdifference across the membrane

Cl- has a tendency to flow from Bto A

As Cl- flows from B to A, anelectrical potential

difference builds up acrossthe membrane (side Abecomes negative withrespect to side B)

This electrical potentialdifference provides a drivingforce for K+ to flow from B to A


12/16



Dr. M. Khachab

GPHY 402

September 13, 20


When Will Cl- and K+ Stop Flowing?

Given enough time, Cl- and K+ will come intoequilibrium

Cl- and K+ will stop flowing

Cl- and K+ will satisfy the Nernst Equation

At equilibrium, both K+ and Cl- mustequal 0

K+ = RTln([K+]A / [K+]B) + F(EA EB) = 0

Cl- = RTln([Cl-]A/[Cl-]B) - F(EA EB) = 0

The Gibbs-Donnan Equation or Donnan

Relation

K = RTln([K+]A / [K

+]B) + F(EA EB) = 0

Cl = RTln([Cl-]A/[Cl

-]B) - F(EA EB) = 0

Add the two equations, divide by RT, take antilogs of bothsides, cross-multiply, and we obtain:

[K+]A [Cl-]A = [K

+]B [Cl-]B

This is the Donnan Relation or the Gibbs-DonnanEquation

Satisfying the Donnan Relation means that both ionsare in equilibrium

Equation holds for any univalent cation and anion pairin equilibrium


13/16



Dr. M. Khachab

GPHY 402

September 13, 20


How Much Cl- and K+ Will Flow to Attainthe Gibbs Donnan Equilibrium?

The Principle of Electroneutralitystates that in any macroscopicvolume of solution, the numberof positive and negative chargesmust be equal

The number of Cl- ions thatflows from B to A must be thesame number of K+ ions thatflows from B to A

In reality, the number of Cl-

ions that flows from B to Awill be infinitesimally largerthan the number of K+ ionsthat flows from B to A

However, in chemical terms,the difference between thenumber of Cl- and K+ ionsthat flow is insignificant

How Much Cl- and K+ Will Flow to Attain

the Gibbs Donnan Equilibrium? Assume A and B have equal

volumes

If the decrease in [Cl-] and[K+] in B are equal to Z, then[K+] and [Cl-] in A increaseby Z

[X-] cannot change

Applying the Gibbs-DonnanEquation:

[K+]A[Cl-]A = [K

+]B[Cl-]B

(0.1 +Z)(Z) =(0.1 Z)2

Z = 0.0333


14/16



Dr. M. Khachab

GPHY 402

September 13, 20


Ion Concentrations After Gibbs DonnanEquilibrium has been Attained

There is a higher [K+] in Athan in B

There is a higher [Cl-] in Bthan in A

If K+ and Cl- are inelectrochemical equilibrium,both ions satisfy the NernstEquation

Applying the NernstEquation results in :

EA - EB = - 18 mV

Osmotic Imbalance The sum of [K+] and [Cl-] in A is

greater than their sum in B

X- present in A, but not in B

Consequently, there is a higherconcentration of osmoticallyactive ions in A than in B

Water will tend to flow byosmosis from B to A until thetotal osmotic pressure of the 2

solutions is equal

Unless it is restrained, all thewater from B will end up on A


15/16



Dr. M. Khachab

GPHY 402

September 13, 20


Osmotic Imbalance

Water can be restrainedfrom moving by enclosingsolution A in a rigidcontainer

As water flows from B toA, pressure will build upin A

A hydrostatic pressureof 2.99 atmospheres isrequired to prevent

water from flowingfrom B to A in theGibbs-Donnanequilibrium

How do Cells Cope with the Osmotic

Consequences of the Gibbs-Donnan Equilibrium?

Plant cells have a rigid cell wall that restrains the movement ofwater

The cell wall allows turgor pressure to build up in the cell

If the cell wall were not present, the Gibbs-Donnan Equilibriumwould cause osmotic pressure in the cytoplasm to build up inexcess of the osmotic pressure in the extracellular fluid

This build-up of pressure would threaten the maintenance of thenormal cellular volume

Animal cells have evolved ion transport processes to deal withthe osmotic consequences of the Gibbs-Donnan equilibrium

The Na+-K+-ATPase in the plasma membrane pumps Na+ out of

the cytoplasm into the extracellular fluid (ECF) The extrusion of Na+ decreases the osmotic pressure of the

cytoplasm and increases the osmotic pressure of the ECF

If ion transport mechanisms were not present, the osmoticimbalance caused by the Gibbs-Donnan Equilibrium would causeanimal cells to swell and finally burst


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Dr. M. Khachab

GPHY 402

September 13, 20

Features of the Gibbs-Donnan Equilibrium

The side with the impermeant anions (that correspondsto the cytosol) has the following features:

A greater concentration of mobile cations

A smaller concentration of mobile anions

A negative electrical potential with respect tothe other side of the membrane

A greater osmotic pressure

References

Chapter 5: Membrane Potentials and Action Potentials in

Textbook of Medical Physiology by Guyton and Hall

Chapter 7: Membrane Potential inPrinciples of NeuralScience. Edited by Eric R. Kandel, James H. Schwartz,and Thomas M. Jessell.

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010 Ionic Equilibria

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