Concise Physical Chemistry

c13 JWBS043-Rogers September 13, 2010 11:27 Printer Name: Yet to Come

RESISTIVITY, CONDUCTIVITY, AND CONDUCTANCE 205

We know that, in general,μ=μ◦+RTlna; thus, in this case, we have

μ◦+α+Fφα+RTlna+α=μ◦+β+Fφβ+RTlna+β

butμ◦+α=μ◦+βbecause we are talking about the same ion, K+, on either side of the

membrane. Differences come, not inμ◦+, which is fixed, but in the potentialsφand

the activitiesa, which are variable:

Fφα+RTlna+α=Fφβ+RTlna+β

F

(

φα−φβ

)

=RTln

a+β

a+α

Fis called theFaraday constant. It is a conversion factor from charge in coulombs

C or millicoulombs mC to moles. The faraday is the charge on 1 mol of electrons,

96,485 C mol−^1 , which may in general be regarded as 1 mol of charge.

Under the stipulation that the concentrations are rather low, the activity coefficients

γare about equal to 1.0 and we may replace the activities by molarities (or molalities)

mto find the membrane potential

(

φα−φβ

)

=

RT

F

ln

m+β

m+α

Nerve cell walls are semipermeable, accounting for an equilibrium membrane

potential of about 70 mV. This equilibrium potential can be disturbed by an electrical

pulse from a neighbor cell, whereupon the pulse is transmitted to another neighbor

cell in a sequence that is part of the mechanism for transmission of information

along a nerve fiber.

13.2 RESISTIVITY, CONDUCTIVITY, AND CONDUCTANCE

If a charge suspended in a fluid medium is subjected to a potential difference
φ,

it will move. Moving charge is called currentI=dQ/dt, whereQis the charge in

coulombs. Typically, a moving charge meets resistanceR.^1 Resistance is proportional

to the lengthlof a resistor and is inversely proportional to its cross-sectional areaA:

R∝

l

A

=ρ

l

A

The proportionality constantρ=RA/lis a characteristic of the resistor material. It

is called theresistivity.

These definitions prompt us to define the inverse of the resistance called the

conductance L≡ 1 /R, along with the inverse of the resistivity, theconductivity

κ≡ 1 /ρ:

L≡

1

R

=

A

ρl

=

κA

l

(^1) We shall not consider superconductive media here.