Modern Engineering Thermodynamics

17.3 Thermodynamics of Biological Cells 697
and that the electrochemical work required to move that same mole from infinity into the cell is
ðw EC Þ ic = μ ic + z i Fϕ ic (17.3)
then, from Eqs. (17.2) and (17.3), we find that the zero current equilibrium electrical potential difference between
the inside and outside of the cell due to the presence of species i is
ðΔϕ e Þ i = ϕ ic − ϕ io = 1
ð
z i F
w EC
Þ ic − ðw EC Þ io − ðμ ic − μ io Þ
(17.4)
The molar chemical potential of species i can be written for isothermal, dilute solutions as
μ i = μ o i + RT ln c i (17.5)
where μ i o is the molar chemical potential when c i = 1:0, R is the universal gas constant, T is the absolute temperature,
and c i is the molar concentration of i. Using Eq. (17.5), we can write Eq. (17.4) as
ðΔϕ e Þ i = 1
ð
z i F
w EC
Þ ic − ðw EC Þ io − μ o ic − μo io
To simplify the algebra, we call the first term on the right side of Eq. (17.6)
when c ic = c io : Then, Eq. (17.6) becomes
RT +
z i F ln c io
c
(17.6)
ic
Δϕ o e
i , which is the value of ð Δϕ e
ðΔϕ e Þ i
= Δϕe
o i + RT
z i F ln c io
c
(17.7)
ic
However, we still cannot measure either ðΔϕ e Þ i
or Δϕe
o : At this point, we arbitrarily assign the electrical potential
outside the cell the value zero, and we define the membrane potential E i due to species i
i
as
which is given by Eq. (17.7) as
E i = ðΔϕ e Þ i
− Δϕe
o i
At 37°C, Eq. (17.8) becomes (recall that 1 coulomb = 1 joule/volt)
E i = RT
z i F ln c io
c ic
(17.8)
Þ i
½
E i ðat 37°CÞ = 8314:3 J/ ð kgmole .KÞŠð37:0 + 273:15 KÞ
z i ð96,487 kilocoulombs/kgmoleÞ
= 26:7 millivolts . ðkgmole electrons/kgmole iÞ
z i
ln c
io
c ic
ln c
io
c ic
(17.9)
where z i is the valence of species i in kgmole of electrons per kgmole of species i. Note that z i can be either positive
or negative in this equation.
WHAT IS SO SPECIAL ABOUT A BODY TEMPERATURE OF 37°C?
As the temperature of an organism increases up to about 40ºC, the speed of its enzyme-catalyzed metabolic reactions
increases, because the molecules collide more frequently due to thermal agitation. But above 40ºC, the weak bonds that
control the functional shape of the enzymes begin to break, and they become ineffective at sustaining metabolism. For
many years, it was thought that life as we know it could not exist at temperatures above about 40ºC.
However, recently hyperthermophilic (“superheat-loving”) bacteria have been found in high-temperature environments,
such as deep sea volcanic hot vents. They grow at temperatures above 80°C and can survive to temperatures up to 113°C.
They are very tough life forms, even surviving temperatures as low as −140°C. It seems possible that they could have been
carried through space on meteoroids to populate planets.