Modern Engineering Thermodynamics

696 CHAPTER 17: Thermodynamics of Biological Systems
Because cell membranes are molecular machines, their
exact structure is not yet completely understood. The
most universally accepted model of a membrane is the
bimolecular lipid leaflet structure shown schematically
in Figure 17.2. In this model, the membrane structure
consists of two parallel rows of phospholipid molecules
oriented with their hydrophobic chains pointing inward
and their hydrophilic (polar) ends pointing outward.
The inner and outer surfaces of the membrane are
covered with various protein layers, and the membrane
thickness is typically 7 to 10 nanometers ð10 −9 mÞ: It is
also felt that the membrane must contain a uniform
distribution of holes, or pores, about 0.8 nm in
diameter, through which water and certain hydrated
ions can pass. Approximately 0.06% of the membrane
area is made up of these pores. The concentration of
materials inside the cell is determined exclusively by
concentration differences across the membrane.
70-100 Å Cell interior Proteins
Cell exterior
Proteins
Nonpolar
ends
Membranes of living cells maintain an electrical potential difference between the inside and outside of the cell.
With very small electrodes, a reasonably constant current can be continuously drawn from a cell. A cell can produce
electricity in this way only if it has a molecular mechanism for maintaining an unequal ion charge difference
across its membrane. Such membranes are known to contain a molecular level ion pump that transports ion
species in only one direction (into or out of the cell).
How much work is required to pump a charged ion from the solution outside the cell into the solution
inside the cell? The answer to this question can be developed by considering the transport process to be carried
out in two steps. First, consider moving an ion from infinity through a vacuum, through the membrane,
andintothecell.Assumeinthisfirststepthatthecellmembranehasnodipolelayer(i.e.,nonetchargeon
its surface) and the inside of the cell is electrically neutral. Now, as the charged ion moves from infinity to
the membrane, it encounters no resistance, so its transport work is zero. As it moves through the cell membrane,
it begins to feel electrostatic ion-solvent and ion-ion interactions. We lump all these interactions
together and call them chemical effects; therefore, the work done against these interactions in moving the
charged ion into the cell is called chemical work. The chemical work done in moving a mole of ions of chemical
species i from infinity into an uncharged cell through a dipole layer–free membrane is equal to the molar
chemical potential μ i of ion species i.
The second step in this process is to allow the membrane to have a dipole layer and allow the cell to have a net
internal charge. We call the work required to move the ion into this system of net charges the electrical work;
therefore, the total work required to move the ion from infinity through a dipole-layered membrane into a
charged cell is the sum of the chemical work plus the electrical work. This total work is called the electrochemical
work of the cell.
Define ϕ k as the electrical potential (in volts) required to transport a unit charge of species i into the cell. Then,
the electrical work required to transport one ion of species i with valence z i kgmole of electrons per kgmole of
species i (and thus a charge z i e)isz i eϕ ic ,wheree is the charge on one electron. The electrical work required to
transport 1 mole of species i into the cell is N o z i eϕ ic = z i Fϕ ic ,whereN o is Avogadro’s numberandF = N o e =
Faraday’s constant = 96,487 kilocoulombs/kgmole of electrons. Let ðw EC Þ ic be the electrochemical work required
to transport 1 mole of species i with valence z i into a cell. Then, we can write
ðw EC Þ ic = ðW EC Þ ic /n i = μ ic + z i Fϕ ic (17.1)
Unfortunately, neither μ ic or ϕ ic is directly measurable. They were introduced as conceptual quantities for the
purpose of separating chemical effects from electrical effects; however, only their combined effect can be
observed in the laboratory.
Whatcanwemeasure?Wecanmeasuretheelectricalpotential difference (i.e., voltage) between the inside and
outside of the cell. Now, as soon as we introduce an electrode into the cell, we set up a current path, so that the
measured potential Δϕ m is not the same as the zero current (no electrode) equilibrium potential Δϕ e : Again, Δϕ e
cannot be measured, but we can get around that as follows. From Eq. (17.1), we find that the electrochemical
work required to move a mole of ions of species i with valence z i from infinity into the solution outside the cell is
Polar
ends
FIGURE 17.2
Schematic of membrane construction.
ðw EC Þ io
= ðW EC Þ io
/n i = μ io + z i Fϕ io (17.2)