For statistical systems, Boltzmann found that the percentage of particles having an energy E is proportional to

Percent (E) ~ e ^{- E / k T},

the "Boltzmann Distribution". This means that observed energies are most likely to be low, since Percent (E_{1}) >> Percent (E_{2}) if E_{2} >> E_{1}, and bunched together, since Percent (E_{1}) ~ Percent (E_{2}) if E_{1} ~ E_{2}. If all of the energy is kinetic energy, we see that the Boltzmann Distribution is a Gaussian function of the velocity, with mean zero and width proportional to the square root of T/m. So on the average, the velocity is zero (particles do not diffuse very quickly, as we saw in the last section), and at higher temperatures, the spread of velocities is larger than at lower temperatures (as we might expect). Similarly, more massive particles have a smaller range of velocities; since their inertia is greater, their velocities are harder to change from the average.

Note that

Percent (E_{1}) / Percent (E_{2}) =

(Percent (E_{1}) N / V) / (Percent (E_{2}) N / V)

= C_{1} / C_{2},

the ratio of concentrations of particles at energies E_{1} and E_{2}. For a reaction with concentrations c_{P} (products) and c_{R} (reactants), the change in "chemical potential" energy ("Gibb's free energy") is

DG = DG_{0} + R T ln (c_{P} / c_{R})

per mol, where DG_{0} is the chemical potential at standard temperature and R is the "gas constant" (k N_{ A}). The relationship to the Boltzmann Distribution is clear if you equate E to the chemical potential.

Harkening back to Chapter 4, we can equate the chemical potential to the electrical (membrane) potential to examine the diffusion of charged ions of a given specie through a membrane:

Z e DV = - k T ln (c_{in} / c_{out}),

the "Nernst" Equation (Z is the ionic charge). This allows us to compute the membrane potential given the intracellular and interstitial concentrations of any given ion specie. The Nernst Equation assumes that all other concentrations are equal in and outside of the membrane, and that the ions are in static (unchanging) equilibrium. We can take multiple species and steady flux into account if we assume that the electric field is constant or slowly varying inside the cell. If P is the permeability (D / membrane thickness, with dimensions of length / time) and c is the concentration in "equivalents" (concentration times |Z|), the "Goldman" Equation allows us to compute the membrane potential under these more realistic conditions:

e DV = k T ln ( (S (P c ) _{positive interstitial} +

S (P c ) _{negative intracellular} ) /

( S (P c ) _{positive intracellular} +

S (P c ) _{negative interstitial} ) ).

Since cells membranes are essentially only permeable to Cl, K and Na ions, the Goldman Equation becomes

e DV = k T ln ( (P_{K} c_{ K interstitial }+ P_{Na }c_{ Na interstitial} + P_{Cl} c _{Cl intracellular} ) /

(P_{K} c _{K intracellular }+ P_{Na} c _{Na intracellular }+ P_{Cl} c _{Cl interstitial} ) ).

Note that since cell membranes are impermeable to calcium ions, the Ca ions make good signalling devices: a chemically gated Ca channel which opens in response to a protein signal allows an extracellular event to switch on an intracellular one via the influx of Ca ions.

The next section is on heat.

If you have stumbled on this page, and the equations look funny (or you just want to know where you are!), see the College Physics for Students of Biology and Chemistry home page.

©1996, Kenneth R. Koehler. All Rights Reserved. This document may be freely reproduced provided that this copyright notice is included.

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