A thermal physics diagram of the Earth, a plant cell and an animal cell. Their macroscopically ordered structure – and anything
else they do or produce – is ultimately maintained or ‘paid for’ by a flux of energy from the sun (at 6000 K) to the night
sky (at 3 K). The entropy produced by these processes is ultimately exported into interstellar space. (A few exceptional ecosystems
on the ocean floor obtain their energy from cosmogenic chemistry rather than from sunlight, but most of us depend on photosynthesis.)
The Earth absorbs solar energy at a rate of about 5 × 1016 W, so the rate of entropy input is about 5 × 1016 W/6000 K ≅ 8 × 1012 WK− 1. The greenhouse effect notwithstanding, the Earth radiates heat at approximately the same rate, so its rate of entropy output
is 5 × 1016 W/255 K ≅ 2 × 1014 WK− 1. Thus the biosphere creates entropy at a rate of nearly 2 × 1014 W K− 1. Living things contribute only a fraction of this entropy, but that is still a huge amount. Despite the disingenuous claims
of some anti‐evolutionists, biological processes are in no danger of violating the Second Law of thermodynamics.
At equilibrium, molecules and their energies are distributed according to the Boltzmann distribution, sketched here. In states,
phases or regions having different energies E, the concentrations or numbers of molecules are proportional to exp (−E/kT), where E is the energy per molecule, k = 1.38 × 10− 23 J K− 1 is the Boltzmann constant and T is the absolute temperature.
This diagram of a polymer, fixed at one end, illustrates how the entropy term in the Gibbs free energy contributes to a force.
Force times change in length gives the work done in stretching the polymer, so we examine states with different lengths. In
the all‐trans configuration, the polymer has a length L0, but there is only one configuration corresponding to that length, so the entropy associated with the internal bending is
k ln 1 = 0. At a shorter length L, there are N possible configurations, of which only a few are shown. The entropy associated with bending is here k ln N > 0. Suppose that the extra potential energy associated with the cis bonds is ΔU, and neglect changes in the molecule's partial volume. The change in G is ΔG = ΔU − TSbend. We consider the case where TSbend > ΔU, where the shorter state is stable (has a lower G) with respect to the longer. In molecular terms, we would say that the Boltzmann distribution indicates that the ratio of
the number of molecules with length L to that with length L0 is N exp(−ΔU/kT), which for this case is > 1. Note that, when we increase the temperature, the length decreases – which also happens when
one heats a rubber band. We could also calculate the (average) force exerted by this molecule, ∂G/∂L, and, from a collection of such molecules, we can find the elastic properties of this (idealized) polymer. Finally, note
that the entropic force here becomes stronger as T increases, because T multiplies S. (In Boltzmann terms, more molecules are in the higher energy state.) This increasing effect with temperature is important
for entropic forces, including for the hydrophobic interaction, discussed later in the text and in the article The Hydrophobic
The Boltzmann distribution explains the equilibrium distribution of permeating positive (+ve) and negative ions. The electric
potential energy per unit charge is called the electric potential or voltage (Φ). The state of lowest energy for the permeating
ions would have all permeating cations on the left (low Φ) and all permeating anions on the right (assuming that we maintain
the potential difference). The highest‐entropy state would have equal concentrations of the permeating ions everywhere. Note
that this distribution applies only to freely permeating molecules that are not ‘actively pumped’. In practice, the total
number of positive charges inside a cell differs only by a very tiny fraction from the total number of negative charges. In
terms of this diagram, many of the extra intracellular anions required to approach electroneutrality are found on nonpermeating
species, while the extracellular solution will also contain many cations that are expelled from the cell by active (energy‐using)
A charged surface in an ionic solution gives a nonuniform ion distribution over a scale of nanometres (a). Positive ions are
attracted towards the interface, negative ions are repelled. The layer of excess positive ions (called a double layer) counteracts
the electrical effect at increasing distance x. (b) and (c) show how the electrical potential Φ and the ionic concentrations c+ and c− vary near the surface, but approach their bulk concentration co at large x.
The electric charge of ions has another effect that contributes to their distribution. The self‐energy or Born energy of an
ion – the energy associated with the field of the ion itself – depends upon the medium, and is lower in polar media such as
water than in nonpolar media such as the hydrophobic regions of membranes or macromolecules. Thus ions are found in only negligible
concentration in the hydrocarbon interior of membranes. This energy is also thought to affect the distribution of ions inside
aqueous pores or near protein molecules: if the ion is sufficiently close (∼nm) to a nonpolar region of membrane or macromolecule,
it has a high Born energy. This energy and the Boltzmann distribution mean that the concentration of ions of either sign is
lower than would be otherwise expected, not only in nonpolar regions, but also in the aqueous solution very near such regions.
Again, the equilibrium distribution will be a compromise between minimizing the Born energy (ions keep away from nonpolar
regions) and maximizing entropy (ions distribute uniformly everywhere).
The membrane shown in (a) is permeable to water but not to solutes. Water flows towards the more concentrated solution (which
has a lower water concentration). This flow stops and hydraulic equilibrium is achieved when the pressure P in the concentrated solution is sufficiently high. The system in (b) has a higher energy than (a), but it also has higher
entropy. Entropy will be maximized if the membrane bursts and the solutions mix freely. Using the Boltzmann distribution with
PV as the energy difference and making the approximations that solute volume and concentrations are small, one obtains the condition
for osmotic equilibrium – the value of the hydrostatic pressure difference required to stop further flow of water into the
solution side. In the case of a solution, the number fraction of water molecules, rather than their concentration, must be
used in eqn . Using the approximation that solute volume and concentration are small, one readily obtains an approximate expression for
the equilibrium pressure difference, Posm ≅ kTcs = RTCs, where the solute concentration is cs in molecules m− 3 or Cs in kmol m− 3 respectively. Note that water concentration is usually much higher (∼50 kmol m− 3) than solute concentrations, and so the proportional difference in water concentration is small. However, rates of diffusion
and permeation depend on the absolute difference. A solution of 1 kmol m− 3 or 1 mol L− 1 of nondissociating solute gives an osmotic pressure of approximately 2 MPa, or 20 atmospheres, the exact value depending
on temperature and on some corrections due to solute–solvent interactions.
The surface free energy of water. In bulk solution, water molecules are transiently hydrogen‐bonded to their neighbours. A
water molecule at an air–water surface or a hydrocarbon–air interface has fewer neighbours with which to form hydrogen bonds,
so its energy is higher. Because molecules will tend to rotate to form bonds with their neighbours, the surface is more ordered
and has a lower entropy per molecule. Together, these effects give the water surface a relatively large free energy per unit
area, which gives rise to the hydrophobic effect. The work done per unit area of new surface created is numerically equal
to the surface tension γ, which is the force per unit length acting in the surface. This is simply shown: if one expands the
surface by drawing a length L a perpendicular distance dx, the force acting along the length L is γL, so the work done is γL dx = γ dA, where dA is the increased area.
The equilibria among a lipid monomer (a), a vesicle containing N molecules (b) and a macroscopic lipid bilayer membrane (c) shows the effect of molecular aggregation and mechanical stress.