## Free energy and the properties of solutions

In biological systems, the largest flow of water is from soil, through plants to the atmosphere - the so-called soil-plant-atmosphere continuum. Under natural conditions, liquid water enters the system following rainfall - it may be distant rainfall with subsequent surface or subsurface flow, but precipitation of liquid water starts the cycle - and water ends up in the gas phase, in the atmosphere. Little (only about 2%; see Munns, 2005) of the water flowing in the system is retained in plants.

### 3.3.1 Free energy and chemical potential

From a thermodynamic point of view, the soil-plant-atmosphere system can be seen as a system that operates at approximately constant temperature and pressure. Atmospheric pressure does vary, but variations are normally small (about 10% of the average atmospheric pressure at sea level, which is 101.3 kPa): the extremes are about 87 and 109 kPa (http://en.wikipedia.org/wiki/Atmospheric_pressure). Temperature also varies (the extremes on the world surface are about -90 to +58°C; http://en.wikipedia.org/wiki/Temperature_extreme), but it is assumed that temperatures are virtually constant for short periods - where a temperature rises from 10 to 40°C over 6 h, then that would be an increase of only about 0.08°C min-1. For a spontaneous change to occur in such a system at constant temperature and pressure, there must be a decrease in free energy. So, if water moves spontaneously from, say, plant to atmosphere, the free energy of the water must decrease.

Free energy - formally, Gibbs free energy (G) - decreases for spontaneous processes at constant temperature and pressure. However, free energy is an extensive property; it depends, like mass, on the quantity of a substance: the bigger the system, the more the free energy. Hence it is important to be able to assess the amount of free energy of a substance independent of its quantity. This free energy per mole of substance is termed the chemical potential (x). For water, its chemical potential (xw) is given by the equation:

where nw represents the number of moles of water, T is the temperature, P is the pressure, E is the electrical potential, h is the height in a gravitational field and nj is the number of moles of other substances. Water will flow spontaneously from high to low chemical potential. The bigger the difference in chemical potential, the

greater the driving force for water flux (the amount moving per unit area per unit time).

Equation 3.1 indicates that the chemical potential of water (in a system such as the soil-plant-atmosphere system) depends on temperature (T), pressure (P; water can be pumped through pipes), interactions between water and solutes (nj; solutes lower / w because they lower the activity of water; see Section 3.3.3), height in a gravitational field (h; it takes work to lift water; water runs freely downhill) and electric fields (E). However, the influence of the latter on the chemical potential of water is insignificant, as water does not carry a net positive or negative charge, and so E can be ignored. Consequently, the chemical potential of water at constant temperature depends on its activity (aw), pressure (P) and height (h) in a gravitational field. This is expressed mathematically as:

where R is the gas constant, T is the absolute temperature, aw is the activity of the water, Vw is the partial molal volume of water (see below), P is the pressure, mw is the mass per mole of water, g is the acceleration due to gravity, h is the height in a gravitational field and / w is an arbitrary chemical potential of water under standard conditions (a constant of integration). The standard state (Eq. 3.2) is defined when aw = 1(RT ln aw = 0), P = 1 and the height in the gravitational field is zero (mwgh = 0). In practical terms this is pure water at atmospheric pressure and the height (and temperature) of the system under consideration. The partial molal volume (Vw) is the rate of change of volume of water with increasing number of moles of water, when the number of moles of other substances, temperature, pressure, electrical potential and height in a gravitational field is kept constant (i.e. Vw = (9 V/dnw)nj,p,tEh); its value is 1.805 x 10-5 m-3 mol-1 at 20°C.

### 3.3.2 Water potential and water potential gradients

Equation 3.2 is not readily usable, but it is possible to derive a more practical form of the relationship, firstly by defining 'water potential' as the difference between the chemical potential of water at any point in a system and that of pure free water in a standard state; that is:

The term Vw (m3 mol-1), the partial molal volume of water, is introduced to convert the units from those of free energy (J mol-1) to pressure (Pa - a Pascal is equivalent to a J m-3).

Next, by substituting Eq. 3.2 into Eq. 3.3, the following relationship is reached:

where n is the osmotic pressure (-(RT/Vw) ln aw) and pw is the density of water (mw/Vw).

Equation 3.4 is commonly written as:

where Vp is the pressure potential (which may be positive, zero or, where water is under tension, negative), Vn is the osmotic potential (a consequence of the presence of solutes and is always negative) and Vg is the gravitational potential (which is negligible in cells and small plants but may be significant in tall trees: it is important in soils as the driving force for deep drainage and for the uptake of water for deep-rooted species; see footnote in Section 3.4). Thus the awkward arbitrary constant of Eq. 3.2 is removed and the units converted from energy per mole to pressure units -those favoured by physiologists since the discovery of osmotic pressure. More details of the derivation of the formulae can be found in Nobel (2005).

### 3.3.3 Osmosis and colligative properties

Solute molecules interact with water to lower its free energy. The effect of a solute in lowering the free energy of water is easily demonstrated using a simple osmometer (Figure 3.1). Here, a concentrated solution is contained within a thistle funnel by a semipermeable membrane and then immersed in water. Water flows from the high potential in the surrounding water (0 MPa) into the solution until the increase in pressure, represented by the height of the water column, raises the free energy of the solution to that of the pure water surrounding the membrane. At this point there is no longer a gradient in free energy and net water movement ceases. At equilibrium, the effect of the positive pressure developed by the height of the water column (turgor

Solution

Semipermeable membrane

Pure water-

### Osmotic pressure

Figure 3.1 Osmosis and the generation of osmotic pressure. A solution is separated from pure water by a semipermeable membrane: solute molecules represented by the single sphere are unable to cross the membrane. There is a net flux of water molecules (three connected spheres) across the membrane until the water potential in the solution increases to 0 MPa, due to the increase in pressure - equivalent to the head of water (osmotic pressure).

pressure in a cell) is equal and opposite to that of the solute in lowering of free energy. Such a situation arises for a membrane that is impermeable to the solute -that is a perfect semipermeable membrane (see Section 3.5 below).

Equation 3.4 includes a term n, the osmotic pressure (= -(RT/Vw) ln aw = ), that depends on the activity of water. If solute and solvent behave ideally, the osmotic pressure can be expressed in terms of the concentration of a solute (or solutes). Under these conditions:

where cs is the osmolality of the solution (a 1 osmolal solution contains 1 mole of osmotically active particles per kilogram of water). Here the concentration is expressed per unit mass of water (the molal scale) rather than the more commonly used basis of a litre of solution (molar scale): molality does not change with temperature and pressure, as mass is independent of these variables. For dilute solutions of low molecular weight solutes, molal and molar concentrations are similar. Above a concentration of about 0.2 M, the two scales diverge and increasingly so, the higher molecular weight of the solute.

So far, the membrane has been considered as effecting a perfect separation between solvent (water) and solute. However, where solute passes through the membrane to some degree, the effective osmotic pressure is reduced. It is easy to imagine the two extremes - a perfectly semipermeable membrane, where all the solute molecules are reflected by the membrane, and a completely permeable membrane, where the solutes pass through the membrane. In the latter case, there would be no osmotic pressure. Membranes can vary in the proportion of solute reflected and can be characterised by their 'reflection coefficient' for a given solute. This is the ratio of the effective osmotic pressure to the theoretical osmotic pressure given by Eq. 3.6 (see also Section 3.6).

The ability of solutes to change the free energy of water means that a number of properties of water change on addition of a solute, for example the vapour pressure, the freezing point and the boiling point as well as the osmotic pressure. These are the so-called colligative properties of solutions, and provided there are no large solute-solute interactions, there is a linear relationship between solute concentration and solution property. For example, the partial pressure of water vapour in equilibrium with a solution is linearly related to the mole fraction of water in the solution (Raoult's law). The mole fraction is the ratio of number of moles of water divided by the total number of moles of water plus solute in the solution. The relationship holds to higher concentrations when expressed on the molal (that is per kilogram of solvent) rather than on the molar (per litre of solution) basis.