generally compared to a standard. For example, sulfur has four stable isotopes, of which the two most common are 32S and 34S. The ratio of their concentrations, , is reported as where is the same ratio for a standard. Because the differences are small, the ratio is multiplied by 1000 to make it parts per thousand (referred to as parts per mil). This is represented by the symbol . Equilibrium Equilibrium fractionation occurs between chemicals or phases that are in equilibrium with each other. In equilibrium fractionation between phases, heavier phases prefer the heavier isotopes. For two phases A and B, the effect can be represented by the factor In the liquid-vapor phase transition for water, at 20 degrees Celsius is 1.0098 for 18O and 1.084 for 2H. In general, fractionation is greater at lower temperatures. At 0 °C, the factors are 1.0117 and 1.111. Kinetic When there is no equilibrium between phases or chemical compounds, kinetic fractionation can occur. For example, at interfaces between liquid water and air, the forward reaction is enhanced if the humidity of the air is less than 100% or the water vapor is moved by a wind. Kinetic fractionation generally is enhanced compared to equilibrium fractionation and depends on factors such as reaction rate, reaction pathway and bond energy. Since lighter isotopes generally have weaker bonds, they tend to react faster and enrich the reaction products. Biological fractionation is a form of kinetic fractionation since reactions tend to be in one direction. Biological organisms prefer lighter isotopes because there is a lower energy cost in breaking energy bonds. In addition to the previously mentioned factors, the environment and species of the organism can have a large effect on the fractionation. Cycles Through a variety of physical and chemical processes, chemical elements change in concentration and move around in what are called geochemical cycles. An understanding of these changes requires both detailed observation and theoretical models. Each chemical compound, element or isotope has a concentration that is a function of position and time, but it is impractical to model the full variability. Instead, in an approach borrowed from chemical engineering, geochemists average the concentration over regions of the Earth called geochemical reservoirs. The choice of reservoir depends on the problem; for example, the ocean may be a single reservoir or be split into multiple reservoirs. In a type of model called a box model, a