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Geochemical Modeling. Bruce Herbert Geology & Geophysics. Introduction. Geochemical models can predict the concentrations (activities) of species present in a system.
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Geochemical Modeling Bruce Herbert Geology & Geophysics
Introduction • Geochemical models can predict the concentrations (activities) of species present in a system. • From this data, they can be used to deduce and quantify the chemical and biologic processes that affect the fate and transport of pollutants in soils and aquifers. • There are two distinct methods that can be used to delineate these processes: • Inverse models uses hydrologic and chemical data to deduce the operative geochemical processes in a particular hydrologic system • Forward models makes a priori predictions of water chemistry based on assumed geochemical processes.
Hierarchy Model of Geochemical Modeling • Special considerations: • Hydrology • Mineralogical and other solid phase • Equilibrium and reaction kinetics
Hydrologic Considerations • Basic strategy is to determine geochemical reactions as they occur along a flow path • Matching sampling locations to the questions being asked is an important component of geochemical modeling • The residence time of water in a system controls the amount of time rock-water reactions have to attain equilibrium Various possible groundwater flow paths and placement of wells at a leaky landfill.
Hydrologic Considerations Various possible groundwater flow paths and placement of wells at a leaky landfill.
Mineralogical and Other Solid Phase Considerations A detailed knowledge of the minerals and natural organic that comprise the solid matrix is needed to determine solid phase reactions • What minerals are present; how abundant are they? • How does mineral abundance vary spatially? • How does mineral composition, including elemental substitutions, ion exchange and crystallinity, vary spatially? • How does mineralogy vary with respect to flow? • Any evidence for secondary minerals? • What type and quantities of organic matter exist?
Equilibrium and Reaction Kinetics Considerations • Thermodynamics is a system of consistent chemical equations used to describe the state of a system. In particular it describes the energy differences between different states • Use of thermodynamics to describe geochemical processes is based on the assumption of equilibrium • Thermodynamics can give information of what reactions may occur (thermodynamically feasible) but not their rate • Kinetics describes the rates of reaction • Typically, water velocity is compared to the rate of reaction. This determines if the system reaches equilibrium
Thermodynamics Thermodynamics describes the differences in energy between two different states of a system. These differences can be used to predict the final equilibrium state of a system. • To achieve equilibrium, an infinite amount of time must pass • Many reactions are fast. Ignoring the slow reactions results in a partial equilibrium model • To include descriptions of slower reactions would result in a description of pseudo-equilibrium • Thermodynamics describes changes in macroscopic properties of a system
Thermodynamic Systems A thermodynamic system is a macroscopic region of solid, liquid and gaseous matter with electrical and gravitational gradients, enclosed by a bounding surface called the thermodynamic wall. The wall separates the system from its reservoirs. A reservoir is a large thermodynamic system whose intensive properties are unaffected by matter and heat flowing into and out of the reservoir.
Thermodynamic Descriptions of Systems: State Variables A thermodynamic system is described or defined in terms of the values of a set of parameters called state variables. • State: the set of numerical values of the system's properties. The values of these properties define the energy of the system at a particular state. • System properties can either be extensive or intensive • Extensive Properties depend on the amount of matter present: mass, volume, energy, and entropy • Intensive Properties are independent of the amount of matter: bulk density, pressure, temperature, and concentration
Thermodynamic Descriptions of Systems: State Variables Other state variables may also have to be defined including magnetic, gravitational, and surface properties
Thermodynamic Descriptions of Systems: Processes • A reversible process occurs if the system passes through equilibrium states. • Changes that take place in a system spontaneously and of their own accord are natural processes. • Because every process is accompanied by a loss of heat to the universe (entropy), natural processes can never be reversed to return the system to its (exact) original condition. • Because of this characteristic of natural processes, they are termed irreversible processes. • An infinitesimal process occurs if the system's properties change extremely small amounts.
Thermodynamic Descriptions of Systems: Processes Thermodynamic processes take place when thermodynamic properties change. This results in a change in state of the system. A reversible process occurs if the system passes through equilibrium states. Changes that take place in a system spontaneously and of their own accord are natural processes. Because every process is accompanied by a loss of heat to the universe (entropy, see below), natural processes can never be reversed to return the system to its (exact) original condition. Because of this characteristic of natural processes, they are termed irreversible processes. An infinitesimal process occurs if the system's properties change infinitesimally.
Thermodynamic Descriptions of Systems: Phases • Thermodynamic systems are defined in terms of their phases (solids, liquids and gases) • Given a mass of soil or aquifer material the homogeneity and heterogeneity is defined: • The material is uniformif its properties exhibit no spatial variability • The material is constantif no temporal variability • If intensive properties are uniform and phases are not differentiated, then system is homogeneous
Thermodynamic Descriptions of Systems: Phases • Thermodynamic systems are defined in terms of their phases (solids, liquids and gases) • Environmental chemistry uses homogeneous mixtures or solutions: uniform intensive properties with no differentiation between phases. • Soil air: phases (O2, N2, Ar, CO2) are mixed uniformly • Groundwaters: aqueous mixture of water and dissolved solids. • Solid phase mixtures: solid solutions of specific minerals, natural organic matter
Thermodynamic Descriptions of Systems: Exchange • Exchange describes the exchange of matter and energy between system and surroundings
Kinetic Descriptions of Systems • Classic geochemical descriptions emphasize equilibrium • Three reasons to consider time dependent process: • Many reactions are slow • Nonequilibrium conditions can exist due to the physical transport of gases, solution or solutes • kinetic data can provide information about reaction mechanisms Residence times of natural waters along with rates of selected reactions
Kinetic Descriptions of Systems • Soil solution and groundwaters may not always be well mixed due to time-dependence of diffusion • This creates concentration variations within small volumes • This may induce a time-dependent behavior • Time-dependent behavior can also exist due to spatial variations in steady-state conditions where advection-diffusion is balanced by reaction Kinetics can describe both chemical reaction rates and physical processes.
Soil Solution Partitioned betweenMacroporeand Micropore (Left) Diagram of soil solution partitioned between macropore and micropore. Graph of solute concentration (C) as a function of position (x). Taken from (Right) A species present in bulk solution is consumed by a chemical reaction on the available surface. (a) Surface chemical reaction is fast relative to molecular transport. (b) Mixed chemical and transport rate control. (c) Transport is fast relative to surface chemical reaction. Taken from Stone and Morgan (1990) and Skopp (1986).
Internal Energy and Thermodynamic Potentials Thermodynamics describes the energy differences between different states of a system. The energy of a system is given by the internal energy, U. • Internal Energy, U, of a system is given by the function Intensive variables (T, P, u) can be written as partial derivatives of one extensive variable to another. In fact, these are the definitions of the intensive variables.
Thermodynamic Potentials Thermodynamic potentials are extensive quantities equivalent to U Each thermodynamic potential is determined by the independent variables of state that are chosen They have the property that the equilibrium state of a system produces a minimum in the potential
Thermodynamic Potentials • Potentials have units of joules (J). Choice of potential depends on which variables are independent. • Since T, P, and composition of an aquifer / soil are easy to control, then G is typically the most useful potential. • At equilibrium, dG is given by the equation on the left, while dG is given by the equation on the right for a spontaneous change (natural process)
Gibbs Phase Rule • We can determine the number of components needed to describe a thermodynamic system by subtracting the number of independent reactions from the number of species: • The more general expression for the number of independent variables (degrees of freedom) is given by the Gibbs Phase Rule. • For a system with Np phases and Nr independent chemical reactions, then the Gibbs Phase Rule can be written as:
Thermal, Mechanical, and Chemical Equilibria • An expression can be written for dU for any infinitesimal (thermodynamic) change. The subscript "s" refers to the sample and "r" to the reservoir. • At equilibrium, dU will be a minimum. Suppose a small (infitesimal) amount of heat was transferred between the sample and the reservoir. Because of the nature of the wall, Eqn (20) would reduce to: • Because the complete system is isolated, dSs = -dSr. Eqn (22) can then be written:
Thermal, Mechanical, and Chemical Equilibria • The criteria for thermal equilibrium between the soil and its thermal reservoir is that the absolute temperatures, T, are equal. • If the system is not in equilibrium, then it is subject to dU<0 for spontaneous change, which would require Eqn (23) to be: • If dSs>0, then thermal energy must be moving into the soil and Ts<Tr, with the vice versa also being true. • This indicates that thermal energy is transferred from the system at higher T to the system at lower T.
Thermal, Mechanical, and Chemical Equilibria • Consider the following system: a single chemical component in a single phase in contact with a thermal, volume, and matter reservoir. Then infinitesimal reactions could be described with: • At equilibrium, dU will be a minimum. Given that the wall is movable, permeable, diathermal, then Eqn (20) would reduce to: • Given an infinitesimal change in m, then dm is arbitrary and infinitesimal and dU=0. Then Eqn (27) can be rearranged to give:
Thermal, Mechanical, and Chemical Equilibria • The criteria for phase equilibrium between the soil and its matter reservoir is that the chemical potentials, u, are equal. • This indicates that matter is transferred from the system at higher u to the system at lower u. • Similar statements can be drawn for mechanical equilibrium. • The criteria for mechanical equilibrium between the soil and its volume reservoir is that the pressures, P, are equal. • This indicates that mechanical energy is transferred from the system at higher P to the system at lower P.
Chemical Potential Chemical Potential: the intensive property that is the criteria for the transfer of matter. Each species in a system has a chemical potential,m, that determines whether that species could be transferred to another phase or change into another species. • At equilibrium (T,P fixed) (31) • where, are the stoichiometric coefficients for species j. • By definition, the stoichiometric coefficients for products are positive and the stoichiometric coefficients for reactants are negative.
Chemical Potentials and Standard States • The chemical potential of a substance is expressed relative to its value in a defined Standard State and its relative fugacity: • Chemical potential is expressed relative to a Standard State. Unlike the case for T and P, there is no way to express an absolute value for chemical potential in the same ways as T and P. • Standard states are defined, in part, by standard temperature and pressure (STP) • The Standard State chemical potentials of a substance is the chemical potential of the most stable phase of that substance under Standard State conditions
Chemical Potential and Activities • Chemical potentials are defined, in part, by describing the fugacity of a substance. The fugacity (from the Latin fugere, to flee) describes the "escaping tendency" of a substance. • The fugacity can be related to other, more familiar, variables including the pressure of a gas and the activity of a solute. Fugacity of solid or liquid is given by its vapor pressure. Fugacity of a gas is given by its partial pressure. (32) , the standard state chemical potential at STP is the same as standard free energies of formation, . The standard free energies of formation is the free energy change when one mole of a chemical in its standard state is formed from its constituent elements at STP.
Standard State Chemical Potentials • Chemical potential is expressed relative to a Standard State. • The Standard State in general is going to define particular values of T, P, and the degree of interaction between species in the system. Standard states are defined, in part, by standard temperature and pressure (STP) • T = 298.15 K P = 101.325 kPa
Chemical Potentials and Activities The chemical potential of any substance in any phase, is expressed relative to its value in a defined Standard State and its relative fugacity Chemical potential of an ideal gases is Chemical potential of a real gas is
Chemical Potentials and Activities Chemical potential of a solute in an ideal solution is Chemical potential of a real solutions is where the activity coefficient, , typically varies between 0 and 1 in dilute waters, but can be greater than 1 in waters of higher ionic strength, I. Activity can also be written in terms of molality, m, or molarity, M, as: or
Equilibrium Constants and Gibbs Free Energy is the Standard Free Energy Change, the change in free energy of a reaction. This relationship is useful because we can use data to calculate , which can then be used to calculate K. can be used to determine the stability of products relative to reactants when all of the compounds are in the standard state. If <0 then the products are more stable than the reactants.
Activities and Activity Coefficients Interactions of metals with materials in different environmental compartments. Most instruments measure a compound's concentration. These techniques include gas chromatography (organics), atomic absorption (metals), and ion chromatography (metals). The exceptions are pH and ion selective electrodes, which measure an ion's activity.
Activities and Activity Coefficients Activities are used in several important equations including: The most useful activity coefficient is defined for a single ion or aqueous solute. Strict thermodynamics doesn't recognize single ions, therefore single ion activity coefficients are extensions of thermodynamics. Because of this, calculation of single-ion and solute activity coefficients are model dependent.
Activity Coefficients and Reference States Like the need to define a Standard State when defining chemical potentials, to define activity coefficients we need to define a Reference State that dictates when the activity coefficient is unity (i.e. equal to one). • The Standard State of a compound is defined by designated values for T and P (i.e. STP) and an activity of 1.0. • The Reference State of a compound is defined by designated values of T and P (i.e. STP) and an activity coefficient equal to 1.0. • Generally, the Standard State and the Reference State are not equal.
Activity Coefficients and Reference States Infinite Dilution Reference State: most often used in fresh water systems Constant Ionic Medium Reference State: commonly used in seawater system I is fixed
Models of Activity Coefficients One model used to calculate single-ion activity coefficients is the Davies Equation. This equation is valid in systems up to I<0.5 M.
Measuring Ionic Strength The effective ionic strength, I, is related to the electrolytic conductivity, k, of a solution. This relationship is given in the Marion-Babcock equation: which is accurate up to ionic strengths of less than or equal to 0.3 mol dm-3. In this equation, I is in units of mol m-3, and k is in units of decisiemens per meter (dS m-1).
Thermodynamics and the Calculation of Speciation Geochemical speciation models predict the distribution of chemicals in different species at equilibrium. The models use databases of chemical potentials to calculate G and equilibrium constants, K. The equilibrium constants are compared to conditional equilibrium constants calculated from observed concentrations. The model simultaneously solves the equilibrium, mass balance and electroneutrality equations to predict species activities at equilibiurm.
