{"id":51,"date":"2022-01-11T01:14:34","date_gmt":"2022-01-11T01:14:34","guid":{"rendered":"https:\/\/books.gw-project.org\/groundwater-microbiology\/chapter\/chemical-equilibria-and-reaction-rates-in-groundwater\/"},"modified":"2022-02-01T18:03:01","modified_gmt":"2022-02-01T18:03:01","slug":"chemical-equilibria-and-reaction-rates-in-groundwater","status":"publish","type":"chapter","link":"https:\/\/books.gw-project.org\/groundwater-microbiology\/chapter\/chemical-equilibria-and-reaction-rates-in-groundwater\/","title":{"raw":"4.1 Chemical Equilibria and Reaction Rates in Groundwater","rendered":"4.1 Chemical Equilibria and Reaction Rates in Groundwater"},"content":{"raw":"
\r\n

Descriptions of chemical processes in groundwater systems are usually formulated in terms of mass action and mass balance considerations, as well as reaction kinetics. The application of these basic aspects of physical chemistry and thermodynamics provides quantitative insight into the extent, direction, and rate of chemical reactions, including those involving microorganisms. A general reaction is shown in Equation 17.<\/a><\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\naA<\/em> + bB<\/em> = cC<\/em> + dD<\/em><\/td>\r\n(17)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

The corresponding mass action equilibrium constant K<\/em> is calculated as shown in Equation 18.<\/a><\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle K=\\frac{\\left\\{C \\right\\}^{c}\\left\\{D \\right\\}^{d}}{\\left\\{A \\right\\}^{a}\\left\\{B \\right\\}^{b}}=e^{\\frac{-\\Delta G^{0}}{RT}}[\/latex] ; [latex]\\Delta G^{0}=-RT\\textup{ln}K[\/latex]<\/td>\r\n(18)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nwhere:\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
{A<\/em>}, {B<\/em>}<\/td>\r\n=<\/td>\r\nactivities of reactants (dimensionless)<\/td>\r\n<\/tr>\r\n
{C<\/em>}, {D<\/em>}<\/td>\r\n=<\/td>\r\nactivities of products (C,D<\/em>) at equilibrium<\/em> with stoichiometric coefficients given in lower case letters (dimensionless)<\/td>\r\n<\/tr>\r\n
R<\/em><\/td>\r\n=<\/td>\r\nuniversal gas constant (ML2<\/sup>)\/(T2 <\/sup>\u00b0K mol)<\/td>\r\n<\/tr>\r\n
T<\/em><\/td>\r\n=<\/td>\r\ntemperature (\u00b0K)<\/td>\r\n<\/tr>\r\n
\u0394G<\/em>0<\/sup><\/td>\r\n=<\/td>\r\nstandard Gibbs energy of reaction \u0394 (ML2<\/sup>)\/(T2<\/sup> mol)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

For a reversible reaction at equilibrium, the concentrations of reactants and products remain constant. This condition requires the rates of the forward (R<\/em>f<\/em><\/sub>) and reverse (R<\/em>r<\/em><\/sub>) reactions to be equal as shown in Equation 19 such that the equilibrium constant can be expressed as Equation 20 with corresponding forward (k<\/em>f<\/em><\/sub>) and reverse (k<\/em>r<\/em><\/sub>) rate coefficients.<\/a><\/a><\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\nR<\/em>f<\/em><\/sub> = k<\/em>f<\/em><\/sub>{A<\/em>}a<\/em><\/sup>{B<\/em>}b<\/em><\/sup> = R<\/em>r<\/em><\/sub> = k<\/em>r<\/em><\/sub>{C<\/em>}c<\/em><\/sup>{D<\/em>}d<\/em><\/sup><\/td>\r\n(19)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle K=\\frac{k_{f}}{k_{r}}=\\frac{\\left\\{C \\right\\}^{c}\\left\\{D \\right\\}^{d}}{\\left\\{A \\right\\}^{a}\\left\\{B \\right\\}^{b}}[\/latex]<\/td>\r\n(20)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

The dependence of the rate coefficients on temperature and reaction activation energy is evident from the Arrhenius relationship (Equation\u00a01<\/a>).<\/p>\r\n

A far more interesting condition, especially for groundwater systems, is when a reaction is not at equilibrium. Here, the Gibbs energy for a reaction (\u0394G<\/em>r<\/em><\/sub>) to occur is described by Equation 21.<\/a><\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle \\Delta G_{r}=\\Delta G^{0}+RT\\textup{ln}\\frac{\\left\\{C \\right\\}^{c}\\left\\{D \\right\\}^{d}}{\\left\\{A \\right\\}^{a}\\left\\{B \\right\\}^{b}}[\/latex]<\/td>\r\n(21)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

For reactant and product activities observed away from equilibrium, defining the reaction quotient of products to reactants as Q<\/em>,<\/em> gives Equation 22.<\/a><\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle \\Delta G_{r}=\\Delta G^{0}+RT\\textup{ln}Q=RT\\textup{ln}\\frac{Q}{K}[\/latex]<\/td>\r\n(22)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

From this expression, one finds that at equilibrium Q<\/em> is the same as K<\/em>, so \u0394G<\/em>r<\/em><\/sub>\u00a0<\/em><\/sub>=\u00a00. If Q<\/em> is less than K<\/em>, \u0394G<\/em>r<\/em><\/sub> is negative. This means the reaction is spontaneous. Conversely, \u0394G<\/em>r<\/em><\/sub>\u00a0<\/em><\/sub>=\u00a00 is positive when Q<\/em> is greater than K<\/em>, indicating the reaction is not possible (unless energy is supplied from another spontaneous reaction). The capacity to use energy-yielding spontaneous reactions to drive energetically unfavorable reactions is a defining characteristic of biosynthetic processes in microbial metabolism.<\/p>\r\n

Comparison of reaction quotients to equilibrium constants is widely applied to the study of mineral dissolution and precipitation reactions; however, subtle differences in terminology and interpretation exist. For dissolution reactions, the equilibrium constant is known as the solubility product constant (K<\/em>sp<\/em><\/sub>) and the reaction quotient is referred to as the ion activity product (IAP<\/em>). The saturation index (S<\/em>I<\/em>) is defined by Equation 23.<\/a><\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle SI=\\textup{log}_{10}\\frac{IAP}{K_{sp}}[\/latex]<\/td>\r\n(23)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

When the IAP<\/em> is equal to K<\/em>s<\/em><\/sub>p<\/em><\/sub>, S<\/em>I<\/em>\u00a0=\u00a00<\/em> and the solution is said to be at equilibrium with respect to the mineral under consideration. Should IAP<\/em> be greater than K<\/em>sp<\/em><\/sub>, the SI<\/em> will be positive, which indicates the solution is oversaturated and mineral precipitation is possible. If the IAP<\/em> is less than K<\/em>sp<\/em><\/sub>, the S<\/em>I<\/em> will be negative, which indicates the solution is undersaturated and mineral dissolution is possible.<\/p>\r\n\r\n<\/div>","rendered":"

\n

Descriptions of chemical processes in groundwater systems are usually formulated in terms of mass action and mass balance considerations, as well as reaction kinetics. The application of these basic aspects of physical chemistry and thermodynamics provides quantitative insight into the extent, direction, and rate of chemical reactions, including those involving microorganisms. A general reaction is shown in Equation 17.<\/a><\/p>\n\n\n\n
<\/td>\naA<\/em> + bB<\/em> = cC<\/em> + dD<\/em><\/td>\n(17)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

The corresponding mass action equilibrium constant K<\/em> is calculated as shown in Equation 18.<\/a><\/p>\n\n\n\n
<\/td>\n\"\displaystyle K=\frac{\left\{C \right\}^{c}\left\{D \right\}^{d}}{\left\{A \right\}^{a}\left\{B \right\}^{b}}=e^{\frac{-\Delta G^{0}}{RT}}\" ; \"\Delta G^{0}=-RT\textup{ln}K\"<\/td>\n(18)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

where:<\/p>\n\n\n\n\n\n\n\n
{A<\/em>}, {B<\/em>}<\/td>\n=<\/td>\nactivities of reactants (dimensionless)<\/td>\n<\/tr>\n
{C<\/em>}, {D<\/em>}<\/td>\n=<\/td>\nactivities of products (C,D<\/em>) at equilibrium<\/em> with stoichiometric coefficients given in lower case letters (dimensionless)<\/td>\n<\/tr>\n
R<\/em><\/td>\n=<\/td>\nuniversal gas constant (ML2<\/sup>)\/(T2 <\/sup>\u00b0K mol)<\/td>\n<\/tr>\n
T<\/em><\/td>\n=<\/td>\ntemperature (\u00b0K)<\/td>\n<\/tr>\n
\u0394G<\/em>0<\/sup><\/td>\n=<\/td>\nstandard Gibbs energy of reaction \u0394 (ML2<\/sup>)\/(T2<\/sup> mol)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

For a reversible reaction at equilibrium, the concentrations of reactants and products remain constant. This condition requires the rates of the forward (R<\/em>f<\/em><\/sub>) and reverse (R<\/em>r<\/em><\/sub>) reactions to be equal as shown in Equation 19 such that the equilibrium constant can be expressed as Equation 20 with corresponding forward (k<\/em>f<\/em><\/sub>) and reverse (k<\/em>r<\/em><\/sub>) rate coefficients.<\/a><\/a><\/p>\n\n\n\n
<\/td>\nR<\/em>f<\/em><\/sub> = k<\/em>f<\/em><\/sub>{A<\/em>}a<\/em><\/sup>{B<\/em>}b<\/em><\/sup> = R<\/em>r<\/em><\/sub> = k<\/em>r<\/em><\/sub>{C<\/em>}c<\/em><\/sup>{D<\/em>}d<\/em><\/sup><\/td>\n(19)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n
<\/td>\n\"\displaystyle K=\frac{k_{f}}{k_{r}}=\frac{\left\{C \right\}^{c}\left\{D \right\}^{d}}{\left\{A \right\}^{a}\left\{B \right\}^{b}}\"<\/td>\n(20)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

The dependence of the rate coefficients on temperature and reaction activation energy is evident from the Arrhenius relationship (Equation\u00a01<\/a>).<\/p>\n

A far more interesting condition, especially for groundwater systems, is when a reaction is not at equilibrium. Here, the Gibbs energy for a reaction (\u0394G<\/em>r<\/em><\/sub>) to occur is described by Equation 21.<\/a><\/p>\n\n\n\n
<\/td>\n\"\displaystyle \Delta G_{r}=\Delta G^{0}+RT\textup{ln}\frac{\left\{C \right\}^{c}\left\{D \right\}^{d}}{\left\{A \right\}^{a}\left\{B \right\}^{b}}\"<\/td>\n(21)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

For reactant and product activities observed away from equilibrium, defining the reaction quotient of products to reactants as Q<\/em>,<\/em> gives Equation 22.<\/a><\/p>\n\n\n\n
<\/td>\n\"\displaystyle \Delta G_{r}=\Delta G^{0}+RT\textup{ln}Q=RT\textup{ln}\frac{Q}{K}\"<\/td>\n(22)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

From this expression, one finds that at equilibrium Q<\/em> is the same as K<\/em>, so \u0394G<\/em>r<\/em><\/sub>\u00a0<\/em><\/sub>=\u00a00. If Q<\/em> is less than K<\/em>, \u0394G<\/em>r<\/em><\/sub> is negative. This means the reaction is spontaneous. Conversely, \u0394G<\/em>r<\/em><\/sub>\u00a0<\/em><\/sub>=\u00a00 is positive when Q<\/em> is greater than K<\/em>, indicating the reaction is not possible (unless energy is supplied from another spontaneous reaction). The capacity to use energy-yielding spontaneous reactions to drive energetically unfavorable reactions is a defining characteristic of biosynthetic processes in microbial metabolism.<\/p>\n

Comparison of reaction quotients to equilibrium constants is widely applied to the study of mineral dissolution and precipitation reactions; however, subtle differences in terminology and interpretation exist. For dissolution reactions, the equilibrium constant is known as the solubility product constant (K<\/em>sp<\/em><\/sub>) and the reaction quotient is referred to as the ion activity product (IAP<\/em>). The saturation index (S<\/em>I<\/em>) is defined by Equation 23.<\/a><\/p>\n\n\n\n
<\/td>\n\"\displaystyle SI=\textup{log}_{10}\frac{IAP}{K_{sp}}\"<\/td>\n(23)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

When the IAP<\/em> is equal to K<\/em>s<\/em><\/sub>p<\/em><\/sub>, S<\/em>I<\/em>\u00a0=\u00a00<\/em> and the solution is said to be at equilibrium with respect to the mineral under consideration. Should IAP<\/em> be greater than K<\/em>sp<\/em><\/sub>, the SI<\/em> will be positive, which indicates the solution is oversaturated and mineral precipitation is possible. If the IAP<\/em> is less than K<\/em>sp<\/em><\/sub>, the S<\/em>I<\/em> will be negative, which indicates the solution is undersaturated and mineral dissolution is possible.<\/p>\n<\/div>\n","protected":false},"author":1,"menu_order":4,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-51","chapter","type-chapter","status-publish","hentry"],"part":111,"_links":{"self":[{"href":"https:\/\/books.gw-project.org\/groundwater-microbiology\/wp-json\/pressbooks\/v2\/chapters\/51","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/books.gw-project.org\/groundwater-microbiology\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/books.gw-project.org\/groundwater-microbiology\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/books.gw-project.org\/groundwater-microbiology\/wp-json\/wp\/v2\/users\/1"}],"version-history":[{"count":5,"href":"https:\/\/books.gw-project.org\/groundwater-microbiology\/wp-json\/pressbooks\/v2\/chapters\/51\/revisions"}],"predecessor-version":[{"id":257,"href":"https:\/\/books.gw-project.org\/groundwater-microbiology\/wp-json\/pressbooks\/v2\/chapters\/51\/revisions\/257"}],"part":[{"href":"https:\/\/books.gw-project.org\/groundwater-microbiology\/wp-json\/pressbooks\/v2\/parts\/111"}],"metadata":[{"href":"https:\/\/books.gw-project.org\/groundwater-microbiology\/wp-json\/pressbooks\/v2\/chapters\/51\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/books.gw-project.org\/groundwater-microbiology\/wp-json\/wp\/v2\/media?parent=51"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/books.gw-project.org\/groundwater-microbiology\/wp-json\/pressbooks\/v2\/chapter-type?post=51"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/books.gw-project.org\/groundwater-microbiology\/wp-json\/wp\/v2\/contributor?post=51"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/books.gw-project.org\/groundwater-microbiology\/wp-json\/wp\/v2\/license?post=51"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}