{"id":114,"date":"2026-04-10T02:53:06","date_gmt":"2026-04-10T02:53:06","guid":{"rendered":"https:\/\/books.gw-project.org\/age-dating-young-groundwater\/chapter\/ch040-xhtml\/"},"modified":"2026-09-02T19:34:42","modified_gmt":"2026-09-02T19:34:42","slug":"5-4-calculation-of-tracer-age","status":"publish","type":"chapter","link":"https:\/\/books.gw-project.org\/age-dating-young-groundwater\/chapter\/5-4-calculation-of-tracer-age\/","title":{"raw":"5.4 Calculation of Tracer Age","rendered":"5.4 Calculation of Tracer Age"},"content":{"raw":"<div id=\"calculation-of-tracer-age-2\" class=\"level1\">\r\n<div data-custom-style=\"First Paragraph\">\r\n\r\nAs a transient tracer, the general concept of [latex]{SF}_{6}[\/latex] dating involves comparing a measured concentration in a water sample with historical concentrations in the atmosphere. Henry\u2019s Law is used to relate the measured aqueous concentration of a sample to the equilibrium gas\u2011phase concentration. Because the Henry Coefficient is temperature dependent, and because the partial pressure of [latex]{SF}_{6}[\/latex] in the atmosphere depends on total pressure, both the recharge temperature and pressure (i.e., elevation) are needed for age dating. Furthermore, if groundwater contains excess air, and therefore some excess SF<sub>6<\/sub> above equilibrium solubility, the amount of excess air is also required.\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nThe temperature and salinity dependence of the Henry Coefficient was investigated by Bullister and others (2002), who provided an empirical equation for calculating the equilibrium constant ([latex]K_{h}[\/latex]) for the solubility form of Henry\u2019s Law as shown by Equation\u00a0<span data-custom-style=\"Hyperlink\">(23)<\/span>.\r\n\r\n<\/div>\r\n<table style=\"width: 85%;\" data-custom-style=\"Table\"><colgroup> <col style=\"width: 8%;\" \/> <col style=\"width: 64%;\" \/> <col style=\"width: 11%;\" \/> <\/colgroup>\r\n<thead>\r\n<tr>\r\n<th><\/th>\r\n<th style=\"text-align: center;\">[latex]C = K_{h}(T,S)\\left( P - e(T) \\right)[\/latex]<\/th>\r\n<th style=\"text-align: right;\"><span id=\"Xefc6fe3e8250cb645ec82722a2955f3ec328c0f\" class=\"anchor\"><\/span>(23)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<\/table>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nwhere:\r\n\r\n<\/div>\r\n<table style=\"width: 85%;\" data-custom-style=\"Table\"><colgroup> <col style=\"width: 11%;\" \/> <col style=\"width: 3%;\" \/> <col style=\"width: 69%;\" \/> <\/colgroup>\r\n<thead>\r\n<tr>\r\n<th style=\"text-align: right;\"><em>C<\/em><\/th>\r\n<th style=\"text-align: center;\">=<\/th>\r\n<th>aqueous concentration (molM<sup>-1<\/sup>, typically in moles\/kg)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>K<sub>h<\/sub><\/em><\/td>\r\n<td style=\"text-align: center;\">=<\/td>\r\n<td>temperature- and salinity-dependent equilibrium constant (molM<sup>\u20112<\/sup>LT<sup>2<\/sup> typically in moles\u00a0kg<sup>\u20111<\/sup>\u00a0atm<sup>\u20111<\/sup>)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>x<\/em><\/td>\r\n<td style=\"text-align: center;\">=<\/td>\r\n<td>dry air mole fraction (dimensionless)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>P<\/em><\/td>\r\n<td style=\"text-align: center;\">=<\/td>\r\n<td>atmospheric pressure (ML<sup>-1<\/sup>T<sup>-2<\/sup>, typically in atm)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>e<\/em><\/td>\r\n<td style=\"text-align: center;\">=<\/td>\r\n<td>temperature dependent vapor pressure of water (ML<sup>-1<\/sup>T<sup>-2<\/sup>, typically in atm)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nValues for [latex]K_{h}[\/latex] and <em>e<\/em> at various temperatures for fresh water are shown in <a href=\"#table-6\"><span data-custom-style=\"Hyperlink\">Table <strong>6<\/strong><\/span><\/a>. If the recharge temperature (<em>T<\/em>), salinity (<em>S<\/em>), and elevation (and hence <em>P<\/em>) are known, Equation\u00a0<span data-custom-style=\"Hyperlink\">(23)<\/span> can be solved for the dry air mole fraction (<em>x<\/em>) and then <em>x<\/em> can be computed for a measured aqueous concentration. The dry air mole fraction can then be compared to the history of [latex]{SF}_{6}[\/latex] in the atmosphere to obtain a recharge date as illustrated in <a href=\"\/age-dating-young-groundwater\/chapter\/5-2-basic-concepts-and-systematics\/#figure-21\"><span data-custom-style=\"Hyperlink\">Figure 21<\/span><\/a>.\r\n\r\n<\/div>\r\n<div data-custom-style=\"Block Text\">\r\n<blockquote>\r\n<div id=\"table-6\" class=\"figcaption-text\">\r\n\r\n<span id=\"Xcc8452be30d820ec398be18a66806259ff028e2\" class=\"anchor\"><\/span><strong>Table 6\u00a0\u2011<\/strong>\u00a0Values of <em>K<sub>h<\/sub><\/em> for SF<sub>6<\/sub> and water vapor (<em>e<\/em>) at various temperatures for fresh water (salinity = 0).\r\n\r\n<\/div><\/blockquote>\r\n<\/div>\r\n<table style=\"width: 44%;\" data-custom-style=\"Table\"><colgroup> <col style=\"width: 13%;\" \/> <col style=\"width: 22%;\" \/> <col style=\"width: 7%;\" \/> <\/colgroup>\r\n<thead>\r\n<tr>\r\n<th style=\"text-align: center;\">Temperature (\u00b0C)<\/th>\r\n<th style=\"text-align: center;\"><em>K<sub>h<\/sub><\/em>\u00a0[mole\u00a0kg<sup>-1<\/sup>\u00a0atm<sup>-1<\/sup>]<\/th>\r\n<th style=\"text-align: center;\"><em>e<\/em>\u00a0[atm]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"text-align: center;\">0<\/td>\r\n<td style=\"text-align: center;\">0.0006216<\/td>\r\n<td style=\"text-align: center;\">0.0060<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">10<\/td>\r\n<td style=\"text-align: center;\">0.0003969<\/td>\r\n<td style=\"text-align: center;\">0.0121<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">20<\/td>\r\n<td style=\"text-align: center;\">0.0002739<\/td>\r\n<td style=\"text-align: center;\">0.0230<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">30<\/td>\r\n<td style=\"text-align: center;\">0.0002023<\/td>\r\n<td style=\"text-align: center;\">0.0418<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">40<\/td>\r\n<td style=\"text-align: center;\">0.0001587<\/td>\r\n<td style=\"text-align: center;\">0.0726<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div data-custom-style=\"Block Text\">\r\n<blockquote><em>K<sub>h<\/sub><\/em> from Bullister and others (2002)<\/blockquote>\r\n<\/div>\r\n<div data-custom-style=\"Block Text\">\r\n<blockquote><em>e<\/em> from Antoine (1888) and Section\u00a02.3 of this book<\/blockquote>\r\n<\/div>\r\n<div data-custom-style=\"First Paragraph\">\r\n\r\nEquation\u00a0<span data-custom-style=\"Hyperlink\">(23)<\/span> represents equilibrium conditions and does not include the effects of excess air. Solomon and others (2010) derived an expression for the dry-air mole fraction\u2014which is essentially equivalent to the mixing ratio\u2014that includes excess air as given by the CE model presented in Section\u00a02.2 and Equation\u00a0<span data-custom-style=\"Hyperlink\">(24)<\/span>.\r\n\r\n<\/div>\r\n<table style=\"width: 85%;\" data-custom-style=\"Table\"><colgroup> <col style=\"width: 8%;\" \/> <col style=\"width: 64%;\" \/> <col style=\"width: 11%;\" \/> <\/colgroup>\r\n<thead>\r\n<tr>\r\n<th><\/th>\r\n<th style=\"text-align: center;\">[latex]x = \\frac{C}{K_{h}(P - e) + \\frac{(1 - F)A}{1 + \\frac{FA}{K_{h}(P - e)}}}[\/latex]<\/th>\r\n<th style=\"text-align: right;\"><span id=\"X6d12a5cdc28158acd826933e004d56817c4e2f1\" class=\"anchor\"><\/span>(24)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<\/table>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nwhere:\r\n\r\n<\/div>\r\n<table style=\"width: 85%;\" data-custom-style=\"Table\"><colgroup> <col style=\"width: 12%;\" \/> <col style=\"width: 3%;\" \/> <col style=\"width: 69%;\" \/> <\/colgroup>\r\n<thead>\r\n<tr>\r\n<th style=\"text-align: right;\"><em>x<\/em><\/th>\r\n<th style=\"text-align: center;\">=<\/th>\r\n<th>dry air mole fraction (dimensionless)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>C<\/em><\/td>\r\n<td style=\"text-align: center;\">=<\/td>\r\n<td>aqueous concentration (molM<sup>-1<\/sup>, typically in mole\/kg)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>K<sub>h<\/sub><\/em><\/td>\r\n<td style=\"text-align: center;\">=<\/td>\r\n<td>temperature\u2011\u00a0and\u00a0salinity\u2011dependent\u00a0equilibrium\u00a0constant\u00a0(molM<sup>\u20112<\/sup>LT<sup>2<\/sup> typically in moles kg<sup>\u20111<\/sup>atm<sup>\u20111<\/sup>)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>P<\/em><\/td>\r\n<td style=\"text-align: center;\">=<\/td>\r\n<td>atmospheric pressure (ML<sup>-1<\/sup>T<sup>-2<\/sup>, typically in atm)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>e<\/em><\/td>\r\n<td style=\"text-align: center;\">=<\/td>\r\n<td>temperature\u2011dependent vapor pressure of water (ML<sup>-1<\/sup>T<sup>-2<\/sup>, typically in atm)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>A<\/em><\/td>\r\n<td style=\"text-align: center;\">=<\/td>\r\n<td>volume of gas trapped in the porous media per unit mass of water (molM<sup>-1<\/sup>, typically in moles\/kg)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>F<\/em><\/td>\r\n<td style=\"text-align: center;\">=<\/td>\r\n<td>fractionation factor (dimensionless)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nAs discussed in Section\u00a02, when gas bubbles completely dissolve, the fractionation factor (<em>F<\/em>) is 0 and the CE model reverts to the UA model; such that Equation\u00a0<span data-custom-style=\"Hyperlink\">(20)<\/span> becomes Equation\u00a0<span data-custom-style=\"Hyperlink\">(25)<\/span>.\r\n\r\n<\/div>\r\n<table id=\"equation-25\" style=\"width: 85%;\" data-custom-style=\"Table\"><colgroup> <col style=\"width: 8%;\" \/> <col style=\"width: 64%;\" \/> <col style=\"width: 11%;\" \/> <\/colgroup>\r\n<thead>\r\n<tr>\r\n<th><\/th>\r\n<th style=\"text-align: center;\">[latex]x = \\frac{C}{K_{h}(P - e) + A'}[\/latex]<\/th>\r\n<th style=\"text-align: right;\"><span id=\"X855942feeac10e911d732f802ca87db9083560f\" class=\"anchor\"><\/span>(25)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<\/table>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nwhere:\r\n\r\n<\/div>\r\n<table style=\"width: 85%;\" data-custom-style=\"Table\"><colgroup> <col style=\"width: 12%;\" \/> <col style=\"width: 3%;\" \/> <col style=\"width: 69%;\" \/> <\/colgroup>\r\n<thead>\r\n<tr>\r\n<th style=\"text-align: right;\"><em>A\u2019<\/em><\/th>\r\n<th style=\"text-align: center;\">=<\/th>\r\n<th>amount of excess air dissolved per unit mass of water (molM<sup>-1<\/sup>, typically in moles\/kg)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<\/table>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nAs discussed in Section\u00a02.6, <em>A<\/em> in Equation\u00a0<span data-custom-style=\"Hyperlink\">(24)<\/span> and <em>A\u2019<\/em> in Equation\u00a0<span data-custom-style=\"Hyperlink\">(25)<\/span> are not generally equivalent except when air bubbles are fully dissolved as represented by <em>F<\/em> = 0.\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nA general procedure for calculating a tracer age using [latex]{SF}_{6}[\/latex] is as follows:\r\n\r\n<\/div>\r\n<ol type=\"1\">\r\n \t<li>Measure the aqueous concentration (<em>C<\/em>) of [latex]{SF}_{6}[\/latex] in a groundwater sample.<\/li>\r\n \t<li>Estimate the recharge elevation (<em>Z<\/em>) of the sample.<\/li>\r\n \t<li>Compute the recharge pressure (<em>P<\/em>) using Equation\u00a0<span data-custom-style=\"Hyperlink\">(5)<\/span>: [latex]P = P_{0}\\exp\\left( - \\frac{Z}{Z_{s}} \\right)[\/latex] where [latex]P_{0}[\/latex] is the pressure at <em>Z<\/em>[latex] = [\/latex]0 (1 atm) and [latex]Z_{s}[\/latex] is a scaling factor ([latex]Z_{s}[\/latex] = 8,300 m for <em>Z<\/em>\u00a0&lt;\u00a01,800\u00a0m).<\/li>\r\n \t<li>Estimate the recharge temperature for the sample. This could utilize noble gas thermometry, if available, or could be taken as the average annual temperature of the recharge zone for the sample.<\/li>\r\n \t<li>Estimate the recharge salinity for the sample. In many cases this is assumed to be zero for fresh water in the recharge area.<\/li>\r\n \t<li>Compute the equilibrium constant ([latex]K_{h}[\/latex]) for the temperature and salinity of the recharge area. This could be computed using the equations of Bullister and others (2002) or could be approximated from the values given in <a href=\"#table-6\"><span data-custom-style=\"Hyperlink\">Table <strong>6<\/strong><\/span><\/a>.<\/li>\r\n \t<li>Compute the vapor pressure of water (<em>e<\/em>) at the recharge temperature. This could be computed using empirical equations such as Antoine (1888; Section\u00a02.3) or could be interpolated from the values given in <a href=\"#table-6\"><span data-custom-style=\"Hyperlink\">Table <strong>6<\/strong><\/span><\/a>.<\/li>\r\n \t<li>Estimate the amount of excess air in the sample. If noble gas data are available, the values of <em>A<\/em> and <em>F<\/em> for the CE model or <em>A\u2019<\/em> for the UA model can be obtained by fitting theoretical values to measured values as discussed in Section\u00a02.6. The units for <em>A<\/em> or <em>A\u2019<\/em> must match those of <em>C<\/em>. If a noble gas model provides <em>A<\/em> or <em>A\u2019<\/em> in units of ccSTP\/g, these can be converted to mole\/g by dividing by the molar volume of an ideal gas (22414\u00a0ml\/mole).<\/li>\r\n \t<li>Compute the mole fraction (x) of [latex]{SF}_{6}[\/latex] in the atmosphere that would have resulted in the measured aqueous concentration using <span data-custom-style=\"Hyperlink\">(24)<\/span> for the CE model or Equation\u00a0<span data-custom-style=\"Hyperlink\">(25)<\/span> for the UA model.<\/li>\r\n \t<li>Compare the computed mole fraction\u2014which is essentially equivalent to the mixing ratio\u2014to the atmospheric history (<a href=\"\/age-dating-young-groundwater\/chapter\/5-1-background-and-historical-development\/#figure-20\"><span data-custom-style=\"Hyperlink\">Figure 20<\/span><\/a>) to derive the estimated year the sample recharged.<\/li>\r\n \t<li>Compute the age as sample date minus recharge year.<\/li>\r\n<\/ol>\r\n<a href=\"\/age-dating-young-groundwater\/chapter\/exercise-7\/\"><span data-custom-style=\"Hyperlink\"><u>Exercise\u00a07<\/u><\/span><\/a>\u2ba7 provides an opportunity to practice calculating tracer age using [latex]{SF}_{6}[\/latex].\r\n\r\n<\/div>","rendered":"<div id=\"calculation-of-tracer-age-2\" class=\"level1\">\n<div data-custom-style=\"First Paragraph\">\n<p>As a transient tracer, the general concept of [latex]{SF}_{6}[\/latex] dating involves comparing a measured concentration in a water sample with historical concentrations in the atmosphere. Henry\u2019s Law is used to relate the measured aqueous concentration of a sample to the equilibrium gas\u2011phase concentration. Because the Henry Coefficient is temperature dependent, and because the partial pressure of [latex]{SF}_{6}[\/latex] in the atmosphere depends on total pressure, both the recharge temperature and pressure (i.e., elevation) are needed for age dating. Furthermore, if groundwater contains excess air, and therefore some excess SF<sub>6<\/sub> above equilibrium solubility, the amount of excess air is also required.<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>The temperature and salinity dependence of the Henry Coefficient was investigated by Bullister and others (2002), who provided an empirical equation for calculating the equilibrium constant ([latex]K_{h}[\/latex]) for the solubility form of Henry\u2019s Law as shown by Equation\u00a0<span data-custom-style=\"Hyperlink\">(23)<\/span>.<\/p>\n<\/div>\n<table style=\"width: 85%;\" data-custom-style=\"Table\">\n<colgroup>\n<col style=\"width: 8%;\" \/>\n<col style=\"width: 64%;\" \/>\n<col style=\"width: 11%;\" \/> <\/colgroup>\n<thead>\n<tr>\n<th><\/th>\n<th style=\"text-align: center;\">[latex]C = K_{h}(T,S)\\left( P - e(T) \\right)[\/latex]<\/th>\n<th style=\"text-align: right;\"><span id=\"Xefc6fe3e8250cb645ec82722a2955f3ec328c0f\" class=\"anchor\"><\/span>(23)<\/th>\n<\/tr>\n<\/thead>\n<\/table>\n<div data-custom-style=\"Body Text\">\n<p>where:<\/p>\n<\/div>\n<table style=\"width: 85%;\" data-custom-style=\"Table\">\n<colgroup>\n<col style=\"width: 11%;\" \/>\n<col style=\"width: 3%;\" \/>\n<col style=\"width: 69%;\" \/> <\/colgroup>\n<thead>\n<tr>\n<th style=\"text-align: right;\"><em>C<\/em><\/th>\n<th style=\"text-align: center;\">=<\/th>\n<th>aqueous concentration (molM<sup>-1<\/sup>, typically in moles\/kg)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"text-align: right;\"><em>K<sub>h<\/sub><\/em><\/td>\n<td style=\"text-align: center;\">=<\/td>\n<td>temperature- and salinity-dependent equilibrium constant (molM<sup>\u20112<\/sup>LT<sup>2<\/sup> typically in moles\u00a0kg<sup>\u20111<\/sup>\u00a0atm<sup>\u20111<\/sup>)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><em>x<\/em><\/td>\n<td style=\"text-align: center;\">=<\/td>\n<td>dry air mole fraction (dimensionless)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><em>P<\/em><\/td>\n<td style=\"text-align: center;\">=<\/td>\n<td>atmospheric pressure (ML<sup>-1<\/sup>T<sup>-2<\/sup>, typically in atm)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><em>e<\/em><\/td>\n<td style=\"text-align: center;\">=<\/td>\n<td>temperature dependent vapor pressure of water (ML<sup>-1<\/sup>T<sup>-2<\/sup>, typically in atm)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-custom-style=\"Body Text\">\n<p>Values for [latex]K_{h}[\/latex] and <em>e<\/em> at various temperatures for fresh water are shown in <a href=\"#table-6\"><span data-custom-style=\"Hyperlink\">Table <strong>6<\/strong><\/span><\/a>. If the recharge temperature (<em>T<\/em>), salinity (<em>S<\/em>), and elevation (and hence <em>P<\/em>) are known, Equation\u00a0<span data-custom-style=\"Hyperlink\">(23)<\/span> can be solved for the dry air mole fraction (<em>x<\/em>) and then <em>x<\/em> can be computed for a measured aqueous concentration. The dry air mole fraction can then be compared to the history of [latex]{SF}_{6}[\/latex] in the atmosphere to obtain a recharge date as illustrated in <a href=\"\/age-dating-young-groundwater\/chapter\/5-2-basic-concepts-and-systematics\/#figure-21\"><span data-custom-style=\"Hyperlink\">Figure 21<\/span><\/a>.<\/p>\n<\/div>\n<div data-custom-style=\"Block Text\">\n<blockquote>\n<div id=\"table-6\" class=\"figcaption-text\">\n<p><span id=\"Xcc8452be30d820ec398be18a66806259ff028e2\" class=\"anchor\"><\/span><strong>Table 6\u00a0\u2011<\/strong>\u00a0Values of <em>K<sub>h<\/sub><\/em> for SF<sub>6<\/sub> and water vapor (<em>e<\/em>) at various temperatures for fresh water (salinity = 0).<\/p>\n<\/div>\n<\/blockquote>\n<\/div>\n<table style=\"width: 44%;\" data-custom-style=\"Table\">\n<colgroup>\n<col style=\"width: 13%;\" \/>\n<col style=\"width: 22%;\" \/>\n<col style=\"width: 7%;\" \/> <\/colgroup>\n<thead>\n<tr>\n<th style=\"text-align: center;\">Temperature (\u00b0C)<\/th>\n<th style=\"text-align: center;\"><em>K<sub>h<\/sub><\/em>\u00a0[mole\u00a0kg<sup>-1<\/sup>\u00a0atm<sup>-1<\/sup>]<\/th>\n<th style=\"text-align: center;\"><em>e<\/em>\u00a0[atm]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"text-align: center;\">0<\/td>\n<td style=\"text-align: center;\">0.0006216<\/td>\n<td style=\"text-align: center;\">0.0060<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">10<\/td>\n<td style=\"text-align: center;\">0.0003969<\/td>\n<td style=\"text-align: center;\">0.0121<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">20<\/td>\n<td style=\"text-align: center;\">0.0002739<\/td>\n<td style=\"text-align: center;\">0.0230<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">30<\/td>\n<td style=\"text-align: center;\">0.0002023<\/td>\n<td style=\"text-align: center;\">0.0418<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">40<\/td>\n<td style=\"text-align: center;\">0.0001587<\/td>\n<td style=\"text-align: center;\">0.0726<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-custom-style=\"Block Text\">\n<blockquote><p><em>K<sub>h<\/sub><\/em> from Bullister and others (2002)<\/p><\/blockquote>\n<\/div>\n<div data-custom-style=\"Block Text\">\n<blockquote><p><em>e<\/em> from Antoine (1888) and Section\u00a02.3 of this book<\/p><\/blockquote>\n<\/div>\n<div data-custom-style=\"First Paragraph\">\n<p>Equation\u00a0<span data-custom-style=\"Hyperlink\">(23)<\/span> represents equilibrium conditions and does not include the effects of excess air. Solomon and others (2010) derived an expression for the dry-air mole fraction\u2014which is essentially equivalent to the mixing ratio\u2014that includes excess air as given by the CE model presented in Section\u00a02.2 and Equation\u00a0<span data-custom-style=\"Hyperlink\">(24)<\/span>.<\/p>\n<\/div>\n<table style=\"width: 85%;\" data-custom-style=\"Table\">\n<colgroup>\n<col style=\"width: 8%;\" \/>\n<col style=\"width: 64%;\" \/>\n<col style=\"width: 11%;\" \/> <\/colgroup>\n<thead>\n<tr>\n<th><\/th>\n<th style=\"text-align: center;\">[latex]x = \\frac{C}{K_{h}(P - e) + \\frac{(1 - F)A}{1 + \\frac{FA}{K_{h}(P - e)}}}[\/latex]<\/th>\n<th style=\"text-align: right;\"><span id=\"X6d12a5cdc28158acd826933e004d56817c4e2f1\" class=\"anchor\"><\/span>(24)<\/th>\n<\/tr>\n<\/thead>\n<\/table>\n<div data-custom-style=\"Body Text\">\n<p>where:<\/p>\n<\/div>\n<table style=\"width: 85%;\" data-custom-style=\"Table\">\n<colgroup>\n<col style=\"width: 12%;\" \/>\n<col style=\"width: 3%;\" \/>\n<col style=\"width: 69%;\" \/> <\/colgroup>\n<thead>\n<tr>\n<th style=\"text-align: right;\"><em>x<\/em><\/th>\n<th style=\"text-align: center;\">=<\/th>\n<th>dry air mole fraction (dimensionless)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"text-align: right;\"><em>C<\/em><\/td>\n<td style=\"text-align: center;\">=<\/td>\n<td>aqueous concentration (molM<sup>-1<\/sup>, typically in mole\/kg)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><em>K<sub>h<\/sub><\/em><\/td>\n<td style=\"text-align: center;\">=<\/td>\n<td>temperature\u2011\u00a0and\u00a0salinity\u2011dependent\u00a0equilibrium\u00a0constant\u00a0(molM<sup>\u20112<\/sup>LT<sup>2<\/sup> typically in moles kg<sup>\u20111<\/sup>atm<sup>\u20111<\/sup>)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><em>P<\/em><\/td>\n<td style=\"text-align: center;\">=<\/td>\n<td>atmospheric pressure (ML<sup>-1<\/sup>T<sup>-2<\/sup>, typically in atm)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><em>e<\/em><\/td>\n<td style=\"text-align: center;\">=<\/td>\n<td>temperature\u2011dependent vapor pressure of water (ML<sup>-1<\/sup>T<sup>-2<\/sup>, typically in atm)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><em>A<\/em><\/td>\n<td style=\"text-align: center;\">=<\/td>\n<td>volume of gas trapped in the porous media per unit mass of water (molM<sup>-1<\/sup>, typically in moles\/kg)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><em>F<\/em><\/td>\n<td style=\"text-align: center;\">=<\/td>\n<td>fractionation factor (dimensionless)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-custom-style=\"Body Text\">\n<p>As discussed in Section\u00a02, when gas bubbles completely dissolve, the fractionation factor (<em>F<\/em>) is 0 and the CE model reverts to the UA model; such that Equation\u00a0<span data-custom-style=\"Hyperlink\">(20)<\/span> becomes Equation\u00a0<span data-custom-style=\"Hyperlink\">(25)<\/span>.<\/p>\n<\/div>\n<table id=\"equation-25\" style=\"width: 85%;\" data-custom-style=\"Table\">\n<colgroup>\n<col style=\"width: 8%;\" \/>\n<col style=\"width: 64%;\" \/>\n<col style=\"width: 11%;\" \/> <\/colgroup>\n<thead>\n<tr>\n<th><\/th>\n<th style=\"text-align: center;\">[latex]x = \\frac{C}{K_{h}(P - e) + A'}[\/latex]<\/th>\n<th style=\"text-align: right;\"><span id=\"X855942feeac10e911d732f802ca87db9083560f\" class=\"anchor\"><\/span>(25)<\/th>\n<\/tr>\n<\/thead>\n<\/table>\n<div data-custom-style=\"Body Text\">\n<p>where:<\/p>\n<\/div>\n<table style=\"width: 85%;\" data-custom-style=\"Table\">\n<colgroup>\n<col style=\"width: 12%;\" \/>\n<col style=\"width: 3%;\" \/>\n<col style=\"width: 69%;\" \/> <\/colgroup>\n<thead>\n<tr>\n<th style=\"text-align: right;\"><em>A\u2019<\/em><\/th>\n<th style=\"text-align: center;\">=<\/th>\n<th>amount of excess air dissolved per unit mass of water (molM<sup>-1<\/sup>, typically in moles\/kg)<\/th>\n<\/tr>\n<\/thead>\n<\/table>\n<div data-custom-style=\"Body Text\">\n<p>As discussed in Section\u00a02.6, <em>A<\/em> in Equation\u00a0<span data-custom-style=\"Hyperlink\">(24)<\/span> and <em>A\u2019<\/em> in Equation\u00a0<span data-custom-style=\"Hyperlink\">(25)<\/span> are not generally equivalent except when air bubbles are fully dissolved as represented by <em>F<\/em> = 0.<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>A general procedure for calculating a tracer age using [latex]{SF}_{6}[\/latex] is as follows:<\/p>\n<\/div>\n<ol type=\"1\">\n<li>Measure the aqueous concentration (<em>C<\/em>) of [latex]{SF}_{6}[\/latex] in a groundwater sample.<\/li>\n<li>Estimate the recharge elevation (<em>Z<\/em>) of the sample.<\/li>\n<li>Compute the recharge pressure (<em>P<\/em>) using Equation\u00a0<span data-custom-style=\"Hyperlink\">(5)<\/span>: [latex]P = P_{0}\\exp\\left( - \\frac{Z}{Z_{s}} \\right)[\/latex] where [latex]P_{0}[\/latex] is the pressure at <em>Z<\/em>[latex]=[\/latex]0 (1 atm) and [latex]Z_{s}[\/latex] is a scaling factor ([latex]Z_{s}[\/latex] = 8,300 m for <em>Z<\/em>\u00a0&lt;\u00a01,800\u00a0m).<\/li>\n<li>Estimate the recharge temperature for the sample. This could utilize noble gas thermometry, if available, or could be taken as the average annual temperature of the recharge zone for the sample.<\/li>\n<li>Estimate the recharge salinity for the sample. In many cases this is assumed to be zero for fresh water in the recharge area.<\/li>\n<li>Compute the equilibrium constant ([latex]K_{h}[\/latex]) for the temperature and salinity of the recharge area. This could be computed using the equations of Bullister and others (2002) or could be approximated from the values given in <a href=\"#table-6\"><span data-custom-style=\"Hyperlink\">Table <strong>6<\/strong><\/span><\/a>.<\/li>\n<li>Compute the vapor pressure of water (<em>e<\/em>) at the recharge temperature. This could be computed using empirical equations such as Antoine (1888; Section\u00a02.3) or could be interpolated from the values given in <a href=\"#table-6\"><span data-custom-style=\"Hyperlink\">Table <strong>6<\/strong><\/span><\/a>.<\/li>\n<li>Estimate the amount of excess air in the sample. If noble gas data are available, the values of <em>A<\/em> and <em>F<\/em> for the CE model or <em>A\u2019<\/em> for the UA model can be obtained by fitting theoretical values to measured values as discussed in Section\u00a02.6. The units for <em>A<\/em> or <em>A\u2019<\/em> must match those of <em>C<\/em>. If a noble gas model provides <em>A<\/em> or <em>A\u2019<\/em> in units of ccSTP\/g, these can be converted to mole\/g by dividing by the molar volume of an ideal gas (22414\u00a0ml\/mole).<\/li>\n<li>Compute the mole fraction (x) of [latex]{SF}_{6}[\/latex] in the atmosphere that would have resulted in the measured aqueous concentration using <span data-custom-style=\"Hyperlink\">(24)<\/span> for the CE model or Equation\u00a0<span data-custom-style=\"Hyperlink\">(25)<\/span> for the UA model.<\/li>\n<li>Compare the computed mole fraction\u2014which is essentially equivalent to the mixing ratio\u2014to the atmospheric history (<a href=\"\/age-dating-young-groundwater\/chapter\/5-1-background-and-historical-development\/#figure-20\"><span data-custom-style=\"Hyperlink\">Figure 20<\/span><\/a>) to derive the estimated year the sample recharged.<\/li>\n<li>Compute the age as sample date minus recharge year.<\/li>\n<\/ol>\n<p><a href=\"\/age-dating-young-groundwater\/chapter\/exercise-7\/\"><span data-custom-style=\"Hyperlink\"><u>Exercise\u00a07<\/u><\/span><\/a>\u2ba7 provides an opportunity to practice calculating tracer age using [latex]{SF}_{6}[\/latex].<\/p>\n<\/div>\n","protected":false},"author":7,"menu_order":25,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-114","chapter","type-chapter","status-publish","hentry"],"part":33,"_links":{"self":[{"href":"https:\/\/books.gw-project.org\/age-dating-young-groundwater\/wp-json\/pressbooks\/v2\/chapters\/114","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/books.gw-project.org\/age-dating-young-groundwater\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/books.gw-project.org\/age-dating-young-groundwater\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/books.gw-project.org\/age-dating-young-groundwater\/wp-json\/wp\/v2\/users\/7"}],"version-history":[{"count":9,"href":"https:\/\/books.gw-project.org\/age-dating-young-groundwater\/wp-json\/pressbooks\/v2\/chapters\/114\/revisions"}],"predecessor-version":[{"id":459,"href":"https:\/\/books.gw-project.org\/age-dating-young-groundwater\/wp-json\/pressbooks\/v2\/chapters\/114\/revisions\/459"}],"part":[{"href":"https:\/\/books.gw-project.org\/age-dating-young-groundwater\/wp-json\/pressbooks\/v2\/parts\/33"}],"metadata":[{"href":"https:\/\/books.gw-project.org\/age-dating-young-groundwater\/wp-json\/pressbooks\/v2\/chapters\/114\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/books.gw-project.org\/age-dating-young-groundwater\/wp-json\/wp\/v2\/media?parent=114"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/books.gw-project.org\/age-dating-young-groundwater\/wp-json\/pressbooks\/v2\/chapter-type?post=114"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/books.gw-project.org\/age-dating-young-groundwater\/wp-json\/wp\/v2\/contributor?post=114"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/books.gw-project.org\/age-dating-young-groundwater\/wp-json\/wp\/v2\/license?post=114"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}