{"id":74,"date":"2026-04-10T02:52:58","date_gmt":"2026-04-10T02:52:58","guid":{"rendered":"https:\/\/books.gw-project.org\/age-dating-young-groundwater\/chapter\/ch015-xhtml\/"},"modified":"2026-05-12T02:05:31","modified_gmt":"2026-05-12T02:05:31","slug":"2-3-atmospheric-equilibrium","status":"publish","type":"chapter","link":"https:\/\/books.gw-project.org\/age-dating-young-groundwater\/chapter\/2-3-atmospheric-equilibrium\/","title":{"raw":"2.3 Atmospheric Equilibrium","rendered":"2.3 Atmospheric Equilibrium"},"content":{"raw":"
\r\n
\r\n\r\nWhen water is in contact with the atmosphere (at a constant pressure, temperature, and humidity) for sufficient time, an equilibrium concentration is reached that can be calculated using Henry\u2019s Law. While the Henry Coefficient is a function of both temperature and salinity, it is not a function of the total atmospheric pressure. However, the equilibrium concentration of air in water is a function of the total gas pressure and hence is a function of elevation. Equation\u00a0(4)<\/span> is a modification of Henry\u2019s Law that gives the concentration of gas [latex]x_{i}[\/latex] in air\u2011saturated water (ASW).\r\n\r\n<\/div>\r\n<\/colgroup>\r\n\r\n\r\n
<\/th>\r\n[latex]C_{i,eq} = \\frac{\\left( P - e(T) \\right)x_{i}}{K_{w,i}(T,S)}[\/latex]<\/th>\r\n<\/span>(4)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<\/table>\r\n
\r\n\r\nwhere:\r\n\r\n<\/div>\r\n<\/colgroup>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
[latex]C_{i,eq}[\/latex]<\/th>\r\n=<\/th>\r\nequilibrium concentration of gas i<\/em> in air\u2011saturated water (L3<\/sup>M-1<\/sup>, typically in ccSTP\/g)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
P<\/em><\/td>\r\n=<\/td>\r\natmospheric pressure (ML-1<\/sup>T-2<\/sup>, typically in atm)<\/td>\r\n<\/tr>\r\n
e<\/em>(T<\/em>)<\/td>\r\n=<\/td>\r\nvapor pressure of water at temperature T<\/em> (ML-1<\/sup>T-2<\/sup>, typically in atm)<\/td>\r\n<\/tr>\r\n
[latex]K_{w,i}(T,S)[\/latex]<\/td>\r\n=<\/td>\r\nHenry Coefficient of gas i<\/em> at temperature T<\/em> and salinity S<\/em> (M2<\/sup>L-4<\/sup>T-2<\/sup>, typically in g\u2011atm\/ccSTP)<\/td>\r\n<\/tr>\r\n
[latex]x_{i}[\/latex]<\/td>\r\n=<\/td>\r\nfraction of gas i<\/em> (mixing ratio) in the dry atmosphere (dimensionless)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n
\r\n\r\nThe atmospheric pressure (P<\/em>) is a function of elevation (Z<\/em>) and can be closely approximated by Equation\u00a0(5)<\/span>.\r\n\r\n<\/div>\r\n<\/colgroup>\r\n\r\n\r\n
<\/th>\r\n[latex]P = P_{0}\\exp\\left( - \\frac{Z}{Z_{s}} \\right)[\/latex]<\/th>\r\n<\/span>(5)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<\/table>\r\n
\r\n\r\nwhere:\r\n\r\n<\/div>\r\n<\/colgroup>\r\n\r\n\r\n\r\n\r\n\r\n\r\n
[latex]P_{\\ }[\/latex]<\/th>\r\n=<\/th>\r\natmospheric pressure (ML-1<\/sup>T-2<\/sup>, typically in atm)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
[latex]P_{0}[\/latex]<\/td>\r\n=<\/td>\r\npressure at elevation (Z<\/em>) = 0 (ML-1<\/sup>T-2<\/sup>, defined as 1 atm)<\/td>\r\n<\/tr>\r\n
[latex]Z[\/latex]<\/td>\r\n=<\/td>\r\nElevation (L)<\/td>\r\n<\/tr>\r\n
[latex]Z_{s}[\/latex]<\/td>\r\n=<\/td>\r\nscaling factor (L)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n
\r\n\r\nUsing a value for [latex]Z_{s}[\/latex] of 8,300\u00a0m this expression is a very good approximation for [latex]Z[\/latex]\u00a0<\/em><\u00a01,800\u00a0m. The vapor pressure of water (e(T)<\/em>) in Equation\u00a0(4)<\/span> is used because the composition of the atmosphere is typically given without water vapor since the vapor pressure of water is highly variable. For groundwater recharge at the water table, the relative humidity of the soil atmosphere is usually assumed to be 100\u00a0percent and then the partial pressure of water vapor depends only on temperature. There are many empirical formulas for computing e<\/em> as a function of temperature. The simple Antoine equation (Antoine, 1888) states the value of e<\/em> as shown in Equation\u00a0(6)<\/span>.\r\n\r\n<\/div>\r\n<\/colgroup>\r\n\r\n\r\n
<\/th>\r\n[latex]e = 10^{A\\ - \\ \\frac{B}{C + T}}[\/latex]<\/th>\r\n<\/span>(6)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<\/table>\r\n
\r\n\r\nwhere:\r\n\r\n<\/div>\r\n<\/colgroup>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
[latex]e_{\\ }[\/latex]<\/th>\r\n=<\/th>\r\nvapor pressure of water in mmHg<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
A<\/em><\/td>\r\n=<\/td>\r\n8.07131<\/td>\r\n<\/tr>\r\n
B<\/em><\/td>\r\n=<\/td>\r\n1,730.63<\/td>\r\n<\/tr>\r\n
C<\/em><\/td>\r\n=<\/td>\r\n233.426<\/td>\r\n<\/tr>\r\n
T<\/em><\/td>\r\n=<\/td>\r\ntemperature in\u00a0\u00b0C<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n
\r\n\r\nIn this calculation, e<\/em> is accurate to better than one\u00a0percent between 0\u00a0\u00b0C and 100\u00a0\u00b0C.\r\n\r\n<\/div>\r\n<\/div>","rendered":"
\n
\n

When water is in contact with the atmosphere (at a constant pressure, temperature, and humidity) for sufficient time, an equilibrium concentration is reached that can be calculated using Henry\u2019s Law. While the Henry Coefficient is a function of both temperature and salinity, it is not a function of the total atmospheric pressure. However, the equilibrium concentration of air in water is a function of the total gas pressure and hence is a function of elevation. Equation\u00a0(4)<\/span> is a modification of Henry\u2019s Law that gives the concentration of gas [latex]x_{i}[\/latex] in air\u2011saturated water (ASW).<\/p>\n<\/div>\n\n\n\n\n<\/colgroup>\n\n\n
<\/th>\n[latex]C_{i,eq} = \\frac{\\left( P - e(T) \\right)x_{i}}{K_{w,i}(T,S)}[\/latex]<\/th>\n<\/span>(4)<\/th>\n<\/tr>\n<\/thead>\n<\/table>\n
\n

where:<\/p>\n<\/div>\n\n\n\n\n<\/colgroup>\n\n\n\n\n\n\n\n
[latex]C_{i,eq}[\/latex]<\/th>\n=<\/th>\nequilibrium concentration of gas i<\/em> in air\u2011saturated water (L3<\/sup>M-1<\/sup>, typically in ccSTP\/g)<\/th>\n<\/tr>\n<\/thead>\n
P<\/em><\/td>\n=<\/td>\natmospheric pressure (ML-1<\/sup>T-2<\/sup>, typically in atm)<\/td>\n<\/tr>\n
e<\/em>(T<\/em>)<\/td>\n=<\/td>\nvapor pressure of water at temperature T<\/em> (ML-1<\/sup>T-2<\/sup>, typically in atm)<\/td>\n<\/tr>\n
[latex]K_{w,i}(T,S)[\/latex]<\/td>\n=<\/td>\nHenry Coefficient of gas i<\/em> at temperature T<\/em> and salinity S<\/em> (M2<\/sup>L-4<\/sup>T-2<\/sup>, typically in g\u2011atm\/ccSTP)<\/td>\n<\/tr>\n
[latex]x_{i}[\/latex]<\/td>\n=<\/td>\nfraction of gas i<\/em> (mixing ratio) in the dry atmosphere (dimensionless)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
\n

The atmospheric pressure (P<\/em>) is a function of elevation (Z<\/em>) and can be closely approximated by Equation\u00a0(5)<\/span>.<\/p>\n<\/div>\n\n\n\n\n<\/colgroup>\n\n\n
<\/th>\n[latex]P = P_{0}\\exp\\left( - \\frac{Z}{Z_{s}} \\right)[\/latex]<\/th>\n<\/span>(5)<\/th>\n<\/tr>\n<\/thead>\n<\/table>\n
\n

where:<\/p>\n<\/div>\n\n\n\n\n<\/colgroup>\n\n\n\n\n\n\n
[latex]P_{\\ }[\/latex]<\/th>\n=<\/th>\natmospheric pressure (ML-1<\/sup>T-2<\/sup>, typically in atm)<\/th>\n<\/tr>\n<\/thead>\n
[latex]P_{0}[\/latex]<\/td>\n=<\/td>\npressure at elevation (Z<\/em>) = 0 (ML-1<\/sup>T-2<\/sup>, defined as 1 atm)<\/td>\n<\/tr>\n
[latex]Z[\/latex]<\/td>\n=<\/td>\nElevation (L)<\/td>\n<\/tr>\n
[latex]Z_{s}[\/latex]<\/td>\n=<\/td>\nscaling factor (L)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
\n

Using a value for [latex]Z_{s}[\/latex] of 8,300\u00a0m this expression is a very good approximation for [latex]Z[\/latex]\u00a0<\/em><\u00a01,800\u00a0m. The vapor pressure of water (e(T)<\/em>) in Equation\u00a0(4)<\/span> is used because the composition of the atmosphere is typically given without water vapor since the vapor pressure of water is highly variable. For groundwater recharge at the water table, the relative humidity of the soil atmosphere is usually assumed to be 100\u00a0percent and then the partial pressure of water vapor depends only on temperature. There are many empirical formulas for computing e<\/em> as a function of temperature. The simple Antoine equation (Antoine, 1888) states the value of e<\/em> as shown in Equation\u00a0(6)<\/span>.<\/p>\n<\/div>\n\n\n\n\n<\/colgroup>\n\n\n
<\/th>\n[latex]e = 10^{A\\ - \\ \\frac{B}{C + T}}[\/latex]<\/th>\n<\/span>(6)<\/th>\n<\/tr>\n<\/thead>\n<\/table>\n
\n

where:<\/p>\n<\/div>\n\n\n\n\n<\/colgroup>\n\n\n\n\n\n\n\n
[latex]e_{\\ }[\/latex]<\/th>\n=<\/th>\nvapor pressure of water in mmHg<\/th>\n<\/tr>\n<\/thead>\n
A<\/em><\/td>\n=<\/td>\n8.07131<\/td>\n<\/tr>\n
B<\/em><\/td>\n=<\/td>\n1,730.63<\/td>\n<\/tr>\n
C<\/em><\/td>\n=<\/td>\n233.426<\/td>\n<\/tr>\n
T<\/em><\/td>\n=<\/td>\ntemperature in\u00a0\u00b0C<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
\n

In this calculation, e<\/em> is accurate to better than one\u00a0percent between 0\u00a0\u00b0C and 100\u00a0\u00b0C.<\/p>\n<\/div>\n<\/div>\n","protected":false},"author":7,"menu_order":3,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-74","chapter","type-chapter","status-publish","hentry"],"part":26,"_links":{"self":[{"href":"https:\/\/books.gw-project.org\/age-dating-young-groundwater\/wp-json\/pressbooks\/v2\/chapters\/74","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":7,"href":"https:\/\/books.gw-project.org\/age-dating-young-groundwater\/wp-json\/pressbooks\/v2\/chapters\/74\/revisions"}],"predecessor-version":[{"id":446,"href":"https:\/\/books.gw-project.org\/age-dating-young-groundwater\/wp-json\/pressbooks\/v2\/chapters\/74\/revisions\/446"}],"part":[{"href":"https:\/\/books.gw-project.org\/age-dating-young-groundwater\/wp-json\/pressbooks\/v2\/parts\/26"}],"metadata":[{"href":"https:\/\/books.gw-project.org\/age-dating-young-groundwater\/wp-json\/pressbooks\/v2\/chapters\/74\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/books.gw-project.org\/age-dating-young-groundwater\/wp-json\/wp\/v2\/media?parent=74"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/books.gw-project.org\/age-dating-young-groundwater\/wp-json\/pressbooks\/v2\/chapter-type?post=74"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/books.gw-project.org\/age-dating-young-groundwater\/wp-json\/wp\/v2\/contributor?post=74"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/books.gw-project.org\/age-dating-young-groundwater\/wp-json\/wp\/v2\/license?post=74"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}