{"id":149,"date":"2026-04-10T02:53:12","date_gmt":"2026-04-10T02:53:12","guid":{"rendered":"https:\/\/books.gw-project.org\/age-dating-young-groundwater\/chapter\/ch067-xhtml\/"},"modified":"2026-05-12T02:02:59","modified_gmt":"2026-05-12T02:02:59","slug":"solution-exercise-1","status":"publish","type":"chapter","link":"https:\/\/books.gw-project.org\/age-dating-young-groundwater\/chapter\/solution-exercise-1\/","title":{"raw":"Solution\u00a0Exercise\u00a01","rendered":"Solution\u00a0Exercise\u00a01"},"content":{"raw":"
\r\n
\r\n\r\nThe solution to Exercise 1 has two parts: a) and b).\r\n\r\n<\/div>\r\n
\r\n\r\na) Equation\u00a0(4)<\/span><\/a> is duplicated here.\r\n\r\n<\/div>\r\n
\r\n\r\n[latex]C_{i,eq} = \\frac{\\left( P - e(T) \\right)x_{i}}{K_{w,i}(T,S)}[\/latex]\r\n\r\n<\/div>\r\n
\r\n\r\nCommon values needed for all gases are P<\/em> and e<\/em>(T<\/em>). We calculate P<\/em> as:\r\n\r\n<\/div>\r\n
\r\n\r\n[latex]P = P_{0}\\exp\\left( - \\frac{Z}{Z_{s}} \\right) = 1\\ atm\\ exp\\left( - \\frac{450\\ m}{8300\\ m} \\right)\\ = \\ 0.947\\ atm[\/latex]\r\n\r\n<\/div>\r\n
\r\n\r\nWe calculate e<\/em> at 10\u00a0\u00b0C and convert to atmospheres as shown in the next equation.\r\n\r\n<\/div>\r\n
\r\n\r\n[latex]e = 10^{A - \\frac{B}{C + T}} = 10^{\\left( 8.07131 - \\ \\frac{1730.63}{233.426 + 10} \\right)} = 9.16\\ mmHg\\ \\frac{1atm}{760\\ mmHg} = 0.0121\\ atm[\/latex]\r\n\r\n<\/div>\r\n
\r\n\r\nUsing Kh,i<\/sub><\/em> at 10\u00a0\u00b0C from Table 1<\/span><\/a> and [latex]{x\\ }_{i} [\/latex]from Table\u00a02<\/span><\/a>, we have the values shown in this table.\r\n\r\n<\/div>\r\n<\/colgroup>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
i<\/em><\/th>\r\nKh,i<\/sub><\/em>\r\n\r\n(atm g\/ccSTP)<\/th>\r\nxi<\/sub><\/em><\/th>\r\nCi,eq<\/sub><\/em>\r\n\r\n(ccSTP\/g)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
He<\/td>\r\n111<\/td>\r\n5.24[latex]\\mathbf{\\times}[\/latex]10\u20136<\/sup><\/td>\r\n[latex]\\frac{\\left( \\mathbf{0.947 - 0.0121} \\right)\\mathbf{5.24}\\mathbf{\\times}\\mathbf{10}^{\\mathbf{- 6}}}{\\mathbf{111}}\\mathbf{= 4.41}\\mathbf{\\times}\\mathbf{10}^{\\mathbf{- 8}}[\/latex]<\/td>\r\n<\/tr>\r\n
Ne<\/td>\r\n89.0<\/td>\r\n1.82[latex]\\mathbf{\\times}[\/latex]10\u20135<\/sup><\/td>\r\n[latex]\\frac{\\left( \\mathbf{0.947 - 0.0121} \\right)\\mathbf{1.82}\\mathbf{\\times}\\mathbf{10}^{\\mathbf{- 5}}}{\\mathbf{89.0}}\\mathbf{= 1.91}\\mathbf{\\times}\\mathbf{10}^{\\mathbf{- 7}}[\/latex]<\/td>\r\n<\/tr>\r\n
N2<\/sub><\/td>\r\n53.2<\/td>\r\n0.781<\/td>\r\n[latex]\\frac{\\left( \\mathbf{0.947 - 0.0121} \\right)\\mathbf{0.781}}{\\mathbf{53.2}}\\mathbf{= 1.37}\\mathbf{\\times}\\mathbf{10}^{\\mathbf{- 2}}[\/latex]<\/td>\r\n<\/tr>\r\n
Ar<\/td>\r\n23.9<\/td>\r\n9.34[latex]\\mathbf{\\times}[\/latex]10\u20133<\/sup><\/td>\r\n[latex]\\frac{\\left( \\mathbf{0.947 - 0.0121} \\right)\\mathbf{9.34}\\mathbf{\\times}\\mathbf{10}^{\\mathbf{- 3}}}{\\mathbf{23.9}}\\mathbf{= 3.65}\\mathbf{\\times}\\mathbf{10}^{\\mathbf{- 4}}[\/latex]<\/td>\r\n<\/tr>\r\n
Kr<\/td>\r\n12.4<\/td>\r\n1.14[latex]\\mathbf{\\times}[\/latex]10\u20136<\/sup><\/td>\r\n[latex]\\frac{\\left( \\mathbf{0.947 - 0.0121} \\right)\\mathbf{1.14}\\mathbf{\\times}\\mathbf{10}^{\\mathbf{- 6}}}{\\mathbf{12.4}}\\mathbf{= 8.60}\\mathbf{\\times}\\mathbf{10}^{\\mathbf{- 8}}[\/latex]<\/td>\r\n<\/tr>\r\n
Xe<\/td>\r\n6.52<\/td>\r\n8.7[latex]\\mathbf{\\times}[\/latex]10\u20138<\/sup><\/td>\r\n[latex]\\frac{\\left( \\mathbf{0.947 - 0.0121} \\right)\\mathbf{8.7}\\mathbf{\\times}\\mathbf{10}^{\\mathbf{- 8}}}{\\mathbf{6.52}}\\mathbf{= 1.25}\\mathbf{\\times}\\mathbf{10}^{\\mathbf{- 8}}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n
\r\n\r\nb) Equation\u00a0(9)<\/span><\/a> is shown here.\r\n\r\n<\/div>\r\n
\r\n\r\n[latex]C_{i} = \\frac{\\left( P - e(T) \\right)x_{i}}{K_{w,i}(T,S)} + A'x_{i}[\/latex]\r\n\r\n<\/div>\r\n
\r\n\r\nValues for [latex]x_{i}[\/latex] are given in Table\u00a02<\/span><\/a>, and are shown in the table in Solution Exercise 1a. The first term in Equation\u00a0(9)<\/span><\/a> was computed in Solution\u00a0Exercise\u00a01a. The second term (with A\u2019\u00a0<\/em>= 0.003 ccSTP\/g), concentration (Ci<\/sub><\/em>), and percent change are shown in the next table.\r\n\r\n<\/div>\r\n<\/colgroup>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
i<\/em><\/th>\r\nA\u2019xi<\/sub><\/em>\r\n\r\n(ccSTP\/g)<\/th>\r\nCi<\/sub><\/em>\r\n\r\n(ccSTP\/g)<\/th>\r\n<\/th>\r\nPercent Change\r\n\r\n[latex]\\frac{\\mathbf{C}_{\\mathbf{i}}\\mathbf{-}\\mathbf{C}_{\\mathbf{i,eq}}}{\\mathbf{C}_{\\mathbf{i,eq}}}\\mathbf{\\ X\\ 100\\%}[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
He<\/td>\r\n1.57\u00a0X\u00a010\u20138<\/sup><\/td>\r\n5.99\u00a0X\u00a010\u20138<\/sup><\/td>\r\n<\/td>\r\n36.0%<\/td>\r\n<\/tr>\r\n
Ne<\/td>\r\n5.45\u00a0X\u00a010\u20138<\/sup><\/td>\r\n2.46\u00a0X 10\u20137<\/sup><\/td>\r\n<\/td>\r\n29.0%<\/td>\r\n<\/tr>\r\n
N2<\/sub><\/td>\r\n2.34\u00a0X 10\u20133<\/sup><\/td>\r\n1.61\u00a0X\u00a010\u20132<\/sup><\/td>\r\n<\/td>\r\n17.0%<\/td>\r\n<\/tr>\r\n
Ar<\/td>\r\n2.80\u00a0X\u00a010\u20135<\/sup><\/td>\r\n3.93\u00a0X\u00a010\u20134<\/sup><\/td>\r\n<\/td>\r\n7.7%<\/td>\r\n<\/tr>\r\n
Kr<\/td>\r\n3.42\u00a0X\u00a010\u20139<\/sup><\/td>\r\n8.94 X\u00a010\u20138<\/sup><\/td>\r\n<\/td>\r\n4.0%<\/td>\r\n<\/tr>\r\n
Xe<\/td>\r\n2.61\u00a0X\u00a010\u201310<\/sup><\/td>\r\n1.27 X\u00a010\u20138<\/sup><\/td>\r\n<\/td>\r\n2.1%<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n
\r\n\r\nAs seen in this table, the addition of excess air has a much larger effect on the low solubility gases such as [latex]He[\/latex] and [latex]Ne[\/latex] than the higher solubility gases such as [latex]Kr[\/latex] and [latex]Xe[\/latex].\r\n\r\n<\/div>\r\n
\r\n\r\nReturn to Exercise\u00a01<\/u><\/span><\/a>\u2ba5\r\n\r\n<\/div>\r\n
\r\n\r\nReturn to where text linked to Exercise\u00a01<\/span><\/a>\u2ba5\r\n\r\n<\/div>\r\n<\/div>","rendered":"
\n
\n

The solution to Exercise 1 has two parts: a) and b).<\/p>\n<\/div>\n

\n

a) Equation\u00a0(4)<\/span><\/a> is duplicated here.<\/p>\n<\/div>\n

\n

[latex]C_{i,eq} = \\frac{\\left( P - e(T) \\right)x_{i}}{K_{w,i}(T,S)}[\/latex]<\/p>\n<\/div>\n

\n

Common values needed for all gases are P<\/em> and e<\/em>(T<\/em>). We calculate P<\/em> as:<\/p>\n<\/div>\n

\n

[latex]P = P_{0}\\exp\\left( - \\frac{Z}{Z_{s}} \\right) = 1\\ atm\\ exp\\left( - \\frac{450\\ m}{8300\\ m} \\right)\\ = \\ 0.947\\ atm[\/latex]<\/p>\n<\/div>\n

\n

We calculate e<\/em> at 10\u00a0\u00b0C and convert to atmospheres as shown in the next equation.<\/p>\n<\/div>\n

\n

[latex]e = 10^{A - \\frac{B}{C + T}} = 10^{\\left( 8.07131 - \\ \\frac{1730.63}{233.426 + 10} \\right)} = 9.16\\ mmHg\\ \\frac{1atm}{760\\ mmHg} = 0.0121\\ atm[\/latex]<\/p>\n<\/div>\n

\n

Using Kh,i<\/sub><\/em> at 10\u00a0\u00b0C from Table 1<\/span><\/a> and [latex]{x\\ }_{i}[\/latex]from Table\u00a02<\/span><\/a>, we have the values shown in this table.<\/p>\n<\/div>\n\n\n\n\n\n<\/colgroup>\n\n\n\n\n\n\n\n\n\n
i<\/em><\/th>\nKh,i<\/sub><\/em><\/p>\n

(atm g\/ccSTP)<\/th>\n

xi<\/sub><\/em><\/th>\nCi,eq<\/sub><\/em><\/p>\n

(ccSTP\/g)<\/th>\n<\/tr>\n<\/thead>\n

He<\/td>\n111<\/td>\n5.24[latex]\\mathbf{\\times}[\/latex]10\u20136<\/sup><\/td>\n[latex]\\frac{\\left( \\mathbf{0.947 - 0.0121} \\right)\\mathbf{5.24}\\mathbf{\\times}\\mathbf{10}^{\\mathbf{- 6}}}{\\mathbf{111}}\\mathbf{= 4.41}\\mathbf{\\times}\\mathbf{10}^{\\mathbf{- 8}}[\/latex]<\/td>\n<\/tr>\n
Ne<\/td>\n89.0<\/td>\n1.82[latex]\\mathbf{\\times}[\/latex]10\u20135<\/sup><\/td>\n[latex]\\frac{\\left( \\mathbf{0.947 - 0.0121} \\right)\\mathbf{1.82}\\mathbf{\\times}\\mathbf{10}^{\\mathbf{- 5}}}{\\mathbf{89.0}}\\mathbf{= 1.91}\\mathbf{\\times}\\mathbf{10}^{\\mathbf{- 7}}[\/latex]<\/td>\n<\/tr>\n
N2<\/sub><\/td>\n53.2<\/td>\n0.781<\/td>\n[latex]\\frac{\\left( \\mathbf{0.947 - 0.0121} \\right)\\mathbf{0.781}}{\\mathbf{53.2}}\\mathbf{= 1.37}\\mathbf{\\times}\\mathbf{10}^{\\mathbf{- 2}}[\/latex]<\/td>\n<\/tr>\n
Ar<\/td>\n23.9<\/td>\n9.34[latex]\\mathbf{\\times}[\/latex]10\u20133<\/sup><\/td>\n[latex]\\frac{\\left( \\mathbf{0.947 - 0.0121} \\right)\\mathbf{9.34}\\mathbf{\\times}\\mathbf{10}^{\\mathbf{- 3}}}{\\mathbf{23.9}}\\mathbf{= 3.65}\\mathbf{\\times}\\mathbf{10}^{\\mathbf{- 4}}[\/latex]<\/td>\n<\/tr>\n
Kr<\/td>\n12.4<\/td>\n1.14[latex]\\mathbf{\\times}[\/latex]10\u20136<\/sup><\/td>\n[latex]\\frac{\\left( \\mathbf{0.947 - 0.0121} \\right)\\mathbf{1.14}\\mathbf{\\times}\\mathbf{10}^{\\mathbf{- 6}}}{\\mathbf{12.4}}\\mathbf{= 8.60}\\mathbf{\\times}\\mathbf{10}^{\\mathbf{- 8}}[\/latex]<\/td>\n<\/tr>\n
Xe<\/td>\n6.52<\/td>\n8.7[latex]\\mathbf{\\times}[\/latex]10\u20138<\/sup><\/td>\n[latex]\\frac{\\left( \\mathbf{0.947 - 0.0121} \\right)\\mathbf{8.7}\\mathbf{\\times}\\mathbf{10}^{\\mathbf{- 8}}}{\\mathbf{6.52}}\\mathbf{= 1.25}\\mathbf{\\times}\\mathbf{10}^{\\mathbf{- 8}}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
\n

b) Equation\u00a0(9)<\/span><\/a> is shown here.<\/p>\n<\/div>\n

\n

[latex]C_{i} = \\frac{\\left( P - e(T) \\right)x_{i}}{K_{w,i}(T,S)} + A'x_{i}[\/latex]<\/p>\n<\/div>\n

\n

Values for [latex]x_{i}[\/latex] are given in Table\u00a02<\/span><\/a>, and are shown in the table in Solution Exercise 1a. The first term in Equation\u00a0(9)<\/span><\/a> was computed in Solution\u00a0Exercise\u00a01a. The second term (with A\u2019\u00a0<\/em>= 0.003 ccSTP\/g), concentration (Ci<\/sub><\/em>), and percent change are shown in the next table.<\/p>\n<\/div>\n\n\n\n\n\n\n<\/colgroup>\n\n\n\n\n\n\n\n\n\n
i<\/em><\/th>\nA\u2019xi<\/sub><\/em><\/p>\n

(ccSTP\/g)<\/th>\n

Ci<\/sub><\/em><\/p>\n

(ccSTP\/g)<\/th>\n

<\/th>\nPercent Change<\/p>\n

[latex]\\frac{\\mathbf{C}_{\\mathbf{i}}\\mathbf{-}\\mathbf{C}_{\\mathbf{i,eq}}}{\\mathbf{C}_{\\mathbf{i,eq}}}\\mathbf{\\ X\\ 100\\%}[\/latex]<\/th>\n<\/tr>\n<\/thead>\n

He<\/td>\n1.57\u00a0X\u00a010\u20138<\/sup><\/td>\n5.99\u00a0X\u00a010\u20138<\/sup><\/td>\n<\/td>\n36.0%<\/td>\n<\/tr>\n
Ne<\/td>\n5.45\u00a0X\u00a010\u20138<\/sup><\/td>\n2.46\u00a0X 10\u20137<\/sup><\/td>\n<\/td>\n29.0%<\/td>\n<\/tr>\n
N2<\/sub><\/td>\n2.34\u00a0X 10\u20133<\/sup><\/td>\n1.61\u00a0X\u00a010\u20132<\/sup><\/td>\n<\/td>\n17.0%<\/td>\n<\/tr>\n
Ar<\/td>\n2.80\u00a0X\u00a010\u20135<\/sup><\/td>\n3.93\u00a0X\u00a010\u20134<\/sup><\/td>\n<\/td>\n7.7%<\/td>\n<\/tr>\n
Kr<\/td>\n3.42\u00a0X\u00a010\u20139<\/sup><\/td>\n8.94 X\u00a010\u20138<\/sup><\/td>\n<\/td>\n4.0%<\/td>\n<\/tr>\n
Xe<\/td>\n2.61\u00a0X\u00a010\u201310<\/sup><\/td>\n1.27 X\u00a010\u20138<\/sup><\/td>\n<\/td>\n2.1%<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
\n

As seen in this table, the addition of excess air has a much larger effect on the low solubility gases such as [latex]He[\/latex] and [latex]Ne[\/latex] than the higher solubility gases such as [latex]Kr[\/latex] and [latex]Xe[\/latex].<\/p>\n<\/div>\n

\n

Return to Exercise\u00a01<\/u><\/span><\/a>\u2ba5<\/p>\n<\/div>\n

\n

Return to where text linked to Exercise\u00a01<\/span><\/a>\u2ba5<\/p>\n<\/div>\n<\/div>\n","protected":false},"author":7,"menu_order":46,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-149","chapter","type-chapter","status-publish","hentry"],"part":45,"_links":{"self":[{"href":"https:\/\/books.gw-project.org\/age-dating-young-groundwater\/wp-json\/pressbooks\/v2\/chapters\/149","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":8,"href":"https:\/\/books.gw-project.org\/age-dating-young-groundwater\/wp-json\/pressbooks\/v2\/chapters\/149\/revisions"}],"predecessor-version":[{"id":445,"href":"https:\/\/books.gw-project.org\/age-dating-young-groundwater\/wp-json\/pressbooks\/v2\/chapters\/149\/revisions\/445"}],"part":[{"href":"https:\/\/books.gw-project.org\/age-dating-young-groundwater\/wp-json\/pressbooks\/v2\/parts\/45"}],"metadata":[{"href":"https:\/\/books.gw-project.org\/age-dating-young-groundwater\/wp-json\/pressbooks\/v2\/chapters\/149\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/books.gw-project.org\/age-dating-young-groundwater\/wp-json\/wp\/v2\/media?parent=149"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/books.gw-project.org\/age-dating-young-groundwater\/wp-json\/pressbooks\/v2\/chapter-type?post=149"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/books.gw-project.org\/age-dating-young-groundwater\/wp-json\/wp\/v2\/contributor?post=149"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/books.gw-project.org\/age-dating-young-groundwater\/wp-json\/wp\/v2\/license?post=149"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}