{"id":7978,"date":"2021-06-08T21:58:25","date_gmt":"2021-06-08T21:58:25","guid":{"rendered":"https:\/\/opentextbc.ca\/introductorychemistry\/chapter\/free-energy-under-nonstandard-conditions\/"},"modified":"2021-10-27T16:27:55","modified_gmt":"2021-10-27T16:27:55","slug":"free-energy-under-nonstandard-conditions","status":"publish","type":"chapter","link":"https:\/\/opentextbc.ca\/introductorychemistry\/chapter\/free-energy-under-nonstandard-conditions\/","title":{"raw":"Free Energy under Nonstandard Conditions","rendered":"Free Energy under Nonstandard Conditions"},"content":{"raw":"
\r\n

Learning Objectives<\/p>\r\n\r\n<\/header>\r\n

\r\n
    \r\n \t
  • To be able to determine free energy at nonstandard conditions using the standard change in Gibbs free energy,\u00a0\u0394G<\/em>\u00b0.<\/li>\r\n \t
  • To understand and use the relationship between \u0394G<\/em>\u00b0 and the equilibrium constant K<\/em>.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\nMany reactions do not occur under standard conditions, and therefore we need some ways of determining the free energy under nonstandard conditions.\r\n

    Using Standard Change in Gibbs Free Energy, \u0394G<\/em>\u00b0<\/h1>\r\nThe change in Gibbs free energy under nonstandard conditions, \u0394G<\/em>, can be determined from the standard change in Gibbs free energy, \u0394G<\/em>\u00b0:\r\n

    [latex]\\Delta G=\\Delta G^{\\circ}+RT \\ln Q[\/latex]<\/p>\r\nwhere R<\/em> is the ideal gas constant 8.314 J\/mol K, Q<\/em> is the reaction quotient, and T<\/em> is the temperature in Kelvin.\r\n\r\nUnder standard conditions, the reactant and product solution concentrations are 1 M, or the pressure of gases is 1 bar, and Q<\/em> is equal to 1. Taking the natural logarithm simplifies the equation to:\r\n

    [latex]\\Delta G=\\Delta G^{\\circ}\\text{ (under standard conditions)}[\/latex]<\/p>\r\nUnder nonstandard conditions, Q<\/em> must be calculated (in a manner similar to the calculation for an equilibrium constant). For gases, the concentrations are expressed as partial pressures in the units of either atmospheres or bars, and solutes in the units of molarity.\r\n\r\nFor the reaction [latex]aA + bB\\leftrightharpoons cC+dD[\/latex]:\r\n

    [latex]Q_{\\text{gases}}=\\dfrac{\\left(P_C\\right)^c\\left(P_D\\right)^d}{\\left(P_A\\right)^a\\left(P_B\\right)^b}\\hspace{2em}\\text{or}\\hspace{2em}Q_{\\text{solutes}}=\\dfrac{[C]^c[D]^d}{[A]^a[B]^b}[\/latex]<\/p>\r\n\r\n

    \r\n

    Example 18.7<\/p>\r\n\r\n<\/header>\r\n

    \r\n\r\nConsider the following reaction:\r\n\r\n4NH3<\/sub>(g) + 5O2<\/sub>(g) \u21cc 6H2<\/sub>O(g) + 4NO(g)\r\n
      \r\n \t
    1. Use the thermodynamic data in the appendix to calculate \u0394G<\/em>\u00b0 at 298 K.<\/li>\r\n \t
    2. Calculate \u0394G <\/em>at 298 K for a mixture of 2.0 bar NH3<\/sub>(g), 1.0 bar O2<\/sub>(g), 1.5 bar H2<\/sub>O(g), and 1.2 bar NO(g).<\/li>\r\n<\/ol>\r\nSolution<\/em>\r\n
        \r\n \t
      1. [latex]\\phantom{start}[\/latex]\r\n[latex]\\begin{align*}\r\n\\Delta G^{\\circ}&=\\underset{\\text{products}}{\\sum n\\Delta G^{\\circ}{}_{\\text{f}}}-\\underset{\\text{reactants}}{\\sum m\\Delta G^{\\circ}{}_{\\text{f}}} \\\\\r\n&=[(6\\times -228.6\\text{ kJ\/mol})+(4\\times 87.6\\text{ kJ\/mol})] \\\\\r\n&\\phantom{=}-[(4\\times -16.4\\text{ kJ\/mol})+(5\\times 0.0\\text{ kJ\/mol})] \\\\\r\n&=(-1021.2\\text{ kJ\/mol})-(-65.6\\text{ kJ\/mol}) \\\\\r\n&=-955.6\\text{ kJ\/mol}=-9.56\\times 10^5\\text{ J\/mol}\r\n\\end{align*}[\/latex]<\/li>\r\n \t
      2. [latex]\\phantom{start}[\/latex]\r\n[latex]\\begin{align*}\r\n\\Delta G&=\\Delta G^{\\circ}+RT \\ln Q \\\\\r\nQ&=\\dfrac{(1.5\\text{ bar})^6(1.2\\text{ bar})^4}{(2.0\\text{ bar})^4(1.0\\text{ bar})^5}=1.5 \\\\ \\\\\r\n\\Delta G&=(-9.56\\times 10^5\\text{ J\/mol})+(8.314\\text{ J\/mol K})(298\\text{ K})\\ln(1.5) \\\\\r\n&=(-9.56\\times 10^5\\text{ J\/mol})+(1.0\\times 10^3\\text{ J\/mol}) \\\\\r\n&=-9.6\\times 10^5\\text{ J\/mol}\r\n\\end{align*}[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n

        The Relationship between \u0394G<\/em>\u00b0 and K<\/em><\/h1>\r\nThere is a direct relationship between \u0394G<\/em>\u2070 and the equilibrium constant K<\/em>. We can establish this relationship by substituting the equilibrium values (\u0394G<\/em> = 0, and K <\/em>= Q) <\/em>into the equation for determining free energy change under nonstandard conditions:\r\n

        [latex]\\begin{align*}\r\n\\Delta G&=\\Delta G^{\\circ}+RT\\ln Q \\\\\r\n0&=\\Delta G^{\\circ}+RT\\ln K \\\\\r\n\\Delta G^{\\circ}&=-RT\\ln K\r\n\\end{align*}[\/latex]<\/p>\r\nWe now have a way of \u00a0relating the equilibrium constant\u00a0directly to changes in enthalpy and entropy. As well, we can now determine the equilibrium constant from thermochemical data tables or determine the standard free energy change from equilibrium constants.\r\n

        \r\n

        Example 18.8<\/p>\r\n\r\n<\/header>\r\n

        \r\n\r\nThe K<\/em>sp<\/sub> for CuI(s) at 25\u00b0C is 1.27 \u00d7 10\u221212<\/sup>. Determine \u0394G<\/em>\u00b0 for the following:\r\n

        Cu+<\/sup>(aq) + I\u2212<\/sup>(aq) \u2192 CuI(s)<\/p>\r\nSolution<\/em>\r\n

        [latex]\\Delta G^{\\circ}=-RT\\ln K[\/latex]<\/p>\r\nThe equation given is in the opposite direction to the definition of K<\/em>sp<\/sub>:\r\n

        [latex]\\begin{align*}\r\nK&=\\dfrac{1}{K_{\\text{sp}}}=\\left(\\dfrac{1}{1.27}\\times 10^{-12}\\right)=7.87\\times 10^{11} \\\\ \\\\\r\n\\Delta G^{\\circ}&=-(8.314\\text{ J\/mol K})(298\\text{ K})\\ln(7.87\\times 10^{11}) \\\\\r\n&=-(8.314\\text{ J\/mol K})(298\\text{ K})(27.4) \\\\\r\n&=-6.79\\times 10^4\\text{ J\/mol}=-67.9\\text{ kJ\/mol} \\\\\r\n\\end{align*}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

        \r\n

        Key Takeaways<\/p>\r\n\r\n<\/header>\r\n

        \r\n
          \r\n \t
        • The free energy at nonstandard conditions can be determined using\u00a0<\/em>\u0394G<\/em>\u00a0=\u00a0\u0394G<\/em>\u00b0 + RT<\/em>\u00a0ln\u00a0Q.<\/em><\/li>\r\n \t
        • There is a direct relationship between\u00a0\u0394G<\/em>\u00b0 and the equilibrium constant\u00a0K:\u00a0<\/em>\u0394G<\/em>\u00b0 = \u2212RT<\/em>\u00a0ln\u00a0K.<\/em><\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>","rendered":"
          \n
          \n

          Learning Objectives<\/p>\n<\/header>\n

          \n
            \n
          • To be able to determine free energy at nonstandard conditions using the standard change in Gibbs free energy,\u00a0\u0394G<\/em>\u00b0.<\/li>\n
          • To understand and use the relationship between \u0394G<\/em>\u00b0 and the equilibrium constant K<\/em>.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n

            Many reactions do not occur under standard conditions, and therefore we need some ways of determining the free energy under nonstandard conditions.<\/p>\n

            Using Standard Change in Gibbs Free Energy, \u0394G<\/em>\u00b0<\/h1>\n

            The change in Gibbs free energy under nonstandard conditions, \u0394G<\/em>, can be determined from the standard change in Gibbs free energy, \u0394G<\/em>\u00b0:<\/p>\n

            \"\Delta G=\Delta G^{\circ}+RT \ln Q\"<\/p>\n

            where R<\/em> is the ideal gas constant 8.314 J\/mol K, Q<\/em> is the reaction quotient, and T<\/em> is the temperature in Kelvin.<\/p>\n

            Under standard conditions, the reactant and product solution concentrations are 1 M, or the pressure of gases is 1 bar, and Q<\/em> is equal to 1. Taking the natural logarithm simplifies the equation to:<\/p>\n

            \"\Delta G=\Delta G^{\circ}\text{ (under standard conditions)}\"<\/p>\n

            Under nonstandard conditions, Q<\/em> must be calculated (in a manner similar to the calculation for an equilibrium constant). For gases, the concentrations are expressed as partial pressures in the units of either atmospheres or bars, and solutes in the units of molarity.<\/p>\n

            For the reaction \"aA + bB\leftrightharpoons cC+dD\":<\/p>\n

            \"Q_{\text{gases}}=\dfrac{\left(P_C\right)^c\left(P_D\right)^d}{\left(P_A\right)^a\left(P_B\right)^b}\hspace{2em}\text{or}\hspace{2em}Q_{\text{solutes}}=\dfrac{[C]^c[D]^d}{[A]^a[B]^b}\"<\/p>\n

            \n
            \n

            Example 18.7<\/p>\n<\/header>\n

            \n

            Consider the following reaction:<\/p>\n

            4NH3<\/sub>(g) + 5O2<\/sub>(g) \u21cc 6H2<\/sub>O(g) + 4NO(g)<\/p>\n

              \n
            1. Use the thermodynamic data in the appendix to calculate \u0394G<\/em>\u00b0 at 298 K.<\/li>\n
            2. Calculate \u0394G <\/em>at 298 K for a mixture of 2.0 bar NH3<\/sub>(g), 1.0 bar O2<\/sub>(g), 1.5 bar H2<\/sub>O(g), and 1.2 bar NO(g).<\/li>\n<\/ol>\n

              Solution<\/em><\/p>\n

                \n
              1. \"\phantom{start}\"\n

                  <\/span>   <\/span>\"\begin{align*}<\/p>\n<\/li>\n

              2. \"\phantom{start}\"\n

                  <\/span>   <\/span>\"\begin{align*}<\/p>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n

                The Relationship between \u0394G<\/em>\u00b0 and K<\/em><\/h1>\n

                There is a direct relationship between \u0394G<\/em>\u2070 and the equilibrium constant K<\/em>. We can establish this relationship by substituting the equilibrium values (\u0394G<\/em> = 0, and K <\/em>= Q) <\/em>into the equation for determining free energy change under nonstandard conditions:<\/p>\n

                \n

                  <\/span>   <\/span>\"\begin{align*}<\/p>\n

                We now have a way of \u00a0relating the equilibrium constant\u00a0directly to changes in enthalpy and entropy. As well, we can now determine the equilibrium constant from thermochemical data tables or determine the standard free energy change from equilibrium constants.<\/p>\n

                \n
                \n

                Example 18.8<\/p>\n<\/header>\n

                \n

                The K<\/em>sp<\/sub> for CuI(s) at 25\u00b0C is 1.27 \u00d7 10\u221212<\/sup>. Determine \u0394G<\/em>\u00b0 for the following:<\/p>\n

                Cu+<\/sup>(aq) + I\u2212<\/sup>(aq) \u2192 CuI(s)<\/p>\n

                Solution<\/em><\/p>\n

                \"\Delta G^{\circ}=-RT\ln K\"<\/p>\n

                The equation given is in the opposite direction to the definition of K<\/em>sp<\/sub>:<\/p>\n

                \n

                  <\/span>   <\/span>\"\begin{align*}<\/p>\n<\/div>\n<\/div>\n

                \n
                \n

                Key Takeaways<\/p>\n<\/header>\n

                \n
                  \n
                • The free energy at nonstandard conditions can be determined using\u00a0<\/em>\u0394G<\/em>\u00a0=\u00a0\u0394G<\/em>\u00b0 + RT<\/em>\u00a0ln\u00a0Q.<\/em><\/li>\n
                • There is a direct relationship between\u00a0\u0394G<\/em>\u00b0 and the equilibrium constant\u00a0K:\u00a0<\/em>\u0394G<\/em>\u00b0 = \u2212RT<\/em>\u00a0ln\u00a0K.<\/em><\/li>\n<\/ul>\n<\/div>\n<\/div>\n","protected":false},"author":90,"menu_order":6,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":["jessie-a-key"],"pb_section_license":"cc-by"},"chapter-type":[],"contributor":[49],"license":[54],"class_list":["post-7978","chapter","type-chapter","status-publish","hentry","contributor-jessie-a-key","license-cc-by"],"part":7966,"_links":{"self":[{"href":"https:\/\/opentextbc.ca\/introductorychemistry\/wp-json\/pressbooks\/v2\/chapters\/7978","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/opentextbc.ca\/introductorychemistry\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/opentextbc.ca\/introductorychemistry\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/opentextbc.ca\/introductorychemistry\/wp-json\/wp\/v2\/users\/90"}],"version-history":[{"count":11,"href":"https:\/\/opentextbc.ca\/introductorychemistry\/wp-json\/pressbooks\/v2\/chapters\/7978\/revisions"}],"predecessor-version":[{"id":9035,"href":"https:\/\/opentextbc.ca\/introductorychemistry\/wp-json\/pressbooks\/v2\/chapters\/7978\/revisions\/9035"}],"part":[{"href":"https:\/\/opentextbc.ca\/introductorychemistry\/wp-json\/pressbooks\/v2\/parts\/7966"}],"metadata":[{"href":"https:\/\/opentextbc.ca\/introductorychemistry\/wp-json\/pressbooks\/v2\/chapters\/7978\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/opentextbc.ca\/introductorychemistry\/wp-json\/wp\/v2\/media?parent=7978"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/opentextbc.ca\/introductorychemistry\/wp-json\/pressbooks\/v2\/chapter-type?post=7978"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/opentextbc.ca\/introductorychemistry\/wp-json\/wp\/v2\/contributor?post=7978"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/opentextbc.ca\/introductorychemistry\/wp-json\/wp\/v2\/license?post=7978"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}