Editing Equilibrium constant (section)

=== A more complex formulation ===
The calculation of ''K'' at a particular temperature from a known ''K'' at another given temperature can be approached as follows if standard thermodynamic properties are available. The effect of temperature on equilibrium constant is equivalent to the effect of temperature on Gibbs energy because:

:<math>\ln K = {{-\Delta_\mathrm{r} G^\ominus} \over {RT}}</math>

where Δ<sub>r</sub>''G''<sup><s>o</s></sup> is the reaction standard Gibbs energy, which is the sum of the standard Gibbs energies of the reaction products minus the sum of standard Gibbs energies of reactants.

Here, the term "standard" denotes the ideal behaviour (i.e., an infinite dilution) and a hypothetical standard concentration (typically 1&nbsp;mol/kg). It does not imply any particular temperature or pressure because, although contrary to IUPAC recommendation, it is more convenient when describing aqueous systems over wide temperature and pressure ranges.<ref>{{cite book|first1=V. |last1=Majer |first2=J. |last2=Sedelbauer |last3=Wood |chapter=Calculations of standard thermodynamic properties of aqueous electrolytes and nonelectrolytes|title=Aqueous Systems at Elevated Temperatures and Pressures: Physical Chemistry of Water, Steam and Hydrothermal Solutions|editor1-first=D. A.|editor1-last=Palmer |editor2-first=R. |editor2-last=Fernandez-Prini|editor3-first=A.|editor3-last=Harvey |publisher=Elsevier|date=2004}}{{page needed|date=November 2011}}</ref>

The standard Gibbs energy (for each species or for the entire reaction) can be represented (from the basic definitions) as:

:<math>G_{T_2}^\ominus = G_{T_1}^\ominus-S_{T_1}^\ominus(T_2-T_1)-T_2 \int^{T_2}_{T_1} {{C_p^\ominus} \over {T}}\,dT + \int^{T_2}_{T_1} C_p^\ominus\,dT</math>

In the above equation, the effect of temperature on Gibbs energy (and thus on the equilibrium constant) is ascribed entirely to heat capacity. To evaluate the integrals in this equation, the form of the dependence of heat capacity on temperature needs to be known.

If the standard molar heat capacity  ''C''{{su|b=''p''|p=<s>o</s>}} can be approximated by some analytic function of temperature (e.g. the [[Shomate equation]]), then the integrals involved in calculating other parameters may be solved to yield analytic expressions for them. For example,  using approximations of the following forms:<ref name= Roberge>{{cite book|first=P. R. |last=Roberge |title=Handbook of Corrosion Engineering |chapter=Appendix F |publisher=McGraw-Hill |page=1037ff|date=November 2011}}</ref>
*For pure substances (solids, gas, liquid): <math display="block">C_p^\ominus \approx A + BT + CT^{-2}</math>
*For ionic species at {{nowrap|''T'' < 200 °C}}: <math display="block">C_p^\ominus \approx (4.186a+b\breve{S}^\ominus_{T_1}) {{(T_2-T_1)} \over {\ln\left(\frac{T_2}{T_1}\right)}}</math>

then the integrals can be evaluated and the following final form is obtained:

:<math>G_{T_2}^\ominus \approx G_{T_1}^\ominus + (C_p^\ominus - S_{T_1}^\ominus)(T_2-T_1) - T_2 \ln\left(\frac{T_2}{T_1}\right)C_p^\ominus</math>

The constants ''A'', ''B'', ''C'', ''a'', ''b'' and the absolute entropy, ''S̆''{{su|b=298&nbsp;K|p=&nbsp;&nbsp;<s>o</s>}}, required for evaluation of ''C''{{su|b=''p''|p=<s>o</s>}}(''T''), as well as the values of ''G''<sub>298&nbsp;K</sub> and ''S''<sub>298&nbsp;K</sub> for many species are tabulated in the literature.