Standard Gibbs Energy and the Equilibrium Constant vs Temperature
For a reaction written with its standard states, the temperature-dependence of the
equilibrium constant is tied to the Gibbs energy of reaction:
\[
\Delta_r G^\circ(T) \;=\; \Delta_r H^\circ \;-\; T\,\Delta_r S^\circ
\]
\[
K(T) \;=\; \exp\!\left[-\dfrac{\Delta_r G^\circ(T)}{R\,T}\right]
\]
Assumptions
- \(\Delta_r H^\circ\) and \(\Delta_r S^\circ\) are treated as temperature-independent over the range of interest.
- Activities are approximated by concentrations/partial pressures (so the numerical \(K\) you compute matches typical textbook values).
- Gas constant \(R = 8.314462618\ \mathrm{J\,mol^{-1}\,K^{-1}}\).
Working Forms Used by the Calculator
Starting from the definitions above, the following equivalent expressions are used:
\[
\Delta_r G^\circ(T) \;=\; \Delta_r H^\circ - T\,\Delta_r S^\circ
\]
\[
\ln K \;=\; -\,\dfrac{\Delta_r G^\circ(T)}{R\,T}
\;=\; -\,\dfrac{\Delta_r H^\circ}{R\,T} \;+\; \dfrac{\Delta_r S^\circ}{R}
\]
\[
\boxed{\,K \;=\; \exp\!\left[-\dfrac{\Delta_r H^\circ - T\,\Delta_r S^\circ}{R\,T}\right]\,}
\]
Solving for temperature from a known \(K\) (algebraic rearrangement):
\[
\Delta_r H^\circ - T\,\Delta_r S^\circ = -R\,T\ln K
\]
\[
\boxed{\,T \;=\; \dfrac{\Delta_r H^\circ}{\Delta_r S^\circ - R\ln K}\,}
\qquad\text{(undefined if } \Delta_r S^\circ = R\ln K\text{)}
\]
Interpretation & Trends
- Spontaneity at a given \(T\): \(\Delta_r G^\circ(T)<0 \Rightarrow K>1\) (products favored);
\(\Delta_r G^\circ(T)>0 \Rightarrow K<1\) (reactants favored).
- Temperature effect (van ’t Hoff-like form):
\(\displaystyle \ln K = -\frac{\Delta_r H^\circ}{R}\left(\frac{1}{T}\right) + \frac{\Delta_r S^\circ}{R}\).
Plotting \(\ln K\) vs \(1/T\) is linear with slope \(-\Delta_r H^\circ/R\).
Endothermic (\(\Delta_r H^\circ>0\)) → \(K\) increases with \(T\); exothermic (\(\Delta_r H^\circ<0\)) → \(K\) decreases with \(T\).
Units & Conversions
- \(\Delta_r H^\circ\): use \(\mathrm{J\,mol^{-1}}\) (convert kJ·mol⁻¹ → J·mol⁻¹ by ×1000).
- \(\Delta_r S^\circ\): use \(\mathrm{J\,mol^{-1}\,K^{-1}}\) (convert kJ·mol⁻¹·K⁻¹ → J·mol⁻¹·K⁻¹ by ×1000).
- \(T\) in kelvin; if °C is provided, the calculator converts via \(T(\mathrm{K})=t(^{\circ}\mathrm{C})+273.15\).
- \(K\) is dimensionless (based on activities), though we often enter numerical values built from concentrations or partial pressures.
How the Calculator Computes
- Normalize inputs: convert \(\Delta_r H^\circ\) and \(\Delta_r S^\circ\) to J-based units and \(T\) to K.
- If solving for \(K\): evaluate \(\Delta_r G^\circ(T)\) then compute \(K=\exp[-\Delta_r G^\circ/(RT)]\). The tool also reports \(\ln K\) and \(\log_{10}K\).
- If solving for \(T\): use \(T=\Delta_r H^\circ/(\Delta_r S^\circ - R\ln K)\). If the denominator is \(0\), no finite \(T\) satisfies the equation under the constant-\(\Delta_r H^\circ,\Delta_r S^\circ\) approximation.
- If solving for \(\Delta_r G^\circ\): compute \(\Delta_r G^\circ(T)=\Delta_r H^\circ - T\Delta_r S^\circ\) and (optionally) infer \(K\).
Sanity Checks
- Large positive \(\Delta_r H^\circ\) and \(\Delta_r S^\circ\) often give small \(K\) at low \(T\) but larger \(K\) at high \(T\) (entropy wins at high \(T\)).
- If \(|\Delta_r S^\circ|\) is small, \(K\) is dominated by enthalpy (\(\ln K \approx -\Delta_r H^\circ/(RT)\)).
- Extremely large/small \(K\) values are displayed in scientific notation for clarity.
Limitations
- The assumption of constant \(\Delta_r H^\circ\) and \(\Delta_r S^\circ\) can break down over wide temperature ranges or when heat capacities change substantially.
- Non-ideal solutions/gases require activities (activity coefficients, fugacity coefficients) for high accuracy.
- Ensure the reaction stoichiometry and standard states correspond to the tabulated \(\Delta_r H^\circ\) and \(\Delta_r S^\circ\) you use.
