Question

How to determine coefficients for equilibrium constants?

The equilibrium constant built in aspen has the following equation:

ln (Keq) = A + B / T

How to determine these coefficients: A, B

Answer 1

Typically, such temperature dependency coefficients are aimed to be derived from published thermochemical data and not to be directly fitted to equilibrium experimental data. Indeed, for the last case, Van't Hoff equation could, in principle, provide a preferable approach.

Note that ΔG = - RT·ln(Keq) and ΔG = ΔH - T·ΔS, from what it can be seen that the temperature dependency expected for ΔH and ΔS imparts on that predicted for Keq. We may, in principle, apply Hess's law to derive ΔH for the reaction at some given temperature by referring also to temperature-dependency correlations for the molar heat capacity, whose fitting coefficients, given in terms of powers of T, can be found tabulated at well-known reference thermochemical data sources.

Such correlations can be integrated between the reference temperature (Tº) and some higher temperature (Kirchhoff's equation) to derive the reaction enthalpy for the last temperature (T). We can also calculate the entropy of reaction (ΔS) for that temperature based on temperature-dependency correlations for the specific heat. For each (i) reactant or product with molar heat capacity Cpi: SiT = Siº + ∫Tº,T (Cpi / T)dT, except when additional entropic contribution(s) arising from phase transitions are also to be accounted for.

It follows that the temperature dependency predicted for Keq can be derived from that tabulated for the molar heat capacity of reactants and products, which is commonly expressed as Cp = a + b·T + c·T2 + d·T-2. Both ∫CpdT and T·∫(Cp/T)dT terms contribute to express the ΔG temperature dependency. Integration leads to ∫CpdT = a·T + (1/2)b·T2 + (1/3)c·T3 - d·T-1 + k1, and T·∫(Cp/T)dT = (a/K)·T·ln(T/K) + b·T2 + (1/2)·c·T3 - (1/2)d·T-1 + k2·T, where k1 and k2 are (integration) constants and K stands for the temperature unit (kelvin). From that, it is show that we can generally expect a ΔG temperature dependency of the kind ΔG = A·T + B·T2 + C·T·ln(T/K) + D·T3 + E·T-1 + F. That yields a -R·ln(Keq) = A + B·T + C·ln(T/K) + D·T2 + E·T-2 + F/T temperature dependency for the equilibrium constant.

This kind of temperature correlation reduces to that proposed at the enunciate by neglecting the contribution of the parcels that express the quadratic dependency on T2 and T-2. That would be the case when both these contributions are small or jointly tend to compensate each other. By neglecting the quadratic contributions for Cp, so that Cp = a + b·T, we are also conducted to the expression given in the enunciate.

For constant Cp, the temperature dependency for the equilibrium constant could just conform to: ln(Keq) = A' + B'·ln(T/K) + C'/T.

Or

If you have data between keq and T then you can plot in excel and you can determine slope and intercept.

If only if you have experimental data on Keq