What is the formula for calculating K at one temperature based on its value at another temperature using van 't Hoff equation?

The van 't Hoff equation provides a relationship between the equilibrium constant (K) of a reaction at two different temperatures and the standard enthalpy change (ΔrH*) of the reaction. The correct formula for calculating K at one temperature based on its value at another temperature is given by:

[ \ln K_2 - \ln K_1 = -\left(\frac{\Delta rH^*}{R}\right)\left(\frac{1}{T_2} - \frac{1}{T_1}\right) ]

This relationship arises from the principle that if the enthalpy change for a reaction is known, it is possible to estimate how the equilibrium position shifts with temperature. The negative sign indicates that if the enthalpy change is positive (endothermic reaction), an increase in temperature results in an increase in K, meaning the equilibrium shifts to favor products. Conversely, for an exothermic reaction (where ΔrH* is negative), increasing the temperature will reduce the value of K.

In this equation, R is the universal gas constant, T_1 and T_2 are the absolute temperatures in Kelvin at which the equilibrium constants K1 and K2 are defined, respectively. The overall