Can ΔG = ΔG° + RT ln Q be used to calculate K at different temperatures?

Question

Specifically, can $\Delta G=\Delta G^{\circ}+RT \ln Q$ be used to calculate reaction quotients if we choose T to be a different temperature than the one provided by ΔG°?

For example, suppose that we are given ΔG° at 25°C and we are tasked with finding the equilibrium constant at 30°C. Do these equations hold? $$0 \stackrel{?}{=} \Delta G^{\circ}+RT \ln K$$ $$K \stackrel{?}{=} e^\frac{-\Delta G^{\circ}}{RT}$$

I believe this is illegal, as it contradicts the fact that some equilibria are less favored at higher temperatures. However, I'm having trouble finding confirmation online, especially since the above equation has no easily searchable name.

Answer 1

As mentioned in the comments, $K$ is itself temperature dependent. That is to say, $K = K(T)$ is a function of temperature, so

$$\Delta_\mathrm{r} G^\circ = -RT\ln K(T)$$

and the temperature dependence in $\Delta_\mathrm{r} G^\circ$ not only arises from the $RT$ term but also the $K$ term.

The dependence of $K$ on temperature is described by the van 't Hoff equation. I've written about it here (a while ago!).

$$\frac{\mathrm d(\ln{K})}{\mathrm dT} = \frac{\Delta_\mathrm{r} H^\circ}{RT^2}$$