The sign of ΔHrxn and ΔSrxn for several different reactions are given.

In which case is the reaction spontaneous at all temperatures?

ΔHrxn0ΔHrxn<0; ΔSrxn0; ΔSrxn0; ΔSrxn>0

The answer is ΔSrxn>0 and ΔHrxn<0

This is because of the thermodynamic equation of the Gibbs Free Energy given by:

ΔG = Δ H - TΔS

First, it is important to note that spontaneous reactions are usually given by ΔG <0, ΔH0. However, ΔG is the most important identifier of a thermodynamic process, so it is best to figure out the conditions that would give a negative ΔG.

For the equation's first term (ΔH), ΔG decreases when ΔH is decreased. For the second term TΔS, it can be observed that ΔG decreases when TΔS is increased due to the negative or minus sign.

So for all temperatures, ΔG is kept negative if ΔH is negative and TΔS is positive (because of the negative sign).