Prebiotic reactions are reactions that might have occurred under the conditions prevalent on the Earth before life emerged, and that can lead to analogs of molecules necessary for life as we now know it. To qualify, a reaction must proceed with favorable rates and equilibrium. M.P. Robertson and S.I. Miller (Science 268:702, 1995) studied the prebiotic synthesis of 5-hydroxymethyl uracil (HMU) under prebiotic conditions by reaction with various nucleophiles (e.g, H₂S, HCN, indole, imidazole). For the reaction of uracil and formaldehyde (HCHO) to form HMU in aqueous solution at pH 7.0, they found the following temperature dependences:

ln k/(L mol^-1 s^-1) = 11.75 – 5488 /( T / K )

ln K = -1.36 + 1794 / ( T / K)

(a) Calculate the rates and equilibrium constants for a selection of possible prebiotic conditions (hint: H₂O needs to be a liquid) and plot them vs. temperature.

(b) Find the activation energy and the standard Gibbs free energy, enthalpy, and entropy of reaction at 25°. (Review Chapter 4 in Laidler & Meiser if you don't recall how to do this). Sketch the energy profile vs. "reaction coordinate" of this reaction at 25°, indicating the known energy quantities.

(c) Is there a set of conditions at which you expect the formation of HMU to be more favorable than at standard conditions? If so, identify the set of conditions and sketch what the energy profile vs. reaction coordinate looks like under these conditions. If not, identify what characteristic(s) of this reaction is (are) making it unfavorable under standard conditions, and speculate under what conditions the reaction might be more favorable. In your answer, you may wish to consider the overall order of reaction that is reflected in the rate and/or the energy and entropy quantities calculated in (b).