In apaperby Bada, Protsch, and Schroeder [Nature 241:394 (1973)], the rate of isomerizatio...

In apaperby Bada, Protsch, and Schroeder [Nature 241:394 (1973)], the rate of isomerization of isoleucine in fossilized bone is used as an indication of the average temperature of the sample since it was deposited. The reaction

produces a nonbiological amino acid, D-alloisoleucine, that can be measured using an automatic amino acid analyzer. At 20 °C, this first-order reaction has a half-life of 125,000 years, and its activation energy is 139.7 kJ mol-1. After a very long time, the ratio allo/iso reaches an equilibrium value of 1.38. You may assume that this equilibrium con stant is temperature independent.

a. For a hippopotamus mandible found near a warm spring in South Africa, the allo/iso ratio was found to be 0.42. Assuming that no allo was present initially, calculate the ratio of the concentration of allo now present to the concentration of allo after a very long time (Note: The correct answer is between 0.40 and 0.60.)

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b. Radiocarbon dating, which is temperature independent, indicated an age of 38,600 years for the hippo tooth. Using the result of part (a), estimate the half-life for the process.

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c. Calculate the average temperature of this specimen during its residence in the ground. (The present mean temperature of the spring is 28 °C.)