In a paper by 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 L-isoleucene ↔D-alloisoleucine, (shortened to iso↔allo), produces the noncanonical amino acid D-alloisoleucine that can be measured using an automatic amino acid analyzer. At 293 K, this first-order reaction has a half-life of 125,000 years, and its activation energy is 139.7 kJ/mol. After a very long time, the ratio of allo/iso reaches an equilibrium value of 1.38. You may assume that this equilibrium constant is temperature independent.
a. For a hippopotamus tooth 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. The correct answer is between 0.4 and 0.6. (Hint: figure the amount of allo present in terms of the amount of iso when the hippo died at the present time and at very long times.)
b. Radiocarbon dating, which is temperature independent, indicated an age of 38,600 years for the hippo tooth. Using the result from part a), estimate the half-life for the process.
c. Calculate the average temperature of this specimen during its residence in the ground. (The present mean temperature of the spring is 301 K.)