Question

(A) The element carbon consists of the stable isotopes ¹²C (98.90 percent of atoms) and ¹³C (1.10 percent of atoms). In addition, carbon contains a small fraction of the radioisotope ¹⁴C (t_1/2= 5730 years), which is continuously formed in the atmosphere by cosmic rays as CO₂. 14C mixes with the isotopes ¹²C and ¹³C via the natural CO₂ cycle. The decay rate of ¹⁴C is described by (N = number of ¹⁴C atoms; t = time; λ = decay constant): 

decay rate = - dN/dt = πN ..........(1)

Integration of (1) leads to the well-known rate law (2) for the radioactive decay: 

N = N₀ eπt .............(2)

  No = number of ¹⁴C atoms at t = 0

(a) What is the mathematical relationship between the parameters α and t_1/2 (= half life)? 

(b) The decay rate of carbon, which is a part of the natural CO₂ cycle, is found to be 13.6 disintegrations per minute and gram of carbon. When a plant (e. g. a tree) dies, it no longer takes part in the CO₂ cycle. As a consequence, the decay rate of carbon decreases. In 1983, a decay rate of 12.0 disintegrations per minute and gram of carbon was measured for a piece of wood which belongs to a ship of the Vikings. In which year was cut the tree from which this piece of wood originated? 

(c) Assume that the error of the decay rate of 12.0 disintegrations per minute and gram of carbon is 0.2 disintegrations per minute and gram of carbon. What is the corresponding error in the age of the wood in question b)? 

d) What is the isotope ¹²C/¹⁴C ratio of carbon, which takes part in the natural CO₂ cycle (1 year = 365 days)? 

(B) The elements strontium and rubidium have the following isotope composition: Strontium: 0.56 % ⁸⁴Sr ; 9.86 % ⁸⁶Sr ; 7.00 % ⁸⁷Sr ; 82.58 % ⁸⁸Sr (these isotopes are all stable). Rubidium: 72.17 % 85Rb (stable) ; 27.83 % ⁸⁷Rb (radioactive; t_1/2 = 4.7 × 10¹⁰ years). The radioactive decay of ⁸⁷Rb leads to ⁸⁷Sr. In Greenland one finds a gneiss (= silicate mineral) containing both strontium and rubidium. 

(a) What is the equation rate law describing the formation of ⁸⁷Sr from ⁸⁷Rb as a function of time? 

(b) Assume that the isotope ratio ⁸⁷Sr/ ⁸⁶Sr (as determined by mass spectrometry) and the isotope ratio ⁸⁷Rb : ⁸⁶Sr are known for the gneiss. What is the mathematical relationship with which one can calculate the age of the gneiss? 

Answer 1

 (B) (a) Equation (2) describes the decay of the 

⁸⁷Rb: ⁸⁷Rb = ⁸⁷Rbo . exp( -λ t) 

 The symbol ⁸⁷Rb stands for the number of atoms of this nuclide. Consequently, one obtains for the formation of ⁸⁷Sr from ⁸⁷Rb: ⁸⁷Sr = ⁸⁷Rbo – ⁸⁷Rb = ⁸⁷Rb . exp(λt) – ⁸⁷Rb 

(b) The formation of the radiogenic ⁸⁷Sr follows equation (a). One has to take into account that at time t = 0, when the mineral was formed, there was some non-radiogenic strontium in it already: 

⁸⁷Sr = (⁸⁷Sr)o + ⁸⁷Rb . [exp(λt) – 1]

The isotope ratio (⁸⁷Sr/ ⁸⁶Sr)_o follows from the isotope composition of strontium. The time t in this equation corresponds to the age of the gneiss.