A power plant emits a carbon containing pollutant X to the atmosphere at a constant rate E (kg s^-1). X is removed from the atmosphere by a chemical reaction with a first-order rate constant k (s^-1), forming the greenhouse gas CO₂. X is not a greenhouse gas, and X in the atmosphere has reached a steady state (e.g. constant mole fractions and completely mixed in the entire atmosphere). The emitted pollutant X has a ¹³C value of -20 (VPDB, ¹³R_VPDB=0.011117). The chemical removal of X fractionates with a fractionation factor a= k¹³/ k = 0.988.

a) Write down the mass balance of the rare isotope in terms of N_c (total number of molecules removed) and N_e (total number of molecules emitted) and the fractionation factor a and the isotope ratio of the emission R_E and the isotope ratio of pollutant X in the atmosphere R_A. Assume a steady state condition

b) Calculate the value of the atmosphere from the previously found mass balance.

c) What is the mass balance of the rare isotope if, in addition to the chemical loss in the atmosphere, the pollutant is also removed by rain, e.g. wet deposition. This last process occurs without fractionation. Consider again a stationary state (in and outflow are equal). The fractional contributions of the chemical loss to the removal is x, and that of wet deposition is y, with x + y = 1.

d) Suppose we observe a ¹³C value of -15 with respect to VPDB in the atmosphere (in pollutant X), and given the isotopic composition of the emissions, what are then the relative contributions of the two removal mechanisms, e.g. what is x and x-1?

e) If, due to global warming, the amount of rain would increase, would that increase or decrease the global warming impact of pollutant X?