Concentration evolution for reactions of different orders

Only unidirectional elementary reactions are considered in this overview chapter because these reactions are relatively simple to treat. More complicated homogeneous reactions are discussed in Chapter 2. 

An example of a zeroth-order reaction is Reaction l-9f, 2H20(aq) H30+(aq) + OH (aq). For zeroth-order reactions, the concentrations of the reactants do not vary (which is why they are zeroth-order reactions). Use the reaction rate progress parameter Then 

There are many examples of first-order reactions. The most often encountered in geochemistry is the radioactive decay of an unstable nuclide. For example, the rate law for the decay of Sm (Reaction 1-2) can be written as 

The concentration of the radioactive nuclide (reactant, such as Sm) decreases exponentially, which is referred to as radioactive decay. The concentration of the daughter nuclides (products, including Nd and He) grows, which is referred to as radiogenic growth. Note the difference between Equations l-47b and l-47c. In the former equation, the concentration of Nd at time t is expressed as a function of the initial Sm concentration. Hence, from the initial state, one can calculate how the Nd concentration would evolve. In the latter equation, the concentration of Nd at time t is expressed as a function of the Sm concentration also at time t. Let s now define time t as the present time. Then [ Nd] is related to the present amount of Sm, the age (time since Sm and Nd were fractionated), and the initial amount of Nd. Therefore, Equation l-47b represents forward calculation, and Equation l-47c represents an inverse problem to obtain either the age, or the initial concentration, or both. Equation l-47d assumes that there are no other ot-decay nuclides. However, U and Th are usually present in a rock or mineral, and their contribution to He usually dominates and must be added to Equation l-47d. 

Similarly, for Reaction 1-1, the concentration evolution with time can be written as 

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