of A at equilibrium. Then: Since: the reaction rate becomes: which results in: . A plot of the negative natural logarithm of the concentration of A in time minus the concentration at equilibrium versus time t gives a straight line with slope k1 + k−1. By measurement of [A]e and [P]e the values of K and the two reaction rate constants will be known. Generalization of simple example If the concentration at the time t = 0 is different from above, the simplifications above are invalid, and a system of differential equations must be solved. However, this system can also be solved exactly to yield the following generalized expressions: When the equilibrium constant is close to unity and the reaction rates very fast for instance in conformational analysis of molecules, other methods are required for the determination of rate constants for instance by complete lineshape analysis in NMR spectroscopy. Consecutive reactions If the rate constants for the following reaction are and ; A -> B -> C , then the rate equation is: For reactant A: For reactant B: For product C: With the individual concentrations scaled by the total population of reactants to become probabilities, linear systems of differential equations such as these can be formulated as a master equation. The differential equations can be solved analytically and the integrated rate equations are The steady state approximation leads to very similar results in an easier way. Parallel or competitive reactions When a substance reacts simultaneously to give two different products, a parallel or competitive reaction is said to take place. Two first order reactions A -> B and A -> C , with constants and and rate equations ; and The integrated rate equations are then ; and . One important relationship in this case is One first order and one second order reaction This can be the case when studying a bimolecular reaction and a simultaneous hydrolysis (which can be treated as pseudo order one) takes place: the hydrolysis complicates the study of the reaction kinetics, because some reactant is being "spent" in a parallel reaction. For example, A reacts with R to give our product C, but meanwhile the hydrolysis reaction takes away an amount of A to give B, a byproduct: A + H2O -> B and A + R -> C . The rate equations are: and , where is the pseudo first order constant.