Chemical Kinetics The Zeroth Order Integrated Rate Equation

The Rate Law for a general reaction a X + b Y  products is k, m, and n are numbers that are determined experimentally!! This is the differential Rate Law. The differential rate law relates the rate of reaction to the concentrations of the various species in the system. In this form, it isn't useful for predicting how much reactant remains in solution or how much product has been formed in a given amount of time. CHEM 3310

We use the integrated form of the rate law for: Testing to see whether a set of data follows 0th, 1st, or 2nd order. Predicting how much reactant remains in solution or how much product has been formed in a given amount of time. Let’s see what the integrated form of the rate law looks like for: Zeroth, First, and Second Order Rate Laws. CHEM 3310

Zeroth Order Rate Law CHEM 3310

This is the zero order differential rate law. Zero Order (m+n=0) For a general reaction that obeys zero order rate law. a X + b Y  products This is the zero order differential rate law. The rate of a zero order reaction is independent of the concentration of the reactant. Increasing the concentration of the reacting species will not speed up the rate of the reaction. (i.e. Doubling concentration has no effect on rate.) CHEM 3310

This is the zero order differential rate law. Zero Order (m+n=0) For a general reaction that obeys zero order rate law. a X + b Y  products This is the zero order differential rate law. The reaction is experimentally determined to proceed at a constant rate. CHEM 3310

This is the zero order differential rate law. For a general reaction that obeys zero order rate law. a X + b Y  products This is the zero order differential rate law. When this equation is rearranged and both sides are integrated we get the following result. CHEM 3310

that obeys zero order rate law. For a general reaction that obeys zero order rate law. a X + b Y  products This is the zero order differential rate law. Separate the variables Integrate This is the zero order integrated rate law. CHEM 3310

This is the zero order integrated rate law. For a general reaction that obeys zero order rate law. a X + b Y  products This is the zero order integrated rate law. [X]0 is the concentration of X at time=0 [X]t is the concentration of X at time=t Rearrange, [X]t = -ak0t + [X]0 A reaction is zeroth order if concentration data is plotted against time and the result yields straight line. The slope of this resulting line is -ak0 The unit of k0 is M s-1 CHEM 3310

that obeys zero order rate law. For a general reaction that obeys zero order rate law. a X + b Y  products Integrated rate law Differential rate law Rearrange, [X]t = -ak0t + [X]0 Slope = -ak0 CHEM 3310

Zero Order Example: A catalyzed reaction Oxygen atom CO Catalyst CO2 (Pt, Pd or Rh) CO2 CHEM 3310

that obeys zero order rate law. For a general reaction that obeys zero order rate law. a X + b Y  products Integrated rate law Half-life,  The half-life, , of a reaction is the time required for the concentration of the reactant to decrease by half. When t= , Half-life of a zero-order reaction is proportional to the initial concentration of X, [X]0. CHEM 3310

 For a general reaction that obeys zero order rate law. Half-life,  a X + b Y  products Half-life,  

Rate Equations For a general reaction, a X + b Y  products Zeroth Order Differential Rate Law Integrated Rate Law Units of k M s-1 Linear Plot [X] vs. t Half-life CHEM 3310

Determine the rate constant in M s-1 for a zero order reaction, The initial concentration is 2.0 M. After one minute, the concentration is 1.0 M. What is the initial half-live? X  products [X]t = 1.0 M and [X]0= 2.0 M, a = 1 t = (1min) (60. sec/1 min) = 60. s 2.0 M - 1.0M = k (60. sec)     k = rate constant = 0.017 M s-1 Note: Since it took 1 min for the initial concentration to drop to half its value. The half-life for this reaction is 1 min.

Rate Equations For a general reaction, a X + b Y  products Zeroth Order First Order Second Order Differential Rate Law Integrated Rate Law Units of k M s-1 s-1 M-1s-1 Linear Plot [X] vs. t ln([X]) vs. t vs. t Half-life CHEM 3310