integrated rate equation

Examples of integrated rate equation in the following topics:

• The Integrated Rate Law

  • The rate law is a differential equation, meaning that it describes the change in concentration of reactant(s) per change in time.
  • Using calculus, the rate law can be integrated to obtain an integrated rate equation that links concentrations of reactants or products with time directly.
  • We can rearrange this equation to combine our variables, and integrate both sides to get our integrated rate law:
  • However, the integrated first-order rate law is usually written in the form of the exponential decay equation.
  • The final version of this integrated rate law is given by:

• Zero-Order Reactions

  • The rate law for a zero-order reaction is rate = k, where k is the rate constant.
  • By rearranging this equation and using a bit of calculus (see the next concept: The Integrated Rate Law), we get the equation:
  • This is the integrated rate law for a zero-order reaction.
  • Note that this equation has the form $y=mx$.
  • Use graphs of zero-order rate equations to obtain the rate constant and the initial concentration data

• Half-Life

  • If we know the integrated rate laws, we can determine the half-lives for first-, second-, and zero-order reactions.
  • Recall that for a first-order reaction, the integrated rate law is given by:
  • If we plug this in for [A] in our integrated rate law, we have:
  • By rearranging this equation and using the properties of logarithms, we can find that, for a first order reaction:
  • The integrated rate law for a zero-order reaction is given by:

• Experimental Determination of Reaction Rates

  • If we know the order of the reaction, we can plot the data and apply our integrated rate laws.
  • In this equation, a is the absorptivity of a given molecules in solution, which is a constant that is dependent upon the physical properties of the molecule in question, b is the path length that travels through the solution, and C is the concentration of the solution.
  • In this case, the rate law is given by:
  • As discussed in a previous concept, plots derived from the integrated rate laws for various reaction orders can be used to determine the rate constant k.
  • The absorbance is directly proportional to the concentration, so this is simply a plot of the rate law, rate = k[C60O3], and the slope of the line is the rate constant, k.

• Rate-Determining Steps

  • Chemists often write chemical equations for reactions as a single step that shows only the net result of a reaction.
  • It is the "how" of the reaction, whereas the overall balanced equation shows only the "what" of the reaction.
  • In kinetics, the rate of a reaction with several steps is determined by the slowest step, which is known as the rate-determining, or rate-limiting, step.
  • The fact that the experimentally-determined rate law does not match the rate law derived from the overall reaction equation suggests that the reaction occurs over multiple steps.
  • A possible mechanism that explains the rate equation is:

• The Rate Law

  • The rate law for a chemical reaction is an equation that relates the reaction rate with the concentrations or partial pressures of the reactants.
  • In this equation, [A] and [B] express the concentrations of A and B, respectively, in units of moles per liter.
  • To reiterate, the exponents x and y are not derived from the balanced chemical equation, and the rate law of a reaction must be determined experimentally.
  • A certain rate law is given as $Rate=k[H_2][Br_2]^\frac{1}{2}$.
  • The rate law equation for this reaction is: $Rate = k[NO]^{1}[O_{3}]^{1}$.

• Rate Laws for Elementary Steps

  • The sum of each elementary step in a reaction mechanism must yield the overall reaction equation.
  • The rate law of the rate-determining step must agree with the experimentally determined rate law.
  • The overall equation suggests that two NO molecules collide with an oxygen molecule, forming NO2.
  • Note that the two steps here add to the overall reaction equation, as the intermediate N2O2 cancels.
  • However, we cannot simply add the rate laws of each elementary step in order to get the overall reaction rate.

• First-Order Reactions

  • Using the Method of Initial Rates to Determine Reaction Order Experimentally
  • The balanced chemical equation for the decomposition of dinitrogen pentoxide is given above.
  • We then measure the new rate at which the N2O5 decomposes.
  • We can now set up a ratio of the first rate to the second rate:
  • Notice that the left side of the equation is simply equal to 2, and that the rate constants cancel on the right side of the equation.

• Chemical Kinetics and Chemical Equilibrium

  • Each reaction also has a reaction rate.
  • The reaction rate involves differential equations, but in non-mathematical terms it is simply the rate of change in the concentrations.
  • Instead, the reaction rate can be accurately modeled by a rate equation.
  • This is an example of a rate equation that might model the above reaction:
  • You can read more about reaction rates and rate laws in the Kinetics unit.

• Second-Order Reactions

  • If the reaction were second-order in either reactant, it would lead to the following rate laws:
  • In order to determine the reaction order for A, we can set up our first equation as follows:
  • Note that on the right side of the equation, both the rate constant k and the term $(0.200)^y$ cancel.
  • Our equation simplifies to:
  • Manipulate experimentally determined second-order rate law equations to obtain rate constants