integrated rate equation

(noun)

Links concentrations of reactants or products with time; integrated from the rate law.

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 theĀ 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
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