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 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:
  • Calculus with Parametric Curves

    • Parametric equations are equations which depend on a single parameter.
    • Writing these equations in parametric form gives a common parameter for both equations to depend on.
    • This makes integration and differentiation easier to carry out as they rely on the same variable.
    • The horizontal velocity is the time rate of change of the $x$ value, and the vertical velocity is the time rate of change of the $y$ value.
    • Use differentiation to describe the vertical and horizontal rates of change in terms of $t$
  • Arc Length and Speed

    • Arc length and speed in parametric equations can be calculated using integration and the Pythagorean theorem.
    • Since there are two functions for position, and they both depend on a single parameter—time—we call these equations parametric equations.
    • This equation is obtained using the Pythagorean Theorem.
    • where the rate of change of the hypotenuse length depends on the rate of change of $x$ and $y$.
    • Calculate arc length by integrating the speed of a moving object with respect to time
  • Logistic Equations and Population Grown

    • A logistic equation is a differential equation which can be used to model population growth.
    • Choosing the constant of integration $e^c = 1$ gives the other well-known form of the definition of the logistic curve:
    • The logistic equation is commonly applied as a model of population growth, where the rate of reproduction is proportional to both the existing population and the amount of available resources, all else being equal.
    • where the constant $r$ defines the growth rate and $K$ is the carrying capacity.
    • In the equation, the early, unimpeded growth rate is modeled by the first term $rP$.
  • Application of Bernoulli's Equation: Pressure and Speed

    • The Bernoulli equation can be derived by integrating Newton's 2nd law along a streamline with gravitational and pressure forces as the only forces acting on a fluid element.
    • Bernoulli's equation can be applied when syphoning fluid between two reservoirs .
    • The Bernoulli equation can be adapted to flows that are both unsteady and compressible.
    • The flow rate out can be determined by drawing a streamline from point ( A ) to point ( C ).
    • Adapt Bernoulli's equation for flows that are either unsteady or compressible
  • 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.
  • Parametric Equations

    • Parametric equations are a set of equations in which the coordinates (e.g., $x$ and $y$) are expressed in terms of a single third parameter.
    • This way of expressing curves is practical as well as efficient; for example, one can integrate and differentiate such curves term-wise.
    • Converting a set of parametric equations to a single equation involves eliminating the variable from the simultaneous equations.
    • If one of these equations can be solved for $t$, the expression obtained can be substituted into the other equation to obtain an equation involving $x$ and $y$ only.
    • In some cases there is no single equation in closed form that is equivalent to the parametric equations.
  • Solving Differential Equations

    • Differential equations are solved by finding the function for which the equation holds true.
    • A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself to its derivatives of various orders.
    • Solving the differential equation means solving for the function $f(x)$.
    • The "order" of a differential equation depends on the derivative of the highest order in the equation.
    • (This is because, in order to solve a differential equation of the $n$th order, you will integrate $n$ times, each time adding a new arbitrary constant.)
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