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:

• 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.)