Chemical Kinetics Flashcards - Rate Laws, Integrated Rate Laws, and Catalysis

Kinetics Study Notes

Rate: Basics and Definitions

• Velocity (rate) is the change in a quantity per unit time:
  • Velocity (rate) = $\text{rate} = \dfrac{\Delta \text{distance}}{\Delta \text{time}}$
  • For chemical reactions, rate is the change in concentration over time:
    $\text{rate} = -\dfrac{1}{a}\dfrac{\Delta[A]}{\Delta t} = -\dfrac{1}{b}\dfrac{\Delta[B]}{\Delta t} = \dfrac{1}{c}\dfrac{\Delta[C]}{\Delta t} = \dfrac{1}{d}\dfrac{\Delta[D]}{\Delta t}$
• Example (balancing): for the reaction $aA + bB \rightarrow cC + dD$
  • Rate definitions use the stoichiometric coefficients a, b, c, d as shown above.
• The rate of a reaction is always taken positive for the purpose of rate expressions; signs come from the definition (consumption of reactants is negative in concentration changes).
• Typical example of a reaction rate from lecture data:
  • For $2\,N<em>2O</em>5(g) \rightarrow 4\,NO<em>2(g) + O</em>2(g)$ the rate expressions relate to each species via the stoichiometric coefficients:
    $-\dfrac{1}{2}\dfrac{d[N<em>2O</em>5]}{dt} = \dfrac{1}{4}\dfrac{d[NO<em>2]}{dt} = \dfrac{d[O</em>2]}{dt}$

Rate Expressions

• Rate expressions (also called rate laws) relate rate to concentrations:

  • For a general reaction $aA + bB \rightarrow cC + dD$
    $\text{rate} = k[A]^m[B]^n\quad\text{where } m,n\text{ are the orders in A and B, and k is the rate constant.}$

• The rate law orders (m, n) are determined experimentally, not from stoichiometry; the overall order is $m+n+\ldots$ (sum of all reactant orders).

• Examples from course content:

  • NO(g) + O2(g) → NO2(g) with rate law $\text{rate} = k[NO]^2[O_2]$ (second order in NO, first order in O2, third order overall).
  • F2(g) + 2ClO2(g) → 2FClO2(g) with rate law $\text{rate} = k[ F<em>2 ][ClO</em>2]$ (first order in F2 and first in ClO2; overall second order).

• For the reaction $2N<em>2O</em>5(g) \rightarrow 4NO<em>2(g) + O</em>2(g)$ the rate expressions in terms of concentration changes are:
  $-\dfrac{1}{2}\dfrac{d[N<em>2O</em>5]}{dt} = \dfrac{1}{4}\dfrac{d[NO<em>2]}{dt} = \dfrac{d[O</em>2]}{dt}$

• Relation between rate expressions and rate laws:

  • Rate expressions are written in terms of concentration changes with time.
  • Rate laws express rate as a function of concentration with a rate constant: $\text{rate} = k[A]^m[B]^n…$

• Practical derivations from data:

  • Example: For a reaction with measured average rate from two time intervals, you compute the instantaneous rate from slope of concentration vs time graphs and use definitions to relate to reactant/product changes.
  • Example: For $2N<em>2O</em>5(g) \rightarrow 4NO<em>2(g) + O</em>2(g)$ with data given in table form, you compute the rate of appearance of products or rate of disappearance of reactants and relate them through stoichiometric coefficients as shown above.

• Average, Instantaneous, and Initial rates:

  • Average rate: change in concentration over a specified time interval.
  • Instantaneous rate: slope at a specific time (tangent to the curve at t).
  • Initial rate: tangent at time zero (t = 0).

• Common exercise style: Given a balanced equation, determine the rate of reaction with respect to a particular species by using the appropriate coefficient. E.g., for $2A + B \rightarrow 3C$, if you know the rate from C, you can relate to the rates of A and B via stoichiometry.

Rate Laws and Orders

• The rate law for a reaction aA + bB → products has the form:
  $\text{rate} = k[A]^m[B]^n\quad\text{where } m,n\text{ are the orders in A and B, respectively.}$

• Important notes:

  • Rate laws are determined experimentally.
  • Reaction order is defined with respect to reactants, not products.
  • The order of a reactant is not necessarily equal to its stoichiometric coefficient.
  • Orders can be 0, 1, 2, fractional, or even negative (less common).
  • Overall order = sum of the individual reactant orders (e.g., for m=2, n=1, overall order = 3).

• Examples from notes:

  • For the reaction $2NO(g) + O<em>2(g) \rightarrow 2NO</em>2(g)$, rate law: $\text{rate} = k[NO]^2[O_2]$. Overall order = 3 (second order in NO, first in O2).
  • For CH3CHO (acetaldehyde) decomposition: $\text{rate} = k[CH_3CHO]^{3/2}$; order in CH3CHO is 3/2, overall order 3/2.
  • For H2O2 + I- + H+ reaction: $\text{rate} = k[H<em>2O</em>2][I^-]$; first order in H2O2, first order in I-, zero order in H+. Overall order = 2.

• Common rate laws and units table (summary):

  • Zero order: Rate = k[A]^0; Units: M s^-1; Integrated: $[A]<em>t = [A]</em>0 - kt$; Graph: [A] vs t is linear.
  • First order: Rate = k[A]^1; Units: s^-1; Integrated: $\ln[A]<em>t = -kt + \ln[A]</em>0$; Graph: ln[A] vs t linear.
  • Second order: Rate = k[A]^2; Units: M^-1 s^-1; Integrated: $\frac{1}{[A]<em>t} = kt + \frac{1}{[A]</em>0}$; Graph: 1/[A] vs t linear.

• Rate constant units depend on overall order; e.g., for second order in a single reactant, k has units M^-1 s^-1.

• Typical problem workflow (method of initial rates):

  • Use data from experiments with varying initial concentrations to determine m and n by observing how the initial rate changes when one reactant concentration changes while others are held constant.
  • Once m and n are found, solve for k using any experiment’s initial rate and concentrations.
  • Example worked problems in notes show extracting m and n by comparing rate changes when [NO], [O2] are varied, then computing k from one data set.

• Rate laws vs rate expressions:

  • Rate expression is the expression of a rate in terms of concentration changes over time for a particular stoichiometry (often the same as a rate law but sometimes presented in differential form with coefficients).
  • Rate law is the mathematical relation rate = f([A],[B],…) with exponents m,n determined experimentally.

Integrated Rate Laws and Graphical Methods

• Integrated rate laws relate concentration to time for different reaction orders:

  • First order: $\ln[A]<em>t = -kt + \ln[A]</em>0$
  • Zero order: $[A]<em>t = [A]</em>0 - kt$
  • Second order: $\frac{1}{[A]<em>t} = kt + \frac{1}{[A]</em>0}$

• How to determine order graphically:

  • If plotting \ln[A] vs t yields a straight line, the reaction is first order in A.
  • If plotting 1/[A] vs t yields a straight line, the reaction is second order in A.
  • If plotting [A] vs t yields a straight line with negative slope, the reaction is zero order in A.

• Example from course: decomposition of N2O5 data is used to illustrate how to determine whether a plot of ln[A] vs t, 1/[A] vs t, or [A] vs t is linear to assign order.

• Integrated rate law graphs provide a linear relationship (y = mx + b) allowing extraction of k (slope) and/or [A]0 (intercept).

• Equation references:

  • For general A → products: $[A]<em>t = [A]</em>0 - kt$ (zero order).
  • For first order: $\ln[A]<em>t = -kt + \ln[A]</em>0$; Slope = -k.
  • For second order: $\frac{1}{[A]<em>t} = kt + \frac{1}{[A]</em>0}$; Slope = k; intercept = 1/[A]_0.

• Example problems and graphs shown in notes include:

  • Integration forms for representative reactions and use of slope-intercept form to determine k.
  • Graphical determination of reaction order for a multi-step example (N2O5 decomposition).

Half-Life and Its Dependence on Order

• Definition: half-life t1/2 is the time required for [A] to fall