4.S: Kinetics (Study Guide)

4.1: Prelude to Kinetics

chemical kinetics – area of chemistry dealing with speeds/rates of reactions

4.2: Reaction Rates

• reaction rate – speed of a chemical reaction

\[\displaystyle \textit{average rate} = \frac{\textit{change #moles B}}{\textit{change in time}}= \frac{\Delta\textit{moles B}}{\Delta t}\textit{ if }A \to B \nonumber \]

\[\Delta\textit{moles B} = \textit{moles B at final time}- \textit{moles B at initial time} \nonumber \]

\[\displaystyle \textit{average rate} = -\frac{\Delta\textit{moles A}}{\Delta t}\textit{ if }A \to B \nonumber \]

• rate calculated in units of M/s
• brackets around a substance indicate the concentration
• instantaneous rate – rate at a particular time
• instantaneous rate obtained from the straight line tangent that touches the curve at a specific point
• slopes give instantaneous rates
• instantaneous rate also referred to as the rate

• for the irreversible reaction \(aA+bB\to cC+dD\)

\[\displaystyle\textit{rate} = -\frac{1}{a}\frac{\Delta [A]}{\Delta t} = -\frac{1}{b}\frac{\Delta [B]}{\Delta t} = \frac{1}{c}\frac{\Delta [C]}{\Delta t} = \frac{1}{d}\frac{\Delta [D]}{\Delta t} \nonumber \]

4.3: Factors that Affect Reaction Rates

• rates of reactions affected by four factors
  1. concentrations of reactants
  2. temperature at which reaction occurs
  3. presence of a catalyst
  4. surface area of solid or liquid reactants and/or catalysts

4.4: Rate Laws

• Rate law – expression that shows that rate depends on concentrations of reactants
• k = rate constant

• Rate = k[reactant 1]^m[reactant 2]ⁿ
• m, n are called reaction orders
• m+n, overall reaction order
• reaction orders do not have to correspond with coefficients in balanced equation
• values of reaction order determined experimentally
• reaction order can be fractional or negative

4.4.1 Reaction Order and Rate Constant Units

• units of rate constant depend on overall reaction order of rate law
• for reaction of second order overall
• units of rate = (units of rate constant)(units of concentration)²
• units of rate constant = M^-1s^-1

• zero order – no change in rate when concentration changed
• first order – change in concentration gives proportional changes in rate
• second order – change in concentration changes rate by the square of the concentration change, such as 2² or 3², etc…
• rate constant does not depend on concentration

4.5: Integrated Rate Laws

• rate laws can be converted into equations that give concentrations of reactants or products

4.5.1 First-Order Reactions

\[\textit{rate} = -\frac{\Delta [A]}{\Delta t} = k[A] \nonumber \]

and in integral form:

\[\ln[A]_t - \ln[A]_0 =-kt \nonumber \]

or

\[\ln\frac{[A]_t}{[A]_0} = -kt \nonumber \]

\[\ln[A]_t = - kt + \ln[A]_0 \nonumber \]

• corresponds to a straight line with \(y = mx + b\)
• equations used to determine:
  1. concentration of reactant remaining at any time
  2. time required for given fraction of sample to react
  3. time required for reactant concentration to reach a certain level

4.5.2 Second-Order Reactions

• rate depends on reactant concentration raised to second power or concentrations of two different reactants each raised to first power

\[\text{Rate} = k[A]^2 \nonumber \]

\[\displaystyle\frac{1}{[A]_t} = kt + \frac{1}{[A]_0} \nonumber \]

\[\displaystyle\textit{half life} = t_{\frac{1}{2}} = \frac{1}{k[A]_0} \nonumber \]

• half life dependent on initial concentration of reactant

4.5.2 Zero-Order Reactions

• A zero-order reaction thus exhibits a constant reaction rate, regardless of the concentration of its reactants.

4.5.4 Half-Life

• half-life of first order reaction

\[\displaystyle t_{\frac{1}{2}} = -\frac{\ln\frac{1}{2}}{k} = \frac{0.693}{k} \nonumber \]

• half-life – time required for concentration of reactant to drop to one-half of initial value
• \(t_{1/2}\) of first order independent of initial concentrations
• half-life same at any given time of reaction
• in first order reaction – concentrations of reactant decreases by ½ in each series of regularly spaced time intervals

4.6: Collision Theory

• collision model – molecules must collide to react
• greater frequency of collisions the greater the reaction rate
• for most reactions only a small fraction of collisions leads to a reaction
• rate constant must increase with increasing temperature, thus increasing the rate of reaction

4.6.1 Activation Energy

• Svante August Arrhenius
• Molecules must have a minimum amount of energy to react
• Energy comes from kinetic energy of collisions
• Kinetic energy used to break bonds
• Activation energy, E_a – minimum energy required to initiate a chemical reaction
• Activated complex or transition state – atoms at the top of the energy barrier
• Rate depends on temperature and E_a
• Lower E_a means faster reaction
• Reactions occur when collisions between molecules occur with enough energy and proper orientation

4.6.1 The Arrhenius Equation

• reaction rate data:
• theArrhenius Equation:

\[\displaystyle k = A e^{\frac{-E_a}{RT}} \nonumber \]

• \(k\) = rate constant, \(E_a\) = activation energy, \(R\) = gas constant (8.314 J/(mol K)), \(T\) = absolute temperature, \(A\) = frequency factor
• \(A\) relates to frequency of collisions, favorable orientations

\[\displaystyle \ln k = -\frac{E_a}{RT} + \ln A \nonumber \]

• the \(\ln k\) vs. \(1/t\) graph (also known as an Arrhenius plot) has a slope \(–E_a/R\) and the y-intercept \(\ln A\)
• for two temperatures:

\[\displaystyle \ln \frac{k_1}{k_2} = \frac{E_a}{R}\left(\frac{1}{T_2} - \frac{1}{T_1}\right) \nonumber \]

• used to calculate rate constant, \(k_1\) and \(T_1\)

4.7: Reaction Mechanisms

• reaction mechanism – process by which a reaction occurs

• elementary steps – each step in a reaction
• molecularity – if only one molecule involved in step
• unimolecular – if only one molecule involved in step
• bimolecular – elementary step involving collision of two reactant molecules
• termolecular – elementary step involving simultaneous collision of three molecules
• elementary steps in multi-step mechanism must always add to give chemical equation of overall process
• intermediate – product formed in one step and consumed in a later step

• if reaction is known to be an elementary step then the rate law is known
• rate of unimolecular step is first order (Rate = k[A])
• rate of bimolecular steps is second order (Rate = k[A][B])
• first order in [A] and [B]
• if double [A] than number of collisions of A and B will double

• rate-determining step – slowest elementary step
• determines rate law of overall reaction

• intermediates are usually unstable, in low concentration, and difficult to isolate
• when a fast step precedes a slow one, solve for concentration of intermediate by assuming that equilibrium is established in fast step

4.8: Catalysis

• catalyst – substance that changes speed of chemical reaction without undergoing a permanent chemical change

• homogeneous catalyst – catalyst that is present in same phase as reacting molecule
• catalysts alter E_a or A
• generally catalysts lowers overall E_a for chemical reaction
• catalysts provides a different mechanism for reaction

• Heterogeneous catalyst exists in different phase from reactants
  • initial step in heterogeneous catalyst is adsorption
  • adsorption – binding of molecules to surface
  • adsorption occurs because ions/atoms at surface of solid extremely reactive

• Enzymes are biological catalysts
  • large protein molecules with molecular weights 10,000 – 1 million amu
  • catalase – enzyme in blood and liver that decomposes hydrogen peroxide into water and oxygen
  • substrates – substances that undergo reaction at the active site
  • lock-and-key model – substrate molecules bind specifically to the active site
  • enzyme-substrate complex – combination of enzyme and substrate
  • binding between enzyme and substrate involves intermolecular forces (dipole-dipole, hydrogen bonding, and London dispersion forces)
  • product from reaction leaves enzyme allowing for another substrate to enter enzyme
  • enzyme inhibitors – molecules that bind strongly to enzymes
  • turnover number – number of catalyzed reactions occurring at a particular active site
  • large turnover numbers = low activation energies