Chemical Kinetics Notes

Kinetics

• Kinetics studies the factors affecting reaction speed and the reaction mechanism.

• Four factors influence reaction speed:

  • Nature of reactants

  • Temperature

  • Catalysts

  • Concentration

Defining Rate

• Rate is the change in quantity over time.

• Example: Car speed is the distance traveled (miles) per time (hour), with units of mi/hr.

Defining Reaction Rate

• Reaction rate is measured by:

  • The decrease in reactant concentration over time.

  • The increase in product concentration over time.

• A negative sign is used for reactants to indicate the decrease in concentration.

Reaction Rate Changes Over Time

• Reaction rate generally slows over time due to decreasing reactant concentration.

• The reaction stops when:

  • Reactants are depleted.

  • The system reaches equilibrium.

Reaction Rate and Stoichiometry

• Balanced equations often have different coefficients: $H<em>2(g) + I</em>2(g) \rightarrow 2HI(g)$

• The change in the number of molecules varies for each substance.

  • For every 1 mole of $H<em>2$ used, 1 mole of $I</em>2$ is used and 2 moles of $HI$ are produced.

• To maintain consistency, the change in concentration is multiplied by 1/coefficient.

Average Rate

• Average rate is the change in measured concentrations over a time period.

• It is a linear approximation of a curve.

• Larger time intervals cause greater deviation from the instantaneous rate.

• Average rate in a given time period = - slope of the line connecting the $[H_2]$ points; and ½ +slope of the line for $[HI]$

• Example:

  • The average rate for the first 10 s is 0.0181 M/s

  • The average rate for the first 40 s is 0.0150 M/s

  • The average rate for the first 80 s is 0.0108 M/s

Instantaneous Rate

• Instantaneous rate is the change in concentration at a specific time.

• It is the slope of a tangent line to the curve at that point.

• It represents the first derivative of the function.

Measuring Reaction Rate

• To measure reaction rate, track the concentration of at least one component over time.

• Approaches:

  • Continuous monitoring for reactions complete in under 1 hour.

  • Sampling for long reactions, with quenching to stop the reaction in the sample.

Continuous Monitoring

• Polarimetry: Measures changes in the rotation of plane-polarized light.

• Spectrophotometry: Measures light absorption by a component over time (complementary color).

• Total pressure: Relates total pressure of a gas mixture to partial pressures of reacting gases.

Sampling

• Gas chromatography: Measures concentrations of volatile components by separation via surface adherence.

• Periodic aliquots with quantitative analysis:

  • Titration.

  • Gravimetric analysis.

Factors Affecting Reaction Rate: Nature of the Reactants

• Nature of reactants includes the type and physical condition of reactant molecules.

  • Small molecules react faster than large molecules.

  • Gases react faster than liquids, which react faster than solids.

  • Powdered solids react faster than blocks due to increased surface area.

  • Certain chemicals are more reactive (e.g., activity series of metals).

  • Ions react faster than molecules because no bonds need to be broken.

Factors Affecting Reaction Rate: Temperature

• Increasing temperature increases reaction rate.

  • Rule of thumb: a 10°C rise doubles the reaction speed (for many reactions).

• Arrhenius discovered a mathematical relationship between absolute temperature and reaction speed.

Factors Affecting Reaction Rate: Catalysts

• Catalysts affect reaction speed without being consumed.

  • Positive catalysts speed up reactions.

  • Negative catalysts slow down reactions.

• Homogeneous catalysts are in the same phase as reactants.

• Heterogeneous catalysts are in a different phase.

Factors Affecting Reaction Rate: Reactant Concentration

• Generally, higher reactant concentration leads to a faster reaction.

  • Increases the frequency of reactant molecule contact.

  • Gas concentration depends on partial pressure (higher pressure = higher concentration).

  • Solution concentration depends on the solute-to-solution ratio (molarity).

The Rate Law

• The rate law is the mathematical relationship between reaction rate and reactant concentrations (and homogeneous catalysts).

• Reaction rate is directly proportional to each reactant's concentration raised to a power.

• For the reaction $aA + bB \rightarrow products$, the rate law is: $Rate = k[A]^n[B]^m$

  • $n$ and $m$ are the orders for each reactant.

  • $k$ is the rate constant.

Reaction Order

• The exponent on each reactant in the rate law is the order with respect to that reactant.

• The sum of the exponents on the reactants is the overall reaction order.

• Example: For the reaction $2NO(g) + O<em>2(g) \rightarrow 2NO</em>2(g)$, $Rate = k[NO]^2[O_2]$

  • Second order with respect to $[NO]$, first order with respect to $[O_2]$, third order overall.

Sample Rate Laws

• Autocatalytic reactions have a product affecting the rate.

• Negative catalysts (e.g. $Hg^{2+}$) slow the reaction when their concentration increases.

Half-Life

• Half-life ($t_{1/2}$) is the time for reactant concentration to fall to half its initial value.

• The half-life depends on the reaction order.

Zero Order Reactions

• $Rate = k[A]^0 = k$

• Constant rate reactions.

• Integrated rate law: $[A] = -kt + [A]_0$

• Graph of $[A]$ vs. time is a straight line with slope $-k$ and y-intercept $[A]_0$.

• Half-life: $t<em>{1/2} = \frac{[A</em>0]}{2k}$

• Units: If $Rate = M/sec$, then $k = M/sec$

First Order Reactions

• $Rate = k[A]$

• Integrated rate law: $ln[A] = -kt + ln[A]_0$

• Graph of $ln[A]$ vs. time is a straight line with slope $-k$ and y-intercept $ln[A]_0$.

• Half-life: $t_{1/2} = \frac{0.693}{k}$

• The half-life of a first-order reaction is constant.

• Units: If $Rate = M/sec$, then $k = sec^{-1}$.

Second Order Reactions

• $Rate = k[A]^2$

• Integrated rate law: $\frac{1}{[A]} = kt + \frac{1}{[A]_0}$

• Graph of $\frac{1}{[A]}$ vs. time is a straight line with slope $k$ and y-intercept $\frac{1}{[A]_0}$.

• Half-life: $t<em>{1/2} = \frac{1}{k[A</em>0]}$

• Units: If $Rate = M/sec$, then $k = M^{-1} \cdot sec^{-1}$.

Determining the Rate Law

• Determined experimentally.

• Graphically:

  • Rate = slope of curve $[A]$ vs. time.

  • If $[A]$ vs time is a straight line, then exponent on A is 0, rate constant = -slope.

  • If $ln[A]$ vs time is a straight line, then exponent on A is 1, rate constant = -slope.

  • If $\frac{1}{[A]}$ vs time is a straight line, exponent on A is 2, rate constant = slope.

• Initial rates: comparing the effect on the rate of changing the initial concentration of reactants one at a time.

Initial Rate Method

• Change the concentration of one reactant and observe the effect on the initial rate.

• Keep concentrations of other reactants constant.

  • Zero order: changing the concentration has no effect on the rate.

  • First order: rate changes by the same factor as the concentration (doubling the initial concentration doubles the rate).

  • Second order: rate changes by the square of the factor the concentration changes (doubling the initial concentration quadruples the rate).

The Effect of Temperature on Rate

• Changing temperature changes the rate constant.

• Arrhenius equation: $k = Ae^{-\frac{E_a}{RT}}$

  • $R$ is the gas constant = 8.314 J/(mol⋅K).

  • $T$ is temperature in Kelvins.

  • $A$ is the frequency factor.

  • $E_a$ is the activation energy.

Activation Energy and the Activated Complex

• Energy barrier to the reaction.

• Energy needed to convert reactants into the activated complex (transition state).

• The activated complex is a chemical species with partially broken and partially formed bonds.

• It's very high in energy because of partial bonds.

The Arrhenius Equation: The Exponential Factor

• The exponential factor in the Arrhenius equation is a number between 0 and 1.

• Represents the fraction of reactant molecules with sufficient energy to overcome the energy barrier.

• Higher energy barrier (larger activation energy) means fewer molecules have sufficient energy.

• Energy comes from converting kinetic energy to potential energy during molecular collisions.

• Increasing temperature increases the average kinetic energy of molecules, increasing the number of molecules with sufficient energy to overcome the barrier.

Arrhenius Plots

• The Arrhenius Equation can be algebraically solved to give the following form: $lnk = -\frac{E_a}{R}(\frac{1}{T}) + lnA$

• This equation is in the form $y = mx + b$ where $y = ln(k)$ and $x = (1/T)$

• A graph of $ln(k)$ vs. $(1/T)$ is a straight line

• $E_a = (-8.314 J/mol \cdot K)(slope \space of \space the \space line)$, (in Joules)

• $y-intercept = A$, (unit is the same as $k$)

Arrhenius Equation: Two-Point Form

• If you only have two $(T,k)$ data points, the following forms of the Arrhenius Equation can be used: $ln(\frac{k<em>2}{k</em>1}) = \frac{E<em>a}{R}(\frac{1}{T</em>1} - \frac{1}{T_2})$

Collision Theory of Kinetics

• For a reaction to take place, reacting molecules must collide.

• Molecules may react or not after collision, depending on:

  • Collision energy to break bonds.

  • Proper orientation for new bond formation.

Effective Collisions

• Effective collisions meet the conditions for reaction.

• The higher the frequency of effective collisions, the faster the reaction rate.

• An activated complex (transition state) is formed during effective collisions.

Effective Collisions: Kinetic Energy Factor

• Molecules must have sufficient kinetic energy to form the activated complex.

Collision Theory and the Arrhenius Equation

• $A$ is the frequency factor.

• Orientation factor $(p)$

• Collision frequency factor $(z)$

Orientation Factor

• Proper orientation aligns atoms for bond breaking and formation.

• More complex molecules collide less frequently with proper orientation.

  • Reactions between atoms generally have $p = 1$.

  • Reactions with symmetry have $p$ slightly less than 1.

• For most reactions, the orientation factor is less than 1.

Reaction Mechanisms

• Reactions usually occur in a series of small steps involving 1, 2, or 3 molecules.

• Describing the series of steps is called a reaction mechanism.

• The rate law helps to understand the steps in the mechanism.

Elements of a Mechanism: Intermediates

• Intermediates are produced in an early step and consumed in a later step.

• They do not appear in the overall reaction.

Molecularity

• The number of reactant particles in an elementary step is its molecularity.

  • Unimolecular: 1 reactant particle.

  • Bimolecular: 2 reactant particles.

  • Termolecular: 3 reactant particles (rare).

Rate Laws for Elementary Steps

• Each step has its own activation energy and rate law.

• The rate law for an overall reaction must be determined experimentally.

• The rate law of an elementary step can be deduced from the equation of the step.

Rate Determining Step

• The slowest step in the mechanism determines the overall reaction rate.

• The slowest step has the largest activation energy.

• The rate law of the rate-determining step determines the rate law of the overall reaction.

Catalysts

• Catalysts affect reaction rate without being consumed.

• They provide an alternative mechanism with a lower activation energy.

• Catalysts are consumed in an early step and regenerated in a later step.

Types of Catalysts

• Homogeneous catalysts: Same phase as reactants (e.g., Cl(g) in $O_3(g)$ destruction).

• Heterogeneous catalysts: Different phase than reactants (e.g., catalytic converter in a car).

Enzymes

• Protein molecules that catalyze biological reactions.