Chapter 14 - Chemical Kinetics and Reaction Rates

The van't Hoff Factor (i)

• Ionic or acidic solutes produce multiple particles for each formula unit/solute molecule.
• Theoretical van't Hoff factor (i) = ratio of moles of solute particles to moles of formula units/molecules dissolved.
  • (i = \frac{moles \ of \ particles \ in \ solution}{moles \ of \ formula \ units/molecules \ dissolved})
  • (i = \frac{# \ of \ ions \ in \ 1 \ unit \ of \ compound}{1})
• Example: 1 mol HF added to 1L H_2O
  • HF dissociates into H^+ and F^-
  • If 50% dissociate:
    • 0.5 mol H^+
    • 0.5 mol F^-
    • 0.5 mol HF
    • Total = 1.5 mol particles
    • i = \frac{1.5 \ mol \ particles}{1 \ mol \ HF} = 1.5

Percent Dissociation:

• Percent \ Dissociation = (\frac{i-1}{i_{max}-1}) \times 100%
• i_{max} = i expected (100% dissociation)
• i{observed} = i{measured}
• i_{measured} can be less than the expected value due to ion pairing.

Electrolytic Solutes

• If the solute dissociates when added to a solvent, van't Hoff is included.
• \Delta Tf = i m Kf
• \Pi = i M R T where:
  • M = Molarity
  • R = gas constant (0.08206 L atm / mol K)
  • T = temperature (K)
• For nonelectrolytic solutes, i = 1.

Chapter 14: Chemical Kinetics

• Chemical kinetics = study of rate/speed of reactions.
• Collision model: a reaction that takes place between particles when they collide.
  • Matter on an atomic level is in constant motion.
  • During a collision, bonds can be broken and new bonds can be made.
  • More collisions = faster rate.
  • Higher energy collisions = faster rate.
  • Any factors that increase the frequency and/or energy of collisions between particles will increase the rate of the reaction.

Factors Impacting Reaction Rates

• Concentration of reactants:
  • Higher concentration = more frequent collisions.
• Temperature of the reaction:
  • Higher temperature = more energetic collisions.
  • Enough energy MUST be supplied to break bonds.
• Structure and relative orientation of colliding particles:
  • Sometimes approach must be in a certain orientation.

Defining a Reaction Rate

• Measured as the change in amount of a reactant or product over time.
• rate = \frac{\Delta \ in \ amount}{\Delta \ in \ time} = \frac{amount{final} - amount{initial}}{time{final} - time{initial}}
• Usually change in molar concentration (molarity).
• Example:
  • X + Y \rightarrow 2Z
  • \frac{\Delta[X]}{\Delta t} = \frac{2.83 - 4.00}{10 - 0} = -0.117 M/min
  • \frac{\Delta[Y]}{\Delta t} = \frac{1.17 - 0}{10 - 0} = -0.117 M/min
  • \frac{\Delta[Z]}{\Delta t} = \frac{2.34 - 0}{10 - 0} = +0.234 M/min

Key Points About Reaction Rates

• Different values for reaction rate depending on what you are measuring
• Signs depend on whether a reactant or product is monitored:
  • Reactants will lead to (-) values.
  • Products lead to (+) values.
• Magnitudes are different due to stoichiometry of reaction.

Generalized Reaction Rate

• Use stoichiometric coefficients to normalize the individual changes in concentration.
• Generic reaction: aA + bB \rightarrow cC + dD
• generalized \ 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}
• Negative signs are used for reactants so the rate is positive.
• Previous example application:
  • Rate = - (\frac{-0.117}{1}) = -(\frac{-0.117}{1}) = \frac{0.234}{2} = 0.117 M/min

Average vs. Instantaneous Rate of Change (ROC)

• Average rate = the rate of reaction over a time interval.
  • Can change as the reaction proceeds.
• Instantaneous Rate = the rate of the reaction at a specific instant of time.
  • Slope of the tangent at that time point on a curve of concentration vs. time.
• Approximating Instantaneous Rate from Average Rate.
  • Average Rate @ 20s to 40s = -0.050 M/s

Rate Law

• Rate law = mathematical equation that expresses the relationship between the rate of reaction and concentration of reactant.
  • Experimentally determined.
  • Order can't be established from balanced coefficients (except with elementary reactions of a reaction mechanism).
  • General form: r = k [A]^n
    • k = rate constant for reaction.
    • [A] = molar concentration of A raised to some power, n.
    • n = order of reactant (commonly 0, 1, or 2).
    • Could be a fraction, or (-) in rare cases.
  • For A + B \rightarrow products, general form: r = k [A]^m [B]^n
    • [A] = A concentration to some power m
    • [B] = B concentration to some power n

Reaction Order

• Reflects rate's sensitivity to changing that reactant's concentration.
  • High order reactions are more sensitive to changes in concentration than rates of reactions with low orders.
• Example: r = k [O_2] [NO]^2
  • first order with respect to O_2
  • second order with respect to NO
  • overall order (1+2) = 3

Rate Constant K

• Depends on specific reaction & temp.
• Rate ∝ k. Doubling k = doubling rate.
• Units of k depend on overall order.
  • 4th order = units 1/M^3 s

Zero Order Reaction

• rate = k[A]^0 \rightarrow r=k
• Units M/s
• Rate is constant as reaction proceeds.
  • Usually reactions occurring on surfaces.

First Order Reaction (n=1)

• r = k[A]^1
• Units 1/s
• Rate directly proportional to reactant.
• Rate decreases as reaction proceeds since concentration of reactant decreases.

Second Order Reaction (n=2)

• r = k[A]^2
• Units 1/M s
• Rate is directly proportional to the square of reactant concentration.
  • Doubling concentration = quadruple rate.
  • Tripling concentration = nine times rate.
• More sensitive to changes in concentration than the rate of first-order reaction.

Reaction Plots

• Zero order reaction: rate = constant
• 1st order = rate slows down linearly
• 2nd-order = rate slows down at a faster pace

Experimentally Determining Rate Laws and Reaction Orders

• Measure initial rates of the reaction under a set of initial concentrations for the reactants.
• Change the initial concentration of only 1 reactant (keeping the concentrations of the other reactants constant) and measure the initial rate under the new conditions.
  • Establish the order with respect to ONLY 1 reactant.
• Go back & repeat the process for all other reactants until order of each is established.
• Once all the orders are established, calculate the value of the rate constant using the data.
• Example:
  • exp1: 0.00347 = k [0.125]^n
  • exp2: 0.0234 = k [0.325]^n
  • \frac{0.0234}{0.00347} = \frac{[0.325]^n}{[0.125]^n}
  • \frac{0.0234}{0.00347} = [\frac{0.325}{0.125}]^n
  • 6.7435 = [2.6]^n
  • ln(6.7435)= n \times ln(2.6)
  • n = \frac{ln(6.7435)}{ln(2.6)} = 2

Example 1 from Lecture 6

• Reaction: A + B -> products
• rate = k[A]^x [B]^y
• Experiments show:
  • exp1: 5.01 \times 10^{-4} = k [0.253]^x [8.43 \times 10^{-2}]^y
  • exp2: 1.67 \times 10^{-4} = k [0.253]^x [2.81 \times 10^{-2}]^y
  • \frac{5.01 \times 10^{-4}}{1.67 \times 10^{-4}} = \frac{[8.43 \times 10^{-2}]^y}{[2.81 \times 10^{-2}]^y}
  • 3 = 3^y, y = 1
  • exp1: 5.01 \times 10^{-4} = k [0.253]^a [8.43 \times 10^{-2}]^b
  • exp3: 3.01 \times 10^{-3} = k [0.380]^a [8.43 \times 10^{-2}]^b
  • \frac{5.01 \times 10^{-4}}{3.01 \times 10^{-3}} = \frac{[0.253]^x}{[0.380]^x}
  • 0.1664 = 0.665^x, x = 3
• Overall order = 1 +3 =4

Example 2

• A -> products, t1/2 = 15.0 s
• What is rate of reaction when [A] = 0.484M?
• r = k[0.484]^x
• k = \frac{ln2}{t_{1/2}} = \frac{ln2}{15.0} 1st order rxn

Differential Rate Law

• Mathematically relates reactant concentration (M) to instantaneous rate (M/s)
• rate=K[A]^n & instantaneous rate

Integrated Rate Law

• (calculus) → Used for determining the amount of reactant at a certain time
• Mathematically relates reactant concentration (M) to reaction time(s, min)
• Different expressions for 0, 1st, 2nd order reactions
• Focus only on one reactant
• A products.
  • 0 order: [A] = -Kt + [A]o
  • 1st order: ln [A] = - kt + ln [A]o
  • 2nd order: 1/[A] = kt + 1/[A]o

Component Explanations

• [A] = concentration of A present @ time t
• K = rate constant
• t = time of interest
• [A]o = Initial concentration of A
• MUST Know order before using integrated rate law

Example Calculation

• in [A] = -ktrin[Ao]
• (-1 -42x10^-3)/t = (ln[1.23]-in[0.625])
  ln [A] - in [A]o=-1-42x10^-3*t t= 476.77

Plots of [A] VS. Time

• Shapes of plots depend on order of reaction
• zero Order; [A] ↓ at a constant rate faster than in first or second order
• second order: A consumed slower than in 1st-order no vino 200010

Linear Relationships of the integrated Rate Laws

• y = mx + b
• [A] = -kt + [A]o zero order
  • Slope = -k
• ln [A] = -kt + ln [A]o first order
  • slope -K
• 1/[A] = kt + 1/[A]o 2nd order
  • Slope = k

Half-Life

• Amount of time it takes for an initial concentration of a species to decrease by 50%
• Depends on reaction order/derived from integrated rate law
  • 0 order: [A] = - kt + [A]o t 1/2
    • t\frac{1}{2} = \frac{[A]0}{2K}
  • 1st order: ln [A] = - kt + in [A]o t 1/2
    • t_\frac{1}{2} = \frac{ln2}{K}
  • 2nd order: 1/[A] = K++ 1/[A]o
    • t\frac{1}{2} = \frac{1}{K[A]0}

Successive half- Lives

• 1st order: t_{\frac{1}{2}} = \frac{ln2}{K} = constant
• 0 order t{\frac{1}{2}} = \frac{[A]0}{2K}
• 2nd order: t{\frac{1}{2}} = \frac{1}{K[A]0}

Examples

• A products t ½ =15 min
• K = 0.0462 min^A1
• rate = 0.0224 M/min

Effect of Temperature & structure / orientation on Rate

• Arrhenius cavation mathemancal relationship. between magnitude of the rate constant & orner factors
• k = A e^{\frac{-E_a}{RT}}
  • K = rate constant
  • A = frequency factor
  • Ea achvanon energy
  • K=8.314 J/mol.k
  • T= temperature (K)

Achvanon Energy (Ea)

• Minimum amount of energy needed for reaction to take place
• Ea is ALWAYS a positve valve since.. It's an amount energy that must be Supplied/absorbed for a reaction to proceed ca
• at a constant temp→ large Ea = SIOW reaction. small Ea = fast reaction

Energy Diagram

• Relative change in energy as rcachon proceeds from reactants +0 products

pachivation energies & overall reaction enthalpies can be determined to no

Frequency Factor (A)

• # of times a reactant APPROACHES an activation barner per unit times
• NOT same as overcoming all que ONLY fraction of approaches have enough e energy to overcome to a
• collision model: # of collisions with the proper onentation that a reactant undergoes per unit time

collision frequency (z)

• # of collisions that occur per unit time

orientan on factor (p)

• the FRACTION of collisions that have a proper onentation for reaction to continue / occur
• value of p ranges oto 1025 15 Nwo P
• low onentation factor reactants VERY STRINGENT ORIENTATION
• high onentation factor = less stringent onentation requirements.

K= AC^(u- Ea/RT)

• recall that A = # of times the e activation barner IS APPROACHED per unit time
• Possible values of the factor?
• large Ea larger (-) exponent small decimal close to 0, smallerk → slower reaction 002600
• 2) Small Ea smaller (-) exponent larger decimal valve (closer to 1) larger k→ sv faster reaction.
• 3)Large Tsmaller (-) exponent larger decimal value (closer to 1) larger k→ faster reaction
• ) Small T→ larger (-) exponent decimal value smaller k
• smaller k sex slower reaction.

Linearized Arrhenius Equation

• K=A= ^((-Ea/RT) →ink=-\frac{E}{R}*\frac{1}{T}+lnA
• 3t Slope cm) - Ea
• R intercept: bin(A)

Two Point Form of Arrhenius Equation

• 3t no effect of temperature on the rate constant - Ea activation energy
• OR = 8 8.314 J/mol.k
• K₂ is rate constant @ T₂ K, is rate constant @T,

Summarizing Collision Model Requirements

• Collision between particles must be "effective" in that the particles must have the neccessary Onientation for a reaction.
  • exact onentation depends on the specific reaction
• Proper orientation isn't enough
• collision must also have enough energy to overcome the activation barrier
• if either of these 2 criteria aren't mer, no reaction will take place in effective collision

chemical Reactions on Molecular level

• Simple reactions proceed in a single step stop as suggested by the balanced equation

• ex) N2 + 3 H₂ → 2NH30MIL

• 4 molecules collide, can't occur in I step simpler steps make-up reaction mechanism

• Reaction mechanisms = simple steps that reactants go through to make final products

elementary reaction

• single reaction that CANNOT BE broken into simpler steps
• particles interact directly through a collision without any Will occuring other steps
  → may also consist of a single particle rearranging or decomposing
• radical
• Must be proposed & validated by experimental evidence
• 2 requirements:
  • 1) when summed, the individual steps of the mechanism must add up to overall balanced reaction. in context of Hess' Law
  • 2) must account for experimentally determined rate law of overall reaction.
    → sometimes a reaction can have more than i plausible mechanism

molecularity of particles

• # of reactant particles that enter into an elementary reachon reps actual # of particles interacting in elementary reaction
• Unimolecular = ONLY 1 particle needed for reaction to proceed bimolecular 2 particles need to collide for the reach on to proceed termolecular = 3 particles need to collide for the reaction occur

Termolecular

• 3 particles needed to collide for the reaction occur

Determing Rate Law of Elementary Regation

• Determined directly from stoichiometry
• Adding Reactions : 2ND + 2 H₂ → N₂ +2 H₂O

Mechanism's Rate Law

• Each elementary step = Rate Determining
• Rate Law direct stoichiometry answers equation with it's rate determining step with highest activation energy to reach experimentally
• 2 I B r12 + 2 HBr HD
• Rate Determing slowest step which will agree with the expermental. In example agrees expermentally which means it's plausible

Mechanisms with Rate Law

Equilibnum. = Keq

each elementary step = own EA, rate constant, rate law, & rate associated With it
the slowest step of the mechanism determines the mechanism. The rate of the entire

Energy Diagram of Reaction Mechanisms

• # of energy numps = # of elementary reactions in mechanism
• largest Eastep noransorate-determining
• catalyst chemical species that T the rate of the reaction axa bosp16163 takes part in reaction but not va ultimately consumed lowers the activation energy of reaction by changing the mechanism