Reaction Rates: Kinetics, Order, and Mechanisms mod 2 lecture 4

• Graphical methods determine the rate constant for first-order reactions
• Half-life is the time it takes for a reactant to reach half of its initial value
• Elementary step, reaction mechanism, reaction intermediate, and rate-determining step are key kinetics terms
• Rate laws can be derived for two types of mechanisms

Initial Rates Method

• This method allows for the deduction of x and y in a rate equation
• The rate constant, k, can be found using this method: k = initial rate / ([A]₀ˣ [B]₀ʸ )
• The integrated rate law serves as an alternative method

First-Order Reactions: Integrated Rate Law

• Relates concentration to time t, and is an alternative method for finding k
• For a first-order reaction, rate = -d[A]/dt = k[A]
• Mathematical manipulation (integration) yields: ln[A]t = ln[A]₀ - kt
• In first-order reactions, knowing the initial concentration [A]₀ and k allows calculation of the concentration of A, [A]t, after time t

Determining Rate Constant or Initial Concentration

• To find the rate constant, k, or the initial concentration [A]₀ if they are unknown, use In[A]t = -kt + In[A]₀
• Rewrite as y = mx + c, which is the equation for a straight line
• By plotting ln[A]t versus t, a linear graph results if the reaction is first order
• A plot of In [A] versus t that is linear indicates a reaction is first order
• The slope equals -k, or the rate constant
• The intercept equals In [A]₀, or the initial concentration

Decomposition of Ethane Example

• For the decomposition of ethane C₂H₆: H₃C-CH₃ → 2°CH₃
• Data obtained at 1000 K shows the reaction is first order, as concentration data is converted to ln form

Half-Life

• Half-life (t½) is the time taken for a reactant to reach half its initial value
• It is a convenient way to describe reaction speed and is common in nuclear physics and medicine
• For a reactant A, half-life happens when [A]t = (1/2)[A]₀
• For first-order reactions, substituting [A]₀/2 into the integrated rate equation allows the equation to be expressed as: In[A]t = In[A]₀ - kt
• For 1st order kinetics, t½ is independent of the initial concentration of A

Half Life and Rate constant

• For a 1st order reaction, the value of [A]₀/2 at t=t½ can be substituted into an integrated rate equation
• For 1st order kinetics, t½ is independent of the initial concentration of A
• Half life may be calculated by: t½ = 0.693/k

Radioactive Decay and Radiocarbon Dating

• Radioactive decay is a first-order process where half-life is useful and at the basis of radio dating methods
• Radioactive ¹⁴C is formed in the atmosphere as ¹⁴CO₂ by collision of neutrons with ¹⁴N
• The ¹⁴C/¹²C ratio in living matter remains constant via CO₂ exchange from the atmosphere through photosynthesis, and this exchange ceases upon death of the organism
• Over time, ¹⁴C decays (¹⁴₆C → ¹⁴₇N + ₋₁⁰β, where t½ = 5780 years), diminishing the ¹⁴C/¹²C ratio
• The time since death can be calculated by measuring amount of remaining ¹⁴C, and using t½ of ¹⁴C, in a useful range of 1,000 to 50,000 years

Reaction Mechanisms

• Reaction mechanisms have elementary reactions in a series of steps
• Overall reaction example: 2ICl + H₂ → 2HCl + I₂
• Elementary reactions usually consist of unimolecular or bimolecular forms
• They may involve reaction intermediates (species that play a role in the reaction, but are not included in the overall equation, e.g., HI) and do not appear in the overall rate law
• A series of elementary reactions describing an overall reaction is called a reaction mechanism

Reaction Mechanism Rate Laws

• In a reaction mechanism, the sum of elementary steps must equal the overall equation
• Rate laws from chemical equations, using stoichiometric coefficients, is allowed ONLY for elementary reactions
• For step 1, rate = k[ICl][H₂]
• For step 2, rate = k[HI][ICl]
• Focus on collecting individual rate laws into an overall rate law, which must agree with the experimentally determined rate law

Deducing Reaction Mechanisms

• Two paths are used to predict the reaction mechanism and overall rate law, experimental measurement and proposed mechanism steps

Rate-Determining Step

• To derive a predicted rate law, proposing a rate-determining step is needed
• The rate-determining step is the slowest elementary step
• It controls the overall reaction rate, acting as a "bottleneck"

First Step is Rate Determining Reaction Example

• Overall reaction: NO₂(g) + CO(g) → NO(g) + CO₂(g)
• Experimental rate law: rate = k[NO₂]²
• Proposed mechanism involves two elementary steps:
• Step 1: 2NO₂ → NO₃ + NO (slow), Step 2: NO₃ + CO → NO₂ + CO₂ (fast)
• Predicted rate law (written down directly from the slow step) is rate = k[NO₂]²
• Requires more confirmation, e.g., detection of NO₃ reaction intermediate

Reaction Energy Profile

• A plot of energy changes during a reaction
• Two peaks correspond to the two elementary reactions for the proposed mechanism
• The slower step (rate-determining) has a higher activation energy Ea

Second Step is Rate Determining Reaction Example

• Overall reaction: 2NO(g) + O₂(g) -> 2NO₂(g)
• Experimental rate law: rate = k[NO]²[O₂]
• Possible Mechanism: fast pre-equilibrium step followed by slow step
• Step 1: NO + O₂ ⇌ NO₃ (fast), Step 2: NO₃ + NO → 2NO₂ (slow)
• If Step 2 is rate-determining, write the rate law down directly: rate = k₂[NO₃] [NO]
• Equilibrium constant expression: K1 = [NO3]/[NO][O2]
• [NO3] = K1[NO][O2]
• Substitute: rate = k2[NO3][NO] = k2K1[NO]²[O2] = k[NO]²[O2]