# Chemical kinetics: accounting for the rate laws

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Chemical kinetics: accounting for the rate laws
자연과학대학 화학과 박영동 교수

The approach to equilibrium
Forward: A → B Rate of formation of B = kr[A] Reverse: B → A Rate of decomposition of B = kr[B] Net rate of formation of B = kr[A] − kr[B] At equilibrium, rate = 0 = kr[A]eq − kr[B] eq

rate and temperature

The net rate Forward: A → B Rate of formation of B = kr[A]
Reverse: B → A Rate of decomposition of B = kr[B] Net rate d[A]/dt = − kr[A] + kr[B] = − kr[A] + kr([A]0 − [A]) d[A]/dt = − (kr + kr) [A] + kr[A]0 at t = 0,

The approach to equilibrium
The approach to equilibrium of a reaction that is first-order in both directions.

Relaxation methods

Consecutive reactions

Reaction Mechanism and elementary reactions
a unimolecular elementary reaction a bimolecular elementary reaction A → P v = kr[A] A + B → P v = kr[A] [B]

The formulation of rate laws
2 NO(g) + O2(g) → 2 NO2(g) ν = kr[NO]2[O2] Step 1. Two NO molecules combine to form a dimer: (a) NO +NO →N2O2 Rate of formation of N2O2 = ka[NO]2 Step 2. The N2O2 dimer decomposes into NO molecules: N2O2 → NO + NO, Rate of decomposition of N2O2 = ka΄[N2O2] Step 3. Alternatively, an O2 molecule collides with the dimer and results in the formation of NO2: N2O2 + O2 → NO2 + NO2, Rate of consumption of N2O2 = kb[N2O2][O2]

2 NO(g) + O2(g) → 2 NO2(g) Rate of formation of NO2 = 2kb[N2O2][O2] Net rate of formation of N2O2 =ka[NO]2 − ka΄[N2O2] − kb[N2O2][O2] = 0 The steady-state approximation: ka[NO]2 − ka΄[N2O2] − kb[N2O2][O2] = 0 [N2O2] = ka[NO]2 /( ka΄+ kb[O2] ) Rate of formation of NO2 = 2kb[N2O2][O2]= 2kakb [NO]2 [O2]/( ka΄+ kb[O2] ) if ka΄[N2O2]>>kb[N2O2][O2] Rate = 2kakb [NO]2 [O2]/( ka΄+ kb[O2] ) = (2kakb/ka΄)[NO]2 [O2] kr= (2kakb/ka΄)

The rate-determining step
The rate-determining step is the slowest step of a reaction and acts as a bottleneck. The reaction profile for a mechanism in which the first step is rate determining.

Unimolecular Reaction and The Lindemann Mechanism
Rate of formation of A* = ka[A]2 A + A → A* + A Rate of deactivation of A* = ka΄[A*][A] A + A* → A + A Rate of formation of P = kb[A*] A* → P Rate of consumption of A* = kb[A*] Net rate of formation of A* =ka[A]2 − ka΄[A*] [A]− kb[A*]= 0 [A*] = ka[A]2 /( ka΄[A]+ kb) Rate of formation of P = kb[A*]=kakb [A]2 /( ka΄[A]+ kb ) if ka΄[A]>>kb Rate = kakb [A]2 /( ka΄[A]+ kb) = (kakb/ka΄)[A] kr= (kakb/ka΄)

Activation control and diffusion control
Rate of formation of AB = kr,d[A][B] A + B → AB Rate of loss of AB = kr,d΄ [AB] AB → A + B Rate of reactive loss of AB = kr,a [AB] AB → P Rate of formation of P = kr[A][B] kr= kr,a kr,d/(kr,a + kr,d΄) i) kr,a >> kr,d΄ kr= kr,d diffusion-controlled limit ii) kr,a << kr,d΄ kr= kr,a kr,d/ kr,d΄ activation-controlled limit

kr,d = 8𝑅𝑇 3𝜂 For a diffusion-controlled reaction in water, for which η = 8.9 × 10−4 kg m−1 s−1 at 25°C. kr,d = 8𝑅𝑇 3𝜂 = 8×( J K −1 mol −1 )×(298K) 3×8.9× 10 −4 kg m −1 s −1 = 8× ×298 J mol −1 3×8.9× 10 −4 kg m −1 s −1 =7.4× kg m 2 s −2 mol −1 kg m −1 s −1 =7.4× m 3 s −1 mol −1 =7.4× dm 3 s −1 mol −1 kr,d = 7.4 × 109 dm3 mol−1 s−1

Diffusion Fick’s first law of diffusion 𝐽=−𝐷 𝑑𝑐 𝑑𝑥
Table 11.1 Diffusion coefficients at 25°C, D/(10−9 m2 s−1) The flux of solute particles is proportional to the concentration gradient.

Diffusion Fick’s second law of diffusion
𝜕𝑐 𝜕𝑡 = ( 𝐽 𝑥 − 𝐽 𝑥+𝑑𝑥 ) 𝑑𝑥 =𝐷 𝜕 2 𝑐 𝜕𝑥 2

Diffusion Einstein–Smoluchowski equation: 𝐷= λ 2 2𝜏 𝐷= 𝐷 0 𝑒 − 𝐸 𝑎 /𝑅𝑇
𝐷= λ 2 2𝜏 𝐷= 𝐷 0 𝑒 − 𝐸 𝑎 /𝑅𝑇 𝐷= 𝑘𝑇 6𝜋𝜂𝑎 𝜂= 𝜂 0 𝑒 𝐸 𝑎 /𝑅𝑇

Diffusion Suppose an H2O molecule moves through one molecular diameter (about 200 pm) each time it takes a step in a random walk. What is the time for each step at 25°C? Einstein–Smoluchowski equation: 𝐷= λ 2 2𝜏 𝜏= λ 2 2𝐷 = (200𝑝𝑚) 2 2(2.26× 10 −9 𝑚 2 𝑠 −1 ) =8.85× 10 −12 𝑠

Catalysis A catalyst acts by providing a new reaction pathway between reactants and products, with a lower activation energy than the original pathway.

Michaelis-Menten kinetics
Mechanism E + S ES → E + P k2 k1 k-1 ES complex is a reaction intermediate

Michaelis-Menten Kinetics

Michaelis-Menten Kinetics

Michaelis-Menten Kinetics
i) When [S] → 0 1st Order Reaction ii) When [S] → ∞ 0th Order Reaction!

Michaelis-Menten Kinetics
V varies with [S] Vmax approached asymptotically V is initial rate (near time zero) Michaelis-Menten Equation

Determining initial rate (when [P] is low)
Ignore the reverse reaction slope=

Range of KM values KM provides approximation of [S] in vivo for many enzymes

Lineweaver-Burk Plot

Allosteric enzyme kinetics
Sigmoidal dependence of V0 on [S], not Michaelis-Menten Enzymes have multiple subunits and multiple active sites Substrate binding may be cooperative

Enzyme inhibition

Kinetics of competitive inhibitor
Increase [S] to overcome inhibition Vmax attainable, KM is increased Ki = dissociation constant for inhibitor

Kinetics of competitive inhibitor

Kinetics of competitive inhibitor

Competitive inhibitor
Slope: increased Vmax unaltered

Kinetics of non-competitive inhibitor
Increasing [S] cannot overcome inhibition Less E available, Vmax is lower, KM remains the same for available E

Kinetics of non-competitive inhibitor

Kinetics of non-competitive inhibitor
Cf. Michaelis-Menten Eqn

Noncompetitive inhibitor
Vmax decreased KM unaltered

Enzyme inhibition by DIPF
Group - specific reagents react with R groups of amino acids diisopropylphosphofluoridate DIPF (nerve gas) reacts with Ser in acetylcholinesterase

Enzyme inhibition by iodoacetamide
A group - specific inhibitor

Affinity inhibitor: covalent modification

Example 11.1 Determining the catalytic efficiency of an enzyme
the hydration of CO2 in red blood cells CO2(g) + H2O(l) → HCO3-(aq) + H+(aq) at [E]0=2.3 nmol dm−3: 1/[CO2]/(mmol dm-3) 0.8 0.4 0.2 0.05 1/v/(mmol dm-3 s-1) 36.0 20.0 12.0 5.99 [CO2]/(mmol dm-3) 1.25 2.5 5 20 v/(mmol dm-3 s-1) 2.78 × 10-2 5.00 × 10-2 8.33 × 10-2 1.67 × 10-1 𝑉𝑚𝑎𝑥= =0.25 mmol dm-3 s-1 y = 40.0 x 𝐾 𝑀 𝑉 𝑚𝑎𝑥 =40.0 𝐾 𝑀 =40.0× =10.0 mmol dm-3 s-1 The Lineweaver–Burke plot

competitive inhibition

noncompetitive inhibition

Explosions The explosion limits of the H2/O2 reaction. In the explosive regions the reaction proceeds explosively when heated homogeneously.

H2(g) + Br2(g) → 2 HBr(g) Step 1. Initiation: Br2 → Br· + Br·
Rate of consumption of Br2 = ka[Br2] Step 2. Propagation: Step 3. Retardation: Step 4. Termination: Br· + ·Br + M → Br2 + M Rate of formation of Br2 = ke[Br]2

Br = 2ka[Br2] − kb[Br][H2] + kc[H][Br2] + kd[H][HBr] − 2ke[Br]2 = 0
H2(g) + Br2(g) → 2 HBr(g) Net rate of formation of HBr HBr = kb[Br][H2] + kc[H][Br2] − kd[H][HBr] H = kb[Br][H2] − kc[H][Br2] − kd[H][HBr] = 0 Br = 2ka[Br2] − kb[Br][H2] + kc[H][Br2] + kd[H][HBr] − 2ke[Br]2 = 0 Rate of formation of HBr

The DHLL for aqueous solution can be written as
Debye-Huckel Limiting Law is a valid approximation when strong electrolyte ions are in the solution at low concentration. Consider the solubility of Hg2(IO3)2(Ks= 1.3×10-18 )in KCl( 0.05 M). 1. Calculate the mean activity coefficient γ ± of the solution. 2. Calculate solubility of Hg2(IO3)2 at this temperature in unit of mol dm-3. The DHLL for aqueous solution can be written as ln γ ± = |z+z-|I1/2. Assume DHLL applies to this solutioon.

Ks= (aHg22+)(aIO3-) 2= (γ±sHg22+)(γ±sIO3-) 2
KCl(s) ⇄ K+ (aq) + Cl−(aq) Hg2(IO3)2(s) ⇄ Hg22+ (aq) + 2 IO3−(aq) I= ½ ( ×[Hg22+]+ [IO3−]) = 0.05 because ×[Hg22+],+[IO3−] << 0.05. Ks= (aHg22+)(aIO3-) 2= (γ±sHg22+)(γ±sIO3-) 2 sHg22+=s; sIO3- =2s Variable Value KCl 0.05 I γ ± Ks 1.3E-18 S E-6 Ks=4 (γ±s)3= 1.3×10-18 4 s3= Ks/(γ±)3 ln γ ± = |2∙1| 0.051/2 s= 1.16×10-6