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Chemical Kinetics

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Chemical Kinetics - Complete Interactive Lesson

Part 1: Rate Laws & Reaction Order

Chemical Kinetics

Part 1 of 5 — Rate Laws & Reaction Order

Reaction Rate

The reaction rate measures how quickly reactants are consumed or products are formed.

For $a\text{A} + b\text{B} \to c\text{C}$ :

$\text{rate} = -\frac{1}{a}\frac{\Delta[\text{A}]}{\Delta t} = -\frac{1}{b}\frac{\Delta[\text{B}]}{\Delta t} = +\frac{1}{c}\frac{\Delta[\text{C}]}{\Delta t}$

Rate is always expressed as a positive number; the minus sign accounts for consumption of reactants.

Rate Law

The rate law relates reaction rate to reactant concentrations:

$\text{rate} = k[\text{A}]^m[\text{B}]^n$

$k$ = rate constant (depends on temperature, not concentration)
$m$ = order with respect to A
$n$ = order with respect to B
Overall reaction order = $m + n$

Rate laws must be determined experimentally — you cannot read them from a balanced equation (unless it is an elementary step).

Determining Order from Experimental Data

Compare experiments where all concentrations except one are held constant:

$\frac{\text{rate}_2}{\text{rate}_1} = \left(\frac{[\text{A}]_2}{[\text{A}]_1}\right)^m$

Example:

Exp.	[A] (M)	[B] (M)	Rate (M/s)
1	0.10	0.10	$2.0 \times 10^{-4}$
2	0.20	0.10	$4.0 \times 10^{-4}$

From Exp 1 vs 2: doubling [A] doubles rate → 1st order in A ( $m = 1$ )

From Exp 1 vs 3: tripling [B] triples rate → 1st order in B ( $n = 1$ )

Overall: 2nd order; rate $= k[\text{A}][\text{B}]$

$k = \frac{\text{rate}}{[\text{A}][\text{B}]} = \frac{2.0 \times 10^{-4}}{(0.10)(0.10)} = 2.0 \times 10^{-2}\text{ M}^{-1}\text{s}^{-1}$

Units of $k$ by Reaction Order

Order	Rate law	Units of $k$
0th	rate = $k$	M/s
1st	rate = $k$ [A]	1/s (s⁻¹)
2nd	rate = $k$ [A]² or $k$ [A][B]

MCAT tip: The units of $k$ tell you the reaction order!

Zero-Order Reactions

Rate is constant, independent of concentration: $\text{rate} = k$

Example: enzyme-catalyzed reactions at substrate saturation; zero-order because the active sites are all occupied.

Rate Laws & Reaction Order 🎯

Key Takeaways — Part 1

Rate laws must be determined experimentally from rate/concentration data
To find order: compare runs where only one concentration changes; use ratio $\text{rate}_2/\text{rate}_1 = ([\text{X}]_2/[\text{X}]_1)^m$

Part 2: Integrated Rate Laws & Half-Life

Chemical Kinetics

Part 2 of 5 — Integrated Rate Laws & Half-Life

Integrated Rate Laws

Integrated rate laws give concentration as a function of time — essential for predicting how much reactant remains after a given time.

Zero-Order Integrated Rate Law

$[\text{A}]_t = [\text{A}]_0 - kt$

Part 3: Activation Energy & Arrhenius Equation

Chemical Kinetics

Part 3 of 5 — Activation Energy, Arrhenius Equation & Collision Theory

Activation Energy ( $E_a$ )

The activation energy is the minimum energy that colliding molecules must have for a reaction to occur. The transition state (activated complex) sits at the energy maximum.

Energy diagram key points:

Reactants → Transition state → Products
$E_a(\text{forward})$ = energy gap from reactants to peak

Part 4: Mechanisms, RDS & Catalysis

Chemical Kinetics

Part 4 of 5 — Reaction Mechanisms, Rate-Determining Step & Catalysis

Elementary Steps & Molecularity

A reaction mechanism is a proposed sequence of elementary steps that sum to give the overall balanced equation.

Molecularity	Number of reactant molecules	Example
Unimolecular	1	$\text{A} \to \text{products}$
Bimolecular	2	$\text{A} + \text{B} \to$ products

Part 5: Mixed MCAT Review

Chemical Kinetics

Part 5 of 5 — Mixed MCAT Review

High-Yield Checklist

✅ Rate law: experimentally determined; rate = $k[\text{A}]^m[\text{B}]^n$
✅ Units of $k$ : 0th order = M/s; 1st order = s⁻¹; 2nd order = M⁻¹s⁻¹
✅ Integrated rate laws: 0th ( vs ), 1st ( vs ), 2nd ( vs )

(0.10)(0.10)2.0×10−4=

Units of

k

: 0th order = M/s; 1st order = s⁻¹; 2nd order = M⁻¹s⁻¹

Zero-order reactions: rate is constant; seen in enzyme kinetics at Vmax

Rate laws are determined experimentally — not from balanced equations (unless elementary step)

Graph: $[\text{A}]$ vs $t$ is a straight line with slope $= -k$ .

Half-life: $t_{1/2} = \frac{[\text{A}]_0}{2k}$ (depends on initial concentration)

First-Order Integrated Rate Law

$[\text{A}]_t = [\text{A}]_0 e^{-kt}$

Equivalently: $\ln[\text{A}]_t = \ln[\text{A}]_0 - kt$

Graph: $\ln[\text{A}]$ vs $t$ is a straight line with slope $= -k$ .

$t_{1/2} = \frac{\ln 2}{k} = \frac{0.693}{k}$

Half-life is constant — independent of $[\text{A}]_0$ . This is the hallmark of 1st-order kinetics.

Applications: Radioactive decay, drug elimination from the body.

Second-Order Integrated Rate Law

$\frac{1}{[\text{A}]_t} = \frac{1}{[\text{A}]_0} + kt$

Graph: $1/[\text{A}]$ vs $t$ is a straight line with slope $= +k$ .

Half-life: $t_{1/2} = \frac{1}{k[\text{A}]_0}$ (depends on initial concentration; increases over time)

Order	Integrated law	Linear plot	Half-life
0	$[\text{A}] = [\text{A}]_0 - kt$	$[\text{A}]$ vs $t$	$[\text{A}]_0/2k$
1	$\ln[\text{A}] = \ln[\text{A}]_0 - kt$	$\ln[\text{A}]$ vs
2	$1/[\text{A}] = 1/[\text{A}]_0 + kt$	$1/[\text{A}]$ vs

Half-Life Applications (MCAT)

For first-order processes:

$\frac{[\text{A}]_t}{[\text{A}]_0} = \left(\frac{1}{2}\right)^n \quad \text{where } n = \frac{t}{t_{1/2}}$

Example: $k = 0.0693\text{ min}^{-1}$ (1st order). What fraction remains after 30 min?

$t_{1/2} = 0.693/0.0693 = 10\text{ min}$ . After 30 min = 3 half-lives.

$\text{fraction remaining} = (1/2)^3 = 1/8 = 12.5\%$

Identifying Reaction Order from Graphs

If this plot is linear...	Order
$[\text{A}]$ vs $t$	0th
$\ln[\text{A}]$ vs $t$	1st
$1/[\text{A}]$ vs $t$	2nd

Integrated Rate Laws & Half-Life 🎯

Key Takeaways — Part 2

0th order: $[\text{A}]$ vs $t$ linear; $t_{1/2} = [\text{A}]_0/2k$ (concentration-dependent)
1st order: $\ln[\text{A}]$ vs $t$ linear; $t_{1/2} = 0.693/k$ (constant)
2nd order: $1/[\text{A}]$ vs $t$ linear; $t_{1/2} = 1/k[\text{A}]_0$ (inversely proportional to concentration)
1st-order fraction remaining: $(1/2)^n$ where $n$ = number of half-lives elapsed
Radioactive decay, drug elimination = 1st-order processes
Linear graph identity: the order of the reaction is identified by which plot gives a straight line

E_a(\text{reverse})

= energy gap from products to peak

\Delta H = E_a(\text{forward}) - E_a(\text{reverse})

Exothermic: products at lower energy than reactants (

\Delta H < 0

)

Endothermic: products at higher energy than reactants (

\Delta H > 0

)

A catalyst lowers $E_a$ (provides an alternative pathway) without changing $\Delta H$ or the equilibrium constant.

The Arrhenius equation describes how the rate constant $k$ depends on temperature:

where $A$ = frequency factor (pre-exponential factor accounting for collision frequency and orientation), $R = 8.314$ J/mol·K, $T$ = temperature in Kelvin.

Logarithmic form: $\ln k = \ln A - \frac{E_a}{RT}$

Two-temperature form (MCAT-useful): $\ln\frac{k_2}{k_1} = \frac{E_a}{R}\left(\frac{1}{T_1} - \frac{1}{T_2}\right)$

Qualitative Rule of Thumb: For many reactions near room temperature, a 10°C rise in temperature approximately doubles the rate constant (though the precise factor depends on $E_a$ ).

For a reaction to occur, molecules must:

Collide with enough energy ( $\geq E_a$ )
Collide with the correct orientation

The fraction of collisions with energy $\geq E_a$ :

This fraction increases exponentially as $T$ increases or $E_a$ decreases, which is why:

Higher temperature → faster rate
Catalyst (lower $E_a$ ) → faster rate

Effect of Temperature on Rate

Factor	Effect on $k$
Increase $T$	Increases $k$ (exponentially)
Decrease $T$	Decreases $k$
Lower $E_a$ (catalyst)	Increases $k$
Higher $E_a$	Decreases $k$

MCAT application: The Arrhenius equation explains why biological enzymes are so effective — they dramatically lower $E_a$ for biochemical reactions.

Transition State Theory

The transition state (activated complex) is at the peak of the reaction coordinate diagram. It is not a stable product and cannot be isolated.

Quantity	What it means
$E_a$	Energy barrier for forward reaction
$\Delta H_{\text{rxn}}$	Overall enthalpy change
Transition state	High-energy, unstable intermediate
Intermediate	Stable (minimum) species between steps

Intermediate vs. transition state: An intermediate lies in an energy well (local minimum) between two transition states. A transition state is at an energy peak (maximum).

Activation Energy, Arrhenius & Transition State 🎯

Key Takeaways — Part 3

$E_a$ = minimum energy for a successful reaction; sits at transition state peak
Arrhenius: $k = Ae^{-E_a/RT}$ ; larger $E_a$ or lower $T$ → smaller $k$
Catalyst lowers $E_a$ , increases rate, but does NOT change $\Delta H$ or $K_{eq}$
Two-temperature Arrhenius: $\ln(k_2/k_1) = (E_a/R)(1/T_1 - 1/T_2)$
Rough rule: 10°C rise ≈ doubles reaction rate (for moderate $E_a$ )
Intermediate: energy valley (local minimum) between two transition states
Transition state: energy peak (maximum), cannot be isolated

Key distinction: For elementary steps only, the rate law can be written directly from the stoichiometry.

Rate-Determining Step (RDS)

The slowest elementary step controls the overall rate. The rate law for the overall reaction reflects the stoichiometry of the RDS.

Mechanism example:

Step 1 (slow, RDS): $\text{NO}_2 + \text{NO}_2 \to \text{NO}_3 + \text{NO}$

Step 2 (fast): $\text{NO}_3 + \text{CO} \to \text{NO}_2 + \text{CO}_2$

From the RDS (Step 1, bimolecular): $\text{rate} = k[\text{NO}_2]^2$

This matches the experimentally observed rate law — the mechanism is consistent.

Checking a Proposed Mechanism

Steps must sum to give the overall balanced equation (intermediates cancel)
The rate law from the RDS must match the experimentally determined rate law
A catalyst that appears in one step must be regenerated in a later step

Intermediates vs. Catalysts in Mechanisms

Intermediate: Produced in one step, consumed in a later step; does not appear in the overall equation
Catalyst: Consumed in one step, regenerated in a later step; does not appear in the overall equation but speeds up the reaction

Homogeneous Catalysis

Catalyst and reactants are in the same phase (e.g., acid catalysis in solution).

Example: Acid-catalyzed ester hydrolysis — $\text{H}^+$ speeds up the reaction and is regenerated.

Heterogeneous Catalysis

Catalyst and reactants are in different phases.

Example: Platinum catalyst in catalytic converters (solid Pt surface, gaseous reactants). Mechanism: reactant adsorption onto metal surface → reaction → product desorption.

Enzymatic Catalysis (Biological)

Enzymes are protein catalysts:

Active site binds substrate (lock-and-key or induced-fit model)
Michaelis-Menten kinetics: $v = V_{\text{max}}[\text{S}]/(K_m + [\text{S}])$
$K_m$ = substrate concentration at which rate = $V_{\text{max}}/2$
Lower $K_m$ → higher affinity for substrate
At very high $[\text{S}]$ : rate = $V_{\text{max}}$ (zero-order in substrate); all active sites saturated

Effect of Inhibitors on Enzyme Kinetics

Inhibitor type	Effect on $V_{\text{max}}$	Effect on $K_m$
Competitive	Unchanged	Increases (less apparent affinity)
Noncompetitive	Decreases	Unchanged
Uncompetitive	Decreases	Decreases

Mechanisms, RDS & Catalysis 🎯

Key Takeaways — Part 4

Rate law of overall reaction = rate law of the slowest elementary step (RDS)
Intermediates: produced then consumed (cancel in overall equation)
Catalysts: consumed then regenerated (cancel in overall equation)
Homogeneous catalysis: same phase; Heterogeneous: different phases (e.g., solid metal + gas)
Enzyme kinetics: Michaelis-Menten; $V_{\text{max}}$ at saturation; $K_m$ = affinity measure
Competitive inhibitor: $K_m$ ↑, $V_{\text{max}}$ unchanged; Noncompetitive: $V_{\text{max}}$ ↓, $K_m$ unchanged
A mechanism is valid only if its predicted rate law matches the observed experimental rate law

✅ First-order half-life:

t_{1/2} = 0.693/k

(constant)

✅ Arrhenius:

k = Ae^{-E_a/RT}

; higher

T

or lower

E_a

→ larger

k

✅ Catalyst: lowers

E_a

; increases

k

; does not change

\Delta H

K_{eq}

✅ RDS: slowest step; determines overall rate law

✅ Intermediates cancel from overall equation; catalysts regenerate

✅ Enzyme kinetics:

V_{\text{max}}

K_m

, competitive vs noncompetitive inhibition

Mixed Kinetics — MCAT-Style Questions 🎯

Kinetics — Complete Topic Summary

Part 1: Rate laws, extracting order from experimental data, units of $k$ , zero-order reactions.

Part 2: Integrated rate laws (0th, 1st, 2nd), graphical identification, half-life formulas, first-order $(1/2)^n$ calculations.

Part 3: Activation energy, Arrhenius equation, collision theory, temperature effect on $k$ , transition state vs. intermediate.

Part 4: Elementary steps, rate-determining step, substituting intermediates into rate law, catalysis types (homo/hetero/enzymatic), Michaelis-Menten kinetics, inhibitor types.

Part 5: Integrated MCAT practice.

Most Common MCAT Pitfalls

Using stoichiometry to write rate laws instead of experimental data
Confusing intermediates with catalysts in mechanisms
Thinking catalysts change $\Delta H$ or the equilibrium constant (they don't)
Forgetting Km is the concentration at half-Vmax (not an affinity in M directly)
Misidentifying reaction order from graphs

Chemical Kinetics

First-Order Integrated Rate Law

Second-Order Integrated Rate Law

Summary Table

Half-Life Applications (MCAT)

Identifying Reaction Order from Graphs

Key Takeaways — Part 2

Arrhenius Equation

Collision Theory

Effect of Temperature on Rate

Transition State Theory

Key Takeaways — Part 3

Rate-Determining Step (RDS)

Checking a Proposed Mechanism

Intermediates vs. Catalysts in Mechanisms

Types of Catalysis

Homogeneous Catalysis

Heterogeneous Catalysis

Enzymatic Catalysis (Biological)

Effect of Inhibitors on Enzyme Kinetics

Key Takeaways — Part 4

Kinetics — Complete Topic Summary

Most Common MCAT Pitfalls

Chemical Kinetics

Chemical Kinetics - Complete Interactive Lesson

Part 1: Rate Laws & Reaction Order

Chemical Kinetics

Reaction Rate

Rate Law

Determining Order from Experimental Data

Units of kkk by Reaction Order

Zero-Order Reactions

Key Takeaways — Part 1

Part 2: Integrated Rate Laws & Half-Life

Chemical Kinetics

Integrated Rate Laws

Zero-Order Integrated Rate Law

Part 3: Activation Energy & Arrhenius Equation

Chemical Kinetics

Activation Energy (EaE_aEa​)

Part 4: Mechanisms, RDS & Catalysis

Chemical Kinetics

Elementary Steps & Molecularity

Part 5: Mixed MCAT Review

Chemical Kinetics

High-Yield Checklist

First-Order Integrated Rate Law

Second-Order Integrated Rate Law

Summary Table

Half-Life Applications (MCAT)

Identifying Reaction Order from Graphs

Key Takeaways — Part 2

Arrhenius Equation

Collision Theory

Effect of Temperature on Rate

Transition State Theory

Key Takeaways — Part 3

Rate-Determining Step (RDS)

Checking a Proposed Mechanism

Intermediates vs. Catalysts in Mechanisms

Types of Catalysis

Homogeneous Catalysis

Heterogeneous Catalysis

Enzymatic Catalysis (Biological)

Effect of Inhibitors on Enzyme Kinetics

Key Takeaways — Part 4

Kinetics — Complete Topic Summary

Most Common MCAT Pitfalls

Units of $k$ by Reaction Order

Activation Energy ( $E_a$ )