General Chemistry 2: Rate of Reaction and Molecular Kinetics

Learning Objectives

Influence of Factors on Reaction Rates: Describe how various environmental and chemical factors influence the rate of a chemical reaction.
Differentiation of Reaction Orders: Differentiate between zero-order, first-order, and second-order reactions.
Collision Theory (Qualitative): Explain chemical reactions qualitatively in terms of molecular collisions.
Activation Energy and Catalysis: Explain activation energy ( $E_a$ ) and describe how a catalyst affects the reaction rate.
Classification of Catalysts: Cite and differentiate between the different types of catalysts.

Introduction to Reaction Rate

Definition: Reaction rate is a measure of how quickly reactants change into products per unit of time.
General Formula: $\text{Rate of Reaction} = \frac{\text{Change in amount of reactant or product}}{\text{Change in time}}$
Calculation Examples:
- Decreasing Reactant Concentration: If the concentration of a reactant decreases from $0.8\,mole/L$ to $0.4\,mole/L$ over a period of $10\,seconds$ , the rate calculation is based on the loss of reactant.
- Increasing Product Concentration: If the concentration of a product increases from $0.5\,mole/L$ to $0.7\,mole/L$ in $10\,seconds$ , the rate calculation reflects the gain of product.

Collision Theory

The Hard Sphere Model: Reactant molecules are assumed to be hard spheres. Chemical reactions occur only when these spheres (molecules) collide with one another.
Purpose of the Theory: It explains why chemical reactions occur and why some are naturally fast while others are slow.
Reaction Mechanism: Reactions occur when particles (atoms, ions, or molecules) collide.
Ineffective vs. Effective Collisions: Not every collision leads to a reaction. Only collisions that possess adequate energy and proper orientation successfully form products.
The Three Requirements for a Successful Reaction:
1. Frequency: Collisions must be frequent. The reaction rate is directly proportional to the number of collisions per unit time ( $\text{Reaction Rate} \propto \text{Number of Collisions / Time}$ ).
2. Orientation: Particles must collide with the correct orientation. Reacting parts of molecules must come together in a specific way. If the alignment is incorrect, no bond formation occurs even if the collision happens.
3. Activation Energy: Particles need enough energy to start the reaction.

Activation Energy and Catalysts

Activation Energy ( $E_a$ ): To start a chemical reaction, existing chemical bonds in reactants must be broken, a process that requires energy. The minimum energy needed to initiate this process is the activation energy ( $E_a$ ).
Collision Theory Governers: Both activation energy and effective collisions govern the overall rate of a reaction.
Catalysts:
- Definition: A substance that increases the reaction rate without being consumed by the reaction itself.
- Mechanism: A catalyst provides an alternative reaction pathway that has a lower activation energy ( $E_a$ ).
- Energy Hill Analogy: On a potential energy diagram, the "energy hill" (the barrier reactants must overcome) is significantly lower when a catalyst is present compared to when it is absent.

Factors Influencing the Rate of Reaction

Concentration:
- Higher Concentration: Greater number of particles leads to more frequent collisions and a higher tendency to react quickly.
- Lower Concentration: Fewer particles mean lesser chances for collision and a longer time to complete the reaction.
- Correlation: Data shows that as the concentration of a reactant (e.g., $HCl$ ) increases, the completion time of the reaction decreases.
Temperature:
- Kinetic Energy: Increasing temperature increases the kinetic energy of reacting particles, causing them to move faster.
- Increased Collisions: Faster movement leads to more frequent and more energetic collisions.
- Observation: Food cooks faster in a pressure cooker because the higher internal temperature (relative to an open pot) increases the reaction rate of the cooking process.
Nature of Reactants:
- Chemical Identity: Rates depend on the identity of the substance. For example, Calcium ( $Ca$ ) reacts moderately with water ( $Ca(s) + 2H_2O(l) \rightarrow Ca(OH)_2(aq) + H_2(g)$ ), whereas Sodium ( $Na$ ) reacts almost explosively ( $2Na(s) + 2H_2O(l) \rightarrow 2NaOH(aq) + H_2(g)$ ).
Surface Area (Particle Size):
- Exposure: Smaller particle sizes (e.g., powdered calcium carbonate) have more surface area exposed to other reactants compared to larger chunks (e.g., marble chips).
- Frequency: Greater surface area exposure allows for more frequent collisions between reactant particles, speeding up the reaction.
Pressure (for Gases):
- Compression: Increasing the pressure of gases compresses the particles together, causing them to collide more frequently and increasing the reaction rate.

Order of Reactions and Rate Laws

Definition of Rate Laws: Mathematical expressions describing the relationship between the reaction rate and the concentration of reactants.
General Equation: $\text{rate} = k[A]^x$
- $k$ = Rate constant.
- $[A]$ = Concentration of reactant in $mole/L$ .
- $x$ = Order of the reaction.

Zeroth-Order Reactions

Characteristics: The rate does not depend on the reactant concentration. It remains constant.
Rate Law: $\text{Rate} = -\frac{d[A]}{dt} = k[A]^0 = k$
Integrated Form: $[A] = [A]_0 - kt$
Half-life ( $t_{1/2}$ ): The time required for the concentration to decrease to half its initial value. $t_{1/2} = \frac{[A]_0}{2k}$
Graph: A plot of $[A]$ vs. $t$ yields a straight line with slope $-k$ .

First-Order Reactions

Characteristics: The reaction rate depends directly on the concentration of a single substance. If the initial concentration doubles, the rate also doubles.
Rate Law: $\text{Rate} = -\frac{d[A]}{dt} = k[A]$
Integrated Form: $\ln[A] = \ln[A]_0 - kt$ Alternatively expressed as: $[A] = [A]_0 e^{-kt}$
Half-life ( $t_{1/2}$ ): Independent of initial concentration. $t_{1/2} = \frac{\ln(2)}{k} = \frac{0.693}{k}$
Graph: A plot of $\ln[A]$ vs. $t$ yields a straight line.

Second-Order Reactions

Characteristics: The sum of exponents in the rate law equals two.
Case 1: Identical Reactants ( $A + A \rightarrow P$ or $2A \rightarrow P$ )
- Rate Law: $Rate = k[A]^2$
- Integrated Form: $\frac{1}{[A]} = \frac{1}{[A]_0} + kt$
- Half-life ( $t_{1/2}$ ): Inversely related to initial concentration. $t_{1/2} = \frac{1}{k[A]_0}$
- Graph: A plot of $\frac{1}{[A]}$ vs. $t$ yields a straight line.
Case 2: Multiple Reactants ( $A + B \rightarrow P$ )
- Rate Law: $Rate = k[A][B]$
- Situation 1 ( $[A]_0 = [B]_0$ ): Since they react 1:1, the rate law effectively becomes $Rate = k[A]^2$ .
- Situation 2 ( $[A]_0 \neq [B]_0$ ): Requires the method of partial fractions for integration. $\ln\left(\frac{[B][A]_0}{[A][B]_0}\right) = k([B]_0 - [A]_0)t$ Or: $\ln\left(\frac{[A]}{[B]}\right) = k([A]_0 - [B]_0)t + \ln\left(\frac{[A]_0}{[B]_0}\right)$

Summary of Reaction Kinetics

Order	Rate Law	Integrated Equation	Half-life Formula	$k$ Units	Identifying Graph
Zero	$rate = k$	$[A] = [A]_0 - kt$	$t_{1/2} = \frac{[A]_0}{2k}$	$mol\,L^{-1}\,s^{-1}$	$[A]$ vs $t$
First	$rate = k[A]$	$\ln([A]/[A]_0) = -kt$	$t_{1/2} = \frac{0.693}{k}$	$s^{-1}$	$\ln[A]$ vs $t$
Second	$rate = k[A]^2$	$1/[A] - 1/[A]_0 = kt$	$t_{1/2} = \frac{1}{k[A]_0}$	$M^{-1}\,s^{-1}$	$1/[A]$ vs $t$

Assessments and Practice Problems

Factor Analysis Scenarios:
- Crushed Medicine Tablet: Which factor is involved? (Surface Area). Will the reaction be faster or slower? (Faster). Why? (Powdered form has more exposed surface for the solvent to contact).
- Marinating Meat: Which factor is involved? (Concentration/Nature of reactants/Time). Why does it cook faster? (Chemical breakdown of fibers starts early via acidic/enzymatic marination).
- H2O2 Decomposition + Manganese Dioxide: Which factor is involved? (Catalyst). Why? ( $MnO_2$ provides a lower energy path for decomposition).
Zero-Order Problems:
- Determine $k$ if $[A]_0 = 1.5\,M$ and after $120\,s$ , $[A] = 0.75\,M$ .
- Calculate initial concentration if $k = 2 \times 10^{-2}\,mol/Ls$ , time is $25\,s$ , and final concentration is $0.5\,M$ .
- Find the time for $75\%$ completion if $50\%$ completion takes $100\,min$ .
First-Order Problems:
- If $3.0\,g$ of substance decomposes and after $36\,min$ , $0.375\,g$ remains, find the half-life.
- Determine decomposition of $H_2SO_4$ using $k = 6.4 \times 10^{-1}\,s^{-1}$ at $600.0\,s$ .
- The half-life of $H_2O_2$ is $17.0\,min$ . Find $k$ and the time for $86\%$ decomposition.
Second-Order Problems:
- If $k = 0.5\,M^{-1}\,s^{-1}$ and $[A]_0 = 2\,M$ , find $[A]$ after $4\,seconds$ .
- For $NO_2$ decomposition at $300^\circ C$ with $[NO_2]_0 = 0.056\,M$ , find the concentration after $1.0\,h$ and the time needed to reach $10\%$ of the initial concentration.