Kinetics and Thermodynamics: Graphical Methods, Spontaneity, and Reaction Order

Graphical Methods for Determining Reaction Order

• To determine the order of a reaction using graphical methods from concentration-over-time data, specific plots are utilized to find a linear relationship:
  • First-Order Reaction: A plot of the natural log of the concentration over time ($\ln([A])$ vs. $t$) yields a straight line.
  • Second-Order Reaction: A plot of one over the concentration over time ($1/[A]$ vs. $t$) yields a straight line.
  • Zeroth-Order Reaction: A plot of the concentration directly over time ($[A]$ vs. $t$) yields a straight line.
• These relationships are recorded on the "integrated rate cheat sheet" which students should reference to identify the correct straight-line graph for each case.

Collision Theory and Chemical Reactions

• Two primary factors may prevent a collision between particles from producing a chemical reaction even if they physically meet:
  • Correct Orientation: The molecules must strike each other in a specific spatial alignment that allows bonds to break and form.
  • Adequate Energy (Activation Energy): There must be enough kinetic energy to overcome the energy barrier of the reaction.
• If either the correct orientation or sufficient energy is missing, the particles will simply bounce off each other and "peel away" without reacting.

Kinetics Problem Solving: Determining Reaction Order and Rates

• Scenario: A simple decomposition reaction involving one reactant. A data table provides concentration values ($M$) at specific time intervals ($min$).
• Data Provided:
  • $0\,min$: $0.458\,M$
  • $45\,min$: $0.370\,M$
  • $107\,min$: $0.292\,M$
  • $600\,min$: $0.114\,M$
• Calculating Reaction Rates: The rate of decomposition is defined as the negative change in concentration over the change in time:
  • $\text{Rate} = - \frac{\Delta [A]}{\Delta t}$
  • For the first interval ($0$ to $45\,min$): $\frac{0.458 - 0.370}{45 - 0} = 0.001955...\,M\,min^{-1}$.
• Determining Rate Law via Ratios: One technique involves calculating the rate for different increments and comparing them to identifying how the rate changes relative to the initial concentration of that increment.
  • If you have rates and concentrations, you can use the ratio method: $\frac{\text{Rate}_2}{\text{Rate}_1} = \frac{k[A]_2^x}{k[A]_1^x}$.
• Strategy for "Ugly" Numbers on Exams:
  • If a calculated value is very close to a whole number (e.g., $1.8$, $1.9$, or $2.1$, $2.2$), round to the nearest whole number (e.g., $2$).
  • If a value falls directly between two whole numbers (e.g., $2.5$), leave it as a decimal.

Integrated Rate Laws and Concentration Over Time

• Problem: Given a second-order reaction starting at $0.458\,M$ with a rate constant $k = 0.0115$, find the concentration after $12\,hours$.
• Step 1: Unit Conversion: Since the data and $k$ are in minutes, convert hours to minutes.
  • $12\,hours \times 60\,min/hour = 720\,min$
• Step 2: Apply Integrated Rate Law: For a second-order reaction:
  • $\frac{1}{[A]_t} = kt + \frac{1}{[A]_0}$
• Calculation:
  • $\frac{1}{[A]_t} = (0.0115 \times 720) + \frac{1}{0.458}$
  • This yields a final concentration of approximately $0.09557\,M$.
• Key Reminder: If an exam question provides time and asks for concentration (or vice versa), the integrated rate law is the required tool.

The Arrhenius Equation and Activation Energy

• Activation Energy Calculation: When given two different rate constants ($k$) at two different temperatures ($T$), use the linear form of the Arrhenius equation to solve for activation energy ($E_a$).
• Variables:
  • $y = \ln(k)$
  • $x = \frac{1}{T}$
  • $\text{Slope} (m) = -\frac{E_a}{R}$
• Mathematical Process:
  • 1. Convert all temperatures to Kelvin ($K$).
  • 2. Calculate $x_1 = \frac{1}{T_1}$ and $x_2 = \frac{1}{T_2}$.
  • 3. Calculate $y_1 = \ln(k_1)$ and $y_2 = \ln(k_2)$.
  • 4. Find the slope: $m = \frac{y_2 - y_1}{x_2 - x_1}$.
  • 5. Multiply the slope by $-R$ (where $R = 8.314\,J\,mol^{-1}\,K^{-1}$) to find $E_a$.
• Frequency Factor ($A$): In the equation $\ln(k) = -\frac{E_a}{R} \times \frac{1}{T} + \ln(A)$, $A$ represents the frequency factor, which accounts for the frequency of collisions and their orientation.

Fundamental Principles of Thermodynamics

• Standard State Symbol: The little circle ($\circ$) in $\Delta G^{\circ}$ signifies that the reaction occurs under standard conditions, usually at a set temperature of $25^{\circ}C$ ($298.15\,K$).
• Spontaneity Criteria:
  • If $\Delta G < 0$: The reaction is spontaneous.
  • If $\Delta G = 0$: The system is at equilibrium.
  • If $\Delta G > 0$: The reaction is non-spontaneous.
• Thermodynamics vs. Kinetics: A spontaneous reaction (e.g., the oxidation of plastics into $CO_2$ and $H_2O$) does not necessarily happen quickly. Plastics persist in the environment because their degradation rate is extremely slow, illustrating that thermodynamics tells us if a reaction will happen, but kinetics tells us how fast.

Thermodynamics Calculation: Phase Changes and Equilibrium

• Boiling Ethanol Problem: Calculate the entropy change ($\Delta S$) for the boiling of ethanol at its boiling point of $78^{\circ}C$, given an enthalpy ($\Delta H$) of $39.3\,kJ$.
  • At the boiling point, the system is at equilibrium, so $\Delta G = 0$.
  • Equation: $0 = \Delta H - T\Delta S$
  • Temperature conversion: $78^{\circ}C + 273.15 = 351.15\,K$
  • $\Delta S = \frac{\Delta H}{T} = \frac{39.3\,kJ}{351.15\,K} = 0.1119\,kJ/K$
• Ammonia Melting Problem:
  • Enthalpy of Fusion ($\Delta H_{fus}$): $5.65\,kJ\,mol^{-1}$
  • Entropy ($\Delta S$): $28.9\,J\,K^{-1}\,mol^{-1}$
  • Part A: Spontaneity at $200\,K$:
    • $\Delta G = 5650\,J - (200\,K \times 28.9\,J/K) = 5650 - 5780 = -130\,J$
    • Since $\Delta G$ is negative, it melts spontaneously at $200\,K$.
  • Part B: Exact Melting Point:
    • Set $\Delta G = 0$: $0 = 5650\,J - T(28.9\,J/K)$
    • $T = \frac{5650}{28.9} = 195.5\,K$.

Multi-Step Reactions and Hess's Law for Entropy

• Hess's Law for Entropy: To find the $\Delta S$ for a target reaction, manipulate side reactions (flip or multiply) and sum their $\Delta S$ values.
• Rules of Manipulation:
  • If you flip a reaction, change the sign of $\Delta S$.
  • If you multiply or divide a reaction by a coefficient, you must multiply or divide the $\Delta S$ value by that same coefficient.
• Example Comparison: For a target reaction with coefficients of $1$, if the sum of manipulated reactions yields coefficients of $2$, the final summed $\Delta S$ must be divided by $2$.

Standard Molar Entropies and Appendix G Calculations

• Appendix G: Uses standardized tables in textbooks to find $\Delta S^{\circ}$ of formation for various species.
• Values for Ammonia Production: $N_2(g) + 3H_2(g) \rightarrow 2NH_3(g)$
  • $N_2(g)$ and $H_2(g)$ in their standard elemental states have specific non-zero entropies (unlike enthalpy of formation, which is zero).
• Equation: $\Delta S^{\circ}_{reaction} = \sum n S^{\circ}(\text{products}) - \sum m S^{\circ}(\text{reactants})$
• Calculation Example: $(2 \times S^{\circ}_{NH_3}) - (S^{\circ}_{N_2} + 3 \times S^{\circ}_{H_2})$.

Questions & Discussion

• Symposium Preparation: Students discussed logistics for an upcoming research symposium.
  • Dress code: Suits or dresses are expected; "probably not sweatpants."
  • Poster Design: Julia demonstrated designing a poster in Canva and submitting it as a PowerPoint.
  • Research Content: A group discussed their "beetle mug shots" taken during cleaning and measuring, and the inclusion of cage images or location pictures.
  • Group Dynamics: Mention of a student named Titus who failed to help collect materials or clean beetles despite multiple requests.
• Lab Anecdotes:
  • A student recounted a heart dissection lab from the previous quarter involving identify heart parts like the aorta and cutting open valves.
  • A story was shared about a student "throwing [parts] around the room" and blood sputtering out when a vein was cut after being moved around.
• Study Materials: A student requested extra practice problems for kinetics and thermodynamics beyond the slides. The instructor promised to send out a worksheet/resource similar to the practice problems provided previous quarters.