particles in a box Chemical Kinetics

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WHAT IS IT?

Understanding Chemical Kinetics in a Box Model Simulation

Chemical kinetics describes how the concentration of a species changes over time. Unlike thermodynamics, which does not explicitly incorporate time, kinetics is essential for understanding biological and chemical processes.

This model simulates the reaction:

1-propanol ⇌ 2-propanol

using NetLogo's boxes-Propanol-Simulation to visualize reaction kinetics and equilibrium.

HOW IT WORKS

Activation Energy and Reaction Progression

A new switch, Ea (Energy of Activation), introduces a barrier between the two compartments. This barrier limits molecular transitions—only molecules with energy greater than Ea can change states.

Initially, observe how the system evolves:
- ΔG (Free Energy Change) over time
- ΔG° (Standard Free Energy Change) over time
- RTln([Product]/[Reactant])—its relationship with ΔG° at equilibrium

Now, introduce the activation barrier by increasing Ea from 0 to 1 kcal. Analyze:
- How ΔG, ΔG°, and RTln([Product]/[Reactant]) change with time
- How the reaction reaches equilibrium
- How temperature affects equilibrium and reaction rates

When equilibrium is reached, increase temperature and observe its effects on ΔG°, ΔG, and RTln([Product]/[Reactant]).

How does the activation barrier influence the system?

REACTION RATE

In a simple reaction, reactant molecules convert into product molecules at a measurable rate. Since each reactant molecule forms exactly one product molecule, forward and reverse reaction rates are equal:

• Forward rate: v₁ = -d[R]/dt = d[P]/dt
  
• Reverse rate: v₋₁ = -d[P]/dt = d[R]/dt
  

Reaction speed can be measured by counting how many molecules transition between compartments per unit time.

To obtain accurate rate measurements, use initial reaction velocity to avoid interference from reverse reactions.

RATE, ACTIVATION ENERGY, AND CONCENTRATION

We analyze how reaction rate changes with reactant concentration.

1. Set temperature to 310 K, Ea to 0 kcal, and advance to 0 (all molecules start as reactants).
   
2. Activate single-molecule tracking (switch “single” = On).
   
3. Adjust reactant concentration (number-of-particles) to 2000, 4000, 6000, 8000, and 10000 molecules.
   
4. Count the number of molecules transitioning to the product state (red), representing v₁.
   
5. Plot v₁ vs. concentration.
   

Now, repeat with Ea = 1 kcal and graph again.

The slope of this graph defines the rate constant (k) for the reaction:
v₁ = k[A]

REACTION RATE AND TEMPERATURE

Exercise

In a box with 10,000 molecules and Ea = 1 kcal, measure initial reaction rates at 400 K, 600 K, and 800 K:
1. Set advance = 0 (all reactants on one side).
2. Activate single-molecule tracking (switch “single” = On).
3. Count how many molecules cross the barrier in one tick (red molecules).
4. Repeat with all molecules initially in the product state (advance = 1). Count green molecules.

Now, calculate the rate constants:

k₁ = v₁ / 10000
k₋₁ = v₋₁ / 10000

EQUILIBRIUM CONSTANT (Keq) AND RATE CONSTANTS

At equilibrium, the reaction balances:
v₁ = v₋₁

Since reaction rates are defined as:
v₁ = k₁[A] and v₋₁ = k₋₁[B]

At equilibrium:
k₁[A] = k₋₁[B]

Rearrange:
k₁/k₋₁ = [B]/[A],

where [B]/[A] represents the equilibrium constant Keq.

Verification

Compare Keq = k₁/k₋₁ with the direct equilibrium calculation from simulation:

Keq = (number of product molecules) / (number of reactant molecules).

Does the model correctly follow this equilibrium relationship?

RATE CONSTANT, ACTIVATION ENERGY, AND ARRHENIUS RELATION

Using measured k₁ values at different temperatures, calculate Ea using the Arrhenius equation:

k = A * exp(-Ea / RT)

Rearrange to determine Ea graphically:

ln(k) = ln(A) - Ea / (R*T)

Exercise

1. Plot ln(k₁) vs. 1/T
2. The slope of the plot gives Ea/R, allowing calculation of Ea.
   
3. Repeat the calculation using k₋₁ values to find reverse activation energy.
   

PRE-EXPONENTIAL FACTOR IN ARRHENIUS EQUATION

In this box model, the pre-exponential factor (A) in the Arrhenius equation corresponds to the ordinate at origin in the plot of ln(k) vs. 1/T.

Physically, this factor represents the frequency of successful molecular transitions between compartments. It can be calculated directly in the model.

Interestingly, the ln(A) is equivalent to the ln of the probability of jumping to the other compartment, which is determined by the width of the destination sector. This reinforces the statistical basis of reaction kinetics, demonstrating that the likelihood of transition is tied to entropy-driven accessibility.

COPYRIGHT AND LICENSE

Copyright 1997 Uri Wilensky.

This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-sa/3.0/ or send a letter to Creative Commons, 559 Nathan Abbott Way, Stanford, California 94305, USA.

Commercial licenses are also available. To inquire about commercial licenses, please contact Uri Wilensky at uri@northwestern.edu.

This model was created as part of the project: CONNECTED MATHEMATICS: MAKING SENSE OF COMPLEX PHENOMENA THROUGH BUILDING OBJECT-BASED PARALLEL MODELS (OBPML). The project gratefully acknowledges the support of the National Science Foundation (Applications of Advanced Technologies Program) -- grant numbers RED #9552950 and REC #9632612.

This model was converted to NetLogo as part of the projects: PARTICIPATORY SIMULATIONS: NETWORK-BASED DESIGN FOR SYSTEMS LEARNING IN CLASSROOMS and/or INTEGRATED SIMULATION AND MODELING ENVIRONMENT. The project gratefully acknowledges the support of the National Science Foundation (REPP & ROLE programs) -- grant numbers REC #9814682 and REC-0126227. Converted from StarLogoT to NetLogo, 2002.

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