# Step-by-Step Solution Let's address each section of your complex problem, breaking down the equations and providing explanations and final answers. --- ## 1. **Energy-dependent Cross-section & Rate Coefficient** **Given:** The differential form of the energy-dependent cross-section \( \sigma(E) \) with a post-exponential temperature dependency \( T^n \): \[ k(T) = \int_^\infty \sigma(E) E \exp\left(-\frac{E}{k_B T}\right) \sqrt{8E/\pi \mu} dE \] **Explanation:** - \( \sigma(E) \): Cross-section as a function of energy - \( E \): Energy - \( k_B \): Boltzmann constant - \( T \): Temperature - \( \mu \): Reduced mass For a **post-exponential** dependency, \( \sigma(E) \propto T^n \). The **integrated rate coefficient** includes the Maxwell-Boltzmann distribution term \( \exp(-E/k_B T) \) and the velocity term \( \sqrt{E} \). --- ### a. **Symbolic Activation Energy (\( E_a \)) via Tolman’s Definition** **Tolman’s formula for activation energy:** \[ E_a = -\frac{d \ln k(T)}{d (1/k_B T)} \] Insert the expression for \( k(T) \): \[ k(T) = \int_^\infty \sigma(E) E \exp\left(-\frac{E}{k_B T}\right) \sqrt{8E/\pi \mu} dE \] Take the logarithmic derivative with respect to \( 1/k_B T \): \[ E_a = -\frac{d}{d(1/k_B T)} \left[ \ln \left( \int_^\infty f(E,T) dE \right) \right] \] Where \( f(E,T) = \sigma(E) E \exp\left(-\frac{E}{k_B T}\right) \sqrt{8E/\pi \mu} \). --- ### b. **Spatial Diagram of the Potential Energy Surface** **Explanation:** The **potential energy surface (PES)** is a multidimensional surface representing the system's energy as a function of nuclear positions (reaction coordinate). **Diagram:** - X-axis: Reaction coordinate (\( \xi \)) - Y-axis: Potential energy A typical diagram will show: - Reactants at a certain energy level - A peak representing the transition state (activation energy \( E_a \)) - Products at a different energy level --- ## 2. **Spatial Distribution of Molar Extent of Reaction (\( \xi \))** **Given PDE:** \[ \frac{\partial \xi}{\partial \tau} = \frac{1}{Pe} \frac{\partial^2 \xi}{\partial z^2} + Da(1 - \xi)^2 \exp \left( -\frac{\beta \xi}{1 + \lambda \xi} \right) \] **Where:** - \( \xi \): Molar extent of reaction - \( \tau \): Dimensionless time - \( z \): Axial coordinate - \( Pe \): Peclet number (convective/axial dispersion ratio) - \( Da \): Damköhler number (reaction/flow rate) - \( \beta, \lambda \): Parameters in the exothermic term ### a. **Bifurcation Point at \( z = .85 \)** **Explanation:** The bifurcation point is where the steady-state solution becomes unstable (thermal runaway initiates). **To determine it:** - At the bifurcation, the rate of heat generation overtakes the rate of heat removal. - Set \( \partial \xi/\partial \tau = \) (steady-state). \[ = \frac{1}{Pe} \frac{\partial^2 \xi}{\partial z^2} + Da(1 - \xi)^2 \exp \left( -\frac{\beta \xi}{1 + \lambda \xi} \right) \] - Numerically or graphically, find the value of \( \xi \) at \( z = .85 \) where multiple steady-state solutions merge (fold point). **Final Answer:** The exact value depends on the parameters, but the method is to solve for \( \xi(z=.85) \) where the above equation's solutions change stability. --- ## 3. **Equilibrium Thermodynamics: Gibbs Minimization** **Given:** Gibbs free energy for \( N \) species and \( M \) elements: \[ G = \sum_{i=1}^N n_i (\mu_i^ + RT \ln \frac{n_i}{n_{total}}) \] **Subject to:** Elemental balance constraints (for each element \( j \)): \[ \sum_{i=1}^N a_{ji} n_i = b_j \quad \text{for } j=1,...,M \] Where \( a_{ji} \) is the number of atoms of element \( j \) in species \( i \), and \( b_j \) is the total amount of element \( j \). --- ### a. **Lagrangian for Constrained Minimization** Define Lagrangian: \[ \mathcal{L} = G + \sum_{j=1}^{M} \lambda_j \left( b_j - \sum_{i=1}^N a_{ji} n_i \right) \] **Set of equations to minimize:** For each species \( i \): \[ \frac{\partial \mathcal{L}}{\partial n_i} = \] Which leads to the **non-linear algebraic equations**: \[ \mu_i^ + RT \ln \left( \frac{n_i}{n_{total}} \right) - \sum_{j=1}^{M} \lambda_j a_{ji} = \] And for each element \( j \): \[ \sum_{i=1}^N a_{ji} n_i = b_j \] --- ### b. **At \( T = 150\,K \), Minimizing \( \Phi \) (Potential Function)** Set up the above system with \( T = 150\,K \) and solve the non-linear equations for \( n_i \). **Schematic Diagram:** - X-axis: Reaction coordinate or composition variable - Y-axis: Gibbs free energy \( G \) - The minimum represents equilibrium composition. --- ## **Summary Table** | Step | Equation/Procedure | Explanation/Diagram | |------|-------------------|---------------------| | 1 | \( k(T) = \int_^\infty \sigma(E) E \exp(-E/k_BT) \sqrt{8E/\pi\mu} dE \), \( E_a = -d\ln k/d(1/k_BT) \) | Activation energy from rate coefficient, PES schematic | | 2 | \( \frac{\partial \xi}{\partial \tau} = \frac{1}{Pe} \frac{\partial^2 \xi}{\partial z^2} + Da(1-\xi)^2 \exp(-\beta\xi/(1+\lambda\xi)) \) | Steady-state at \( z=.85 \): bifurcation analysis | | 3 | \( G = \sum n_i(\mu_i^ + RT\ln n_i/n_{tot}) \), Lagrange minimization | Non-linear system for equilibrium at \( T=150\,K \), Gibbs free energy vs. composition schematic | --- ## **Final Schematic Diagrams** ### 1. **Potential Energy Surface** ``` Potential Energy ^ | __ | / \ |___Reactants_/ \___Products |---Reaction coordinate (\xi)--> ``` ### 2. **Gibbs Free Energy Minimization** ``` Gibbs Free Energy (G) ^ | | __ | / \ <- Minimum at equilibrium composition |____/ \______ |---Composition (\xi)--> ``` --- **Conclusion:** You've now established the mathematical relationships for activation energy, spatial distribution of reaction extent, and equilibrium composition for a multi-component system, along with associated schematic diagrams. For explicit numerical solutions, parameter values and numerical techniques (e.g., Newton-Raphson) would be needed.