# Step-by-Step Solution ## a. Separable Kinetics and Types of Decay Separable kinetics refers to a reaction rate law where the rate of reaction can be expressed as a product of a function of concentration and a function of time, i.e., \( r = f(C)g(t) \). For many reactions, especially first-order decay, the rate law can be written as \( \frac{dC}{dt} = -kC \), where \( k \) is the rate constant. Solving this differential equation gives \( C(t) = C_ e^{-kt} \), where \( C_ \) is the initial concentration. Two types of decay commonly encountered are first-order decay (as above) and zero-order decay, where \( \frac{dC}{dt} = -k \), which leads to \( C(t) = C_ - kt \). ## b. Diffusion and Molar Flux For equimolar counter diffusion, two species A and B diffuse in opposite directions with equal molar flux but opposite signs, so the net total molar flux is zero. The molar flux of A, \( N_A \), can be given by Fick's law as \( N_A = -D_{AB} \frac{dC_A}{dz} \), where \( D_{AB} \) is the diffusion coefficient, and \( \frac{dC_A}{dz} \) is the concentration gradient of A. For diffusion through a stagnant film, one component (say, B) is non-diffusing or stagnant, and only A diffuses through the film. The molar flux of A, \( N_A \), is given by \( N_A = -D_{AB} \frac{dC_A}{dz} \). However, the total flux now is not zero because B does not counter-diffuse, so the movement of A is through a stationary medium. ## c. Conditions for Equality of Space-Time and Mean Residence Time, and RTD for PFR The space-time, \( \tau \), is defined as the reactor volume divided by the volumetric flow rate (\( \tau = V/Q \)). The mean residence time, \( t_m \), is the average time that a molecule spends in the reactor. \( \tau \) and \( t_m \) are equal under the following conditions: (1) the flow is steady and there is no dead volume or bypassing, and (2) the density and volumetric flow rate are constant throughout the reactor (e.g., for incompressible fluids). For a plug flow reactor (PFR), the residence time distribution (RTD) function is a Dirac delta function: \( E(t) = \delta(t - \tau) \), indicating that all molecules spend exactly the same amount of time (\( \tau \)) in the reactor before exiting. --- **Final Answers:** a. Separable kinetics allows the reaction rate to be written as a product of functions of concentration and time; for first-order decay, \( C(t) = C_ e^{-kt} \), and for zero-order decay, \( C(t) = C_ - kt \). b. In equimolar counter diffusion, \( N_A = -D_{AB} \frac{dC_A}{dz} \) and the net molar flux is zero. In diffusion through a stagnant film, only one component diffuses and \( N_A = -D_{AB} \frac{dC_A}{dz} \). c. Space-time and mean residence time are equal when there is steady, ideal flow (no bypassing or dead zones) and constant volumetric flow rate. The RTD for a PFR is \( E(t) = \delta(t - \tau) \).

# Step-by-Step Solution ## a. Separable Kinetics and Types of Decay Separable kinetics describes a situation where the reaction rate can be expressed as a product of a function of concentration and a function of time. The general form is given by \( r = f(C)g(t) \). For a first-order reaction, the rate law can be expressed as: \[ \frac{dC}{dt} = -kC \] where \( k \) is the rate constant. Solving this differential equation yields: \[ C(t) = C_0 e^{-kt} \] Here, \( C_0 \) is the initial concentration. Two common types of decay are: 1. **First-order decay**: Characterized by the equation above. 2. **Zero-order decay**: Described by: \[ \frac{dC}{dt} = -k \] This leads to: \[ C(t) = C_0 - kt \] ## b. Diffusion and Molar Flux In the case of **equimolar counter diffusion**, two species, A and B, diffuse in opposite directions with equal molar fluxes, thus the total molar flux is zero. The molar flux of A is given by Fick's law: \[ N_A = -D_{AB} \frac{dC_A}{dz} \] where \( D_{AB} \) is the diffusion coefficient and \( \frac{dC_A}{dz} \) is the concentration gradient of A. For **diffusion through a stagnant film**, one species (B) remains stationary while A diffuses through the film. The molar flux of A remains the same: \[ N_A = -D_{AB} \frac{dC_A}{dz} \] In this scenario, the total flux is not zero since B is not diffusing, resulting in A moving through a stagnant medium. ## c. Conditions for Equality of Space-Time and Mean Residence Time, and RTD for PFR Space-time, denoted as \( \tau \), is defined as the reactor volume divided by the volumetric flow rate: \[ \tau = \frac{V}{Q} \] Mean residence time, \( t_m \), is the average time molecules spend in the reactor. The conditions under which \( \tau \) and \( t_m \) are equal include: 1. The flow is steady, with no dead volume or bypassing. 2. The density and volumetric flow rate are constant throughout the reactor, applicable for incompressible fluids. For a plug flow reactor (PFR), the residence time distribution (RTD) is represented by a Dirac delta function: \[ E(t) = \delta(t - \tau) \] This indicates that all molecules exit the reactor after spending exactly the same time \( \tau \). --- **Final Answers:** a. Separable kinetics allows the reaction rate to be expressed as a product of functions of concentration and time. The first-order decay is represented by \( C(t) = C_0 e^{-kt} \), and zero-order decay is \( C(t) = C_0 - kt \). b. In equimolar counter diffusion, the molar flux is \( N_A = -D_{AB} \frac{dC_A}{dz} \) with a net flux of zero. For diffusion through a stagnant film, the molar flux remains \( N_A = -D_{AB} \frac{dC_A}{dz} \), but the total flux is not zero. c. Space-time and mean residence time are equal under steady flow with no bypassing and constant volumetric flow rates. The RTD for a PFR is given by \( E(t) = \delta(t - \tau) \).