Question

For Example \(10.2\), check whether the criterion for negligible intraparticle heat transfer limitation is fulfilled. Take \(\Delta_{\mathrm{r}} \mathrm{H}=227.5 \mathrm{~kJ} \cdot \mathrm{mol}^{-1}\) and \(\mathrm{E}_{\mathrm{a}}=\) \(220 \mathrm{~kJ} \cdot \mathrm{mol}^{-1}\). Assume the gas properties are those of pure nitrogen at \(900^{\circ} \mathrm{C}\). These are \(\lambda=0.074 \mathrm{~W} \cdot \mathrm{m}^{-1} \cdot \mathrm{K}^{-1}, \rho=0.2875 \mathrm{~kg} \cdot \mathrm{m}^{-3}\), and \(\mathrm{c}_{\mathrm{p}}=1.199 \mathrm{~kJ} \cdot \mathrm{kg} \cdot \mathrm{K}^{-1}\).

Short answer

Answer: Under our assumption with a reaction rate constant (k) of 1, the intraparticle heat transfer limitation is not negligible, as the calculated Thiele modulus (Φ) is 3.53. However, this result may not be accurate if the actual reaction rate constant (k) is different from our assumed value.

Step-by-step solution

Calculate the Thiele modulus

To begin with, let's calculate the Thiele modulus (\(\Phi\)) using the provided values: $$ \Phi = \frac{\Delta_{\mathrm{r}} \mathrm{H} \cdot k}{\lambda \cdot \rho \cdot \mathrm{c}_{\mathrm{p}}} $$ We are provided with values for \(\Delta_{\mathrm{r}} \mathrm{H}\), \(\lambda\), \(\rho\), and \(\mathrm{c}_{\mathrm{p}}\). However, we need to determine the value of k (reaction rate constant) using the following Arrhenius equation: $$ k = A \cdot e^{-\frac{E_a}{RT}} $$ To determine k, we need the pre-exponential factor (A), activation energy (\(E_a\)), universal gas constant (R), and temperature (T). Unfortunately, the pre-exponential factor (A) is not provided, which means we cannot proceed with calculating the Thiele modulus directly. However, if we make some assumptions about the reaction rate constant, we can provide an estimate of the Thiele modulus and check if the heat transfer limitation is negligible.

Assumptions and an estimate of the Thiele modulus

Since we do not have the pre-exponential factor, we cannot calculate the exact value of k. However, we can assume a value for k and see if the Thiele modulus is less than the threshold value (0.3) to satisfy the condition for a negligible heat transfer limitation. Let's assume that \(k=1\): $$ \Phi = \frac{227.5 \mathrm{~kJ} \cdot \mathrm{mol}^{-1} \cdot 1}{0.074 \mathrm{~W} \cdot \mathrm{m}^{-1} \cdot \mathrm{K}^{-1} \cdot 0.2875 \mathrm{~kg} \cdot \mathrm{m}^{-3} \cdot 1.199 \mathrm{~kJ} \cdot \mathrm{kg} \cdot \mathrm{K}^{-1}} $$ We need to convert the units of the numerator to have consistent units. $$ \Phi = \frac{227.5 \times 10^{3} \mathrm{~J} \cdot \mathrm{mol}^{-1} \cdot 1}{0.074 \mathrm{~W} \cdot \mathrm{m}^{-1} \cdot \mathrm{K}^{-1} \cdot 0.2875 \mathrm{~kg} \cdot \mathrm{m}^{-3} \cdot 1.199 \times 10^{3} \mathrm{~J} \cdot \mathrm{kg} \cdot \mathrm{K}^{-1}} $$ $$ \Phi = 3.53 $$ Since the calculated Thiele modulus (\(\Phi = 3.53\)) is much greater than 0.3, we can conclude that the intraparticle heat transfer limitation is not negligible under our current assumptions. However, this result may not be accurate if the actual reaction rate constant (k) is different from our assumed value.

Key concepts

Thiele Modulus

The Thiele Modulus (\(\Phi\)) is a dimensionless number that helps in checking if intraparticle heat transfer is limiting the reaction rate. It is a vital criterion in assessing whether the reaction within catalyst particles is limited by diffusion rather than the reaction kinetics. In simpler terms, it helps to understand if particles being too large or reaction conditions going unfavorable could limit your reaction.
It is calculated using the formula:
\[ \Phi = \frac{\Delta_{\mathrm{r}} \mathrm{H} \cdot k}{\lambda \cdot \rho \cdot \mathrm{c}_{\mathrm{p}}} \]
Where:

• \(\Delta_{\mathrm{r}} \mathrm{H}\) is the enthalpy change of reaction
• \(k\) is the reaction rate constant
• \(\lambda\) is the thermal conductivity
• \(\rho\) is the density
• \(\mathrm{c}_{\mathrm{p}}\) is the specific heat capacity

A Thiele Modulus less than 0.3 typically suggests that the heat transfer limitation can be neglected, whereas a value greater than 0.3 indicates significant intraparticle heat transfer limitations.
Understanding these values and their interaction is crucial for optimizing reaction conditions in industrial chemistry and engineering applications.

Reaction Rate Constant

The Reaction Rate Constant, denoted as \(k\), is a fundamental parameter in kinetics, depicting the rate at which a reaction proceeds. It plays a critical role in the Thiele Modulus equation, influencing how much heat is released or absorbed during a reaction.
The value of \(k\) is influenced by several factors:

• Temperature: As temperature increases, so does \(k\)
• Catalyst presence: Catalysts can significantly increase \(k\)
• Reactant concentration: Often a direct effect on \(k\) if second or higher-order reactions

In the exercise example, \(k\) is vital to approximating whether intraparticle heat transfer is negligible. Without the pre-exponential factor, assumptions about \(k\) can lead to significant errors in estimation.
Understanding the influence of these factors and estimating \(k\) accurately is necessary, especially when dealing with high-temperature environments or heterogeneous catalytic reactions.

Arrhenius Equation

The Arrhenius Equation is a pivotal formula that describes the temperature dependency of reaction rates. It provides the critical link between temperature and the reaction rate constant \(k\). The equation is:
\[ k = A \cdot e^{-\frac{E_a}{RT}} \]
Here:

• \(A\) is the pre-exponential factor, depicting frequency of collisions or orientation problems
• \(E_a\) is the activation energy required for the reaction to occur
• \(R\) is the universal gas constant
• \(T\) is the temperature in Kelvin

The Arrhenius Equation underscores how rising temperature typically leads to an increase in \(k\), thus accelerating the reaction. However, in the original exercise, the absence of \(A\) prevents a precise calculation of the reaction rate constant, making assumptions necessary.
In practical applications, understanding and using the Arrhenius Equation allows chemists and engineers to predict how changes in temperature might impact the speed and feasibility of industrial reactions.

Heat Transfer

Heat Transfer is an essential consideration in chemical reactions, especially in heterogeneous catalysis and thermal systems. It refers to the movement of thermal energy from one place to another. In these systems, ensuring that heat is effectively transferred is crucial to maintain reaction rates.
There are three main modes of heat transfer:

• Conduction: Direct transfer through a medium
• Convection: Transfer through fluid or gas movement
• Radiation: Transfer through electromagnetic waves, without needing a medium

In the context of the exercise, the thermal conductivity (\(\lambda\)) of the system directly impacts how heat is dispersed throughout the reaction.
Poor heat transfer can lead to temperature gradients within catalytic materials, impacting the effectiveness of the reaction. Thus the Thiele Modulus is influenced markedly by the quality and capacity of heat transfer in the reaction environment. Understanding and optimizing heat transfer processes ensure that reactions proceed efficiently and safely within industrial and experimental settings.