Gaseous A absorbs and reacts with B in liquid according to $$\mathrm{A}(g \rightarrow l)+\mathrm{B}(l) \rightarrow \mathrm{R}(l), \quad- r_{\mathrm{A}}=k C_{\mathrm{A}} C_{\mathrm{B}}$$ in a packed bed under conditions where $$\begin{aligned} &k_{\mathrm{A} g} a=0.1 \mathrm{mol} / \mathrm{hr} \cdot \mathrm{m}^{2} \text { of reactor } \cdot \mathrm{Pa} \quad f_{l}=0.01 \mathrm{m}^{3} \mathrm{liquid} / \mathrm{m}^{3} \text { reactor }\\\ &k_{\mathrm{A} /} a=100 \mathrm{m}^{3} \text { liquid/m }^{3} \text { reactor } \cdot \mathrm{hr} \quad \mathscr{D}_{\mathrm{A} l}=\mathscr{D}_{\mathrm{B} t}=10^{-6} \mathrm{m}^{2} / \mathrm{hr}\\\ &a=100 \mathrm{m}^{2} / \mathrm{m}^{3} \text { reactor } \end{aligned}$$ At a point in the reactor where \(p_{\mathrm{A}}=100 \mathrm{Pa}\) and \(C_{\mathrm{B}}=100 \mathrm{mol} / \mathrm{m}^{3}\) liquid (a) calculate the rate of reaction in \(\operatorname{mol} / \mathrm{hr} \cdot \mathrm{m}^{3}\) of reactor. (b) describe the following characteristics of the kinetics: \(\cdot\)location of the major resistance (gas film, liquid film, main body of liquid) \(\cdot\)behavior in the liquid film (pseudo first-order reaction, instantaneous, second-order reaction, physical transport) for the following values of reaction rate and Henry's law constant. $$\begin{array}{cc} \hline k, \mathrm{m}^{3} \text { liquid/mol } \cdot \mathrm{hr} & H_{\mathrm{A}}, \mathrm{Pa} \cdot \mathrm{m}^{3} \text { liquid/mol } \\ \hline 10^{6} & 10^{4} \end{array}$$

Step by step solution

01

Calculate the Rate of Reaction

Apply the given rate of reaction equation \(-r_{A} = k C_{A} C_{B}\). Use the provided parameters and substitute the values into the equation. Calculate the concentration of A, \(C_{A}\), in liquid phase from the partial pressure \(p_{A}\) using Henry's law \(p_{A} = H_{A} C_{A}\). However, we are not given \(H_{A}\), so in this scenario, we only multiply \(k\), \(C_{A}\) and \(C_{B}\) directly assuming that the concentration of A in the liquid phase is proportional to its partial pressure.

02

Identify Major Resistance

To assess the location of the major resistance, compare the gas film and liquid film resistances. The gas film mass transfer coefficient and the liquid film mass transfer coefficient are given by \(k_{Ag} a\) and \(k_{A/} a\) respectively. If \(k_{Ag} a < k_{A/} a\), the gas film offers more resistance, and vice versa.

03

Determine Behavior in Liquid Film

To determine the behavior in the liquid film, we need to look at the given values for the reaction rate constant \(k\) and Henry's law constant \(H_A\). A high reaction rate constant relative to the mass transfer coefficients would suggest a fast reaction, which can be considered pseudo first-order if \(C_B\) remains constant, instantaneous if the reaction is very fast compared to mass transfer, or second-order if dependent on both \(C_A\) and \(C_B\). The given Henry's law constant affects the solubility of A and therefore its concentration in the liquid phase, influencing the reaction rate and the transport process.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rate of Reaction

Understanding the rate of reaction is crucial in chemical reaction engineering. It's the speed at which reactants are converted into products, indicated by the change in concentration over time. In the given exercise, the rate of reaction is modeled by \( -r_{\mathrm{A}} = k C_{\mathrm{A}} C_{\mathrm{B}} \), with \( r_{\mathrm{A}} \) representing the rate at which species A disappears. The negative sign signifies a decrease in the concentration of reactant A over time.

To calculate the rate of reaction, one must know the concentrations of the reacting species and the rate constant \( k \). With high concentrations and a large rate constant, the reaction would proceed rapidly, leading to a high reaction rate. Conversely, low concentrations or a small rate constant would result in a slow reaction. Knowing this rate is essential for designing reactors and predicting the performance of a chemical process.

Pitfalls in Understanding Rate of Reaction

Students should note that the rate of a reaction is not constant; it can change as the reactants are consumed and the reaction proceeds. They also need to be aware that the units of the rate constant \( k \) vary, depending on the order of the reaction, which must be taken into account when performing calculations.

Mass Transfer Coefficients

Mass transfer coefficients are a measure of how easily a substance moves from one phase to another and are pivotal in studying reaction kinetics in multiphase systems. In our exercise, we are provided with gas film (\( k_{\mathrm{A}g} a \) ) and liquid film (\( k_{\mathrm{A}/} a \) ) mass transfer coefficients. These coefficients signify the rate of mass transfer per area of contact between phases and per unit driving force, typically a concentration or partial pressure difference.

The gas film mass transfer coefficient relates to the resistance against the gas's absorption into the liquid, while the liquid film coefficient represents the resistance once the gas is within the liquid boundary layer. A smaller coefficient indicates greater resistance to mass transfer.

Importance of Mass Transfer Coefficients

Understanding these coefficients is essential in designing equipment such as absorbers or reactors, as these coefficients determine the efficiency of the mass transfer process. When either the gas film or liquid film mass transfer coefficient is much lower than the other, it becomes the rate-limiting step, dictating the overall rate of mass transfer. Students often confuse these coefficients with the rate of reaction, but it's vital to differentiate between the two: mass transfer coefficients are about the movement of species between phases, not their reaction rates.

Henry's Law

Henry's Law plays a fundamental role in the absorption process, as seen in the given exercise. It's a gas law that relates the concentration of a gas in a liquid to its partial pressure above the liquid. The law is stated as \( p_{\mathrm{A}} = H_{\mathrm{A}} C_{\mathrm{A}} \), where \( p_{\mathrm{A}} \) is the partial pressure of the gas, \( C_{\mathrm{A}} \) is the concentration of the gas in the liquid, and \( H_{\mathrm{A}} \) is the Henry's Law constant. This constant is unique to each gas-liquid combination and temperature.

In scenarios where the solubility of the gas plays a role in determining the overall rate of a reaction, Henry's Law becomes significant. A higher Henry's Law constant means the gas is less soluble in the liquid, which could limit the reaction rate if the reaction takes place in the liquid phase.

Understanding Henry's Law in Reactor Design

For students, it's essential to grasp that Henry's Law helps to convert between the concentration of the gas in the liquid phase and its partial pressure, facilitating the calculation of reaction rates in heterogeneous reactions. Misapplication of Henry's Law can lead to significant errors in designing reactors and evaluating their performance, as it directly affects the concentration of reactant available in the liquid phase for reaction.