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Chapter 14: Problem 32

A platinum catalytic reactor in an automobile is used to convert carbon monoxide to carbon dioxide in an oxidation reaction of the form \(2 \mathrm{CO}+\mathrm{O}_{2} \rightarrow 2 \mathrm{CO}_{2}\). Species transfer between the catalytic surface and the exhaust gases may be assumed to occur by diffusion in a film of thickness \(L=10 \mathrm{~mm}\). Consider an exhaust gas that has a pressure of \(1.2\) bars, a temperature of \(500^{\circ} \mathrm{C}\), and a \(\mathrm{CO}\) mole fraction of \(0.0012\). If the reaction rate constant of the catalyst is \(k_{1}^{\prime \prime}=0.005 \mathrm{~m} / \mathrm{s}\) and the diffusion coefficient of \(\mathrm{CO}\) in the mixture is \(10^{-4} \mathrm{~m}^{2} / \mathrm{s}\), what is the molar concentration of \(\mathrm{CO}\) at the catalytic surface? What is the rate of removal of \(\mathrm{CO}\) per unit area of the catalyst? What is the removal rate if \(k_{1}^{\prime \prime}\) is adjusted to render the process diffusion limited?

Short Answer

Expert verified
The molar concentration of CO at the catalytic surface is calculated using the Ideal Gas Law and the given mole fraction, pressure, and temperature values, and is found to be \(C_{\mathrm{CO}} = \frac{0.0012 * P}{R T}\). The rate of removal of CO per unit area of the catalyst is determined using the given reaction rate constant and the molar concentration of CO, which results in \(-k_{1}^{\prime\prime} * C_{\mathrm{CO}}\). To adjust the reaction rate constant for a diffusion limited process, we equate the rate of removal per unit area to the rate limited, given as \(\frac{D_{\mathrm{CO}} * C_{\mathrm{CO}}}{L}\), and solve for the adjusted reaction rate constant, resulting in \(k_{1\,adjusted}^{\prime\prime} * C_{\mathrm{CO}} = \frac{D_{\mathrm{CO}} * C_{\mathrm{CO}}}{L}\).

Step by step solution

01

Calculate the molar concentration of CO

First, let's calculate the molar concentration of CO at the catalytic surface. Since we know the mole fraction of CO (0.0012) and the pressure and temperature of the gas (1.2 bars and 500 掳C), we can use the ideal gas law to find the concentration. The ideal gas law states that \(P V=n R T\), where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant and T is the temperature. First, convert the given units to S.I. units: Pressure: \(P=1.2\,bars=1.2*10^{5}\,Pa\) Temperature: \(T=500^{\circ} C = 773.15\: K\) Solve for the molar concentration of the exhaust gas: \[n/V = \frac{P}{RT}\] Using the given value for the CO mole fraction (0.0012) and the calculated exhaust gas concentration, we can now find the molar concentration of CO: \[C_{\mathrm{CO}} = \frac{0.0012 * P}{R T}\]
02

Calculate the rate of removal of CO per unit area of the catalyst

Now let's calculate the rate of removal of CO per unit area of the catalyst, using the given reaction rate constant (\(k_{1}^{\prime \prime}\)) and the molar concentration of CO obtained in step 1: Using the relation, \[rate\, of\, removal\, per\, unit\, area = -k_{1}^{\prime\prime} * C_{\mathrm{CO}}\]
03

Adjust the reaction rate constant for the diffusion limited process

Finally, let's adjust the reaction rate constant to make the process diffusion limited. The rate of removal is determined by the diffusion of CO in this case. We can use the diffusion coefficient of CO (\(10^{-4} \mathrm{~m}^{2} / \mathrm{s}\)) and the film thickness (\(L=10 \mathrm{~mm}\)): The relation for the diffusion limited rate is given as follows: \[rate\,limited = \frac{D_{\mathrm{CO}} * C_{\mathrm{CO}}}{L}\] By setting the rate of removal per unit area equal to the rate limited, we can solve for the adjusted reaction rate constant (\(k_{1\,adjusted}^{\prime\prime}\)), as follows: \[k_{1\,adjusted}^{\prime\prime} * C_{\mathrm{CO}} = \frac{D_{\mathrm{CO}} * C_{\mathrm{CO}}}{L}\] Now we can solve for the adjusted reaction rate constant (\(k_{1\,adjusted}^{\prime\prime}\)) and provide the results for the molar concentration of CO, the rate of removal of CO per unit area of the catalyst, and the removal rate for the diffusion limited process.

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Key Concepts

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

Diffusion in Catalytic Reactors
Understanding how substances like carbon monoxide (CO) move in catalytic reactors is vital for enhancing reaction efficiency. This movement, known as diffusion, is the spontaneous spread of particles from regions of higher concentration to areas of lower concentration. In an automobile's platinum catalytic reactor, diffusion becomes critical as exhaust gases interact with the catalyst surface.

In our example, the diffusion of CO in a film of given thickness is what facilitates the reaction to convert it to carbon dioxide (CO鈧�). The rate of diffusion is influenced by factors such as the diffusion coefficient, which represents how easily CO molecules move through the exhaust gas mixture, and the film thickness, which is the space through which these molecules must diffuse to reach the catalytic surface.

To optimize reactions within catalytic reactors, it's important to balance the reaction rate with the rate of diffusion. When the process is 'diffusion limited', it means that the speed at which CO molecules reach the surface controls the overall rate of the reaction, not the intrinsic speed of the reaction itself. This scenario is particularly crucial in exhaust gas treatment where quick and efficient conversion of toxic gases is required.
Ideal Gas Law Calculations
The ideal gas law is a cornerstone of chemical engineering, allowing us to relate variables such as pressure, temperature, volume, and the number of moles of a gas. The law is typically expressed as the equation, \( PV=nRT \) where \( P \) is pressure, \( V \) is volume, \( n \) is the amount of substance (moles), \( R \) is the ideal gas constant, and \( T \) is temperature.

In our problem, we calculate the molar concentration of CO in the gas phase using the ideal gas law. To do this, we first ensure our temperature and pressure are in the correct units (kelvin and pascals). The CO mole fraction then helps us find the specific concentration of CO at a given pressure and temperature. Calculations like this are essential for engineers to design reactors and control the conditions under which chemical reactions occur efficiently.
Reaction Rate Constant
A reaction rate constant, in the context of a catalytic reactor, is a quantifiable measure of how quickly a reaction proceeds. It factors into the equation describing the rate at which a reactant is consumed or a product is formed over time. In our example, \( k_{1}^{\prime \prime} \) represents this constant for the conversion of CO to CO鈧� on the catalyst surface.

It's crucial to understand that the rate constant is influenced by various conditions, such as temperature and the presence of a catalyst. When we consider that the process may be diffusion limited, we adjust the rate constant to reflect the slower, diffusion-controlled rate. This means that even if the catalyst is highly effective, the overall rate at which CO is removed from the exhaust gas could be limited by how fast CO molecules can diffuse to the catalytic surface.

The rate constant is a key component in the design and optimization of reactors, ensuring that they operate efficiently under the given conditions. Without accurate knowledge of these constants, predicting and controlling the outcome of a chemical reaction would be much more challenging.

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Most popular questions from this chapter

If an amount of energy \(Q_{o}^{\prime \prime}\left(\mathrm{J} / \mathrm{m}^{2}\right)\) is released instantaneously, as, for example, from a pulsed laser, and it is absorbed by the surface of a semi-infinite medium, with no attendant losses to the surroundings, the subsequent temperature distribution in the medium is $$ T(x, t)-T_{i}=\frac{Q_{o}^{\prime \prime}}{\rho c(\pi \alpha t)^{1 / 2}} \exp \left(-x^{2} / 4 \alpha t\right) $$ where \(T_{i}\) is the initial, uniform temperature of the medium. Consider an analogous mass transfer process involving deposition of a thin layer of phosphorous (P) on a silicon (Si) wafer at room temperature. If the wafer is placed in a furnace, the diffusion of \(\mathrm{P}\) into Si is significantly enhanced by the high-temperature environment. A Si wafer with \(1-\mu \mathrm{m}\)-thick P film is suddenly placed in a furnace at \(1000^{\circ} \mathrm{C}\), and the resulting distribution of \(P\) is characterized by an expression of the form $$ C_{\mathrm{P}}(x, t)=\frac{M_{\mathrm{P}, o}^{\prime \prime}}{\left(\pi D_{\mathrm{P}-\mathrm{Si}} t\right)^{1 / 2}} \exp \left(-x^{2} / 4 D_{\mathrm{P}-\mathrm{Si}} t\right) $$ where \(M_{\mathrm{P}, o}^{\prime \prime}\) is the molar area density \(\left(\mathrm{kmol} / \mathrm{m}^{2}\right)\) of \(\mathrm{P}\) associated with the film of concentration \(C_{\mathrm{P}}\) and thickness \(d_{o}\). (a) Explain the correspondence between variables in the analogous temperature and concentration distributions. (b) Determine the mole fraction of \(P\) at a depth of \(0.1 \mu \mathrm{m}\) in the Si after \(30 \mathrm{~s}\). The diffusion coefficient is \(D_{\mathrm{P}-\mathrm{Si}}=1.2 \times 10^{-17} \mathrm{~m}^{2} / \mathrm{s}\). The mass densities of \(\mathrm{P}\) and \(\mathrm{Si}\) are 2000 and \(2300 \mathrm{~kg} / \mathrm{m}^{3}\), respectively, and their molecular weights are \(30.97\) and \(28.09 \mathrm{~kg} / \mathrm{kmol}\).

An experiment is designed to measure the partition coefficient, \(K\), associated with the transfer of a pharmaceutical product through a polymer material. The partition coefficient is defined as the ratio of the densities of the species of interest (the pharmaceutical) on either side of an interface. In the experiment, liquid pharmaceutical \(\left(\rho_{p}=1250 \mathrm{~kg} / \mathrm{m}^{3}\right)\) is injected into a hollow polymer sphere of inner and outer diameters \(D_{i}=5 \mathrm{~mm}\) and \(D_{o}=5.1 \mathrm{~mm}\), respectively. The sphere is exposed to convective conditions for which the density of the pharmaceutical at the outer surface is zero. After one week, the sphere's mass is reduced by \(\Delta M=8.2 \mathrm{mg}\). What is the value of the partition coefficient if the mass diffusivity is \(D_{\mathrm{AB}}=0.2 \times 10^{-11} \mathrm{~m}^{2} / \mathrm{s}\) ?

Pulverized coal pellets, which may be approximated as carbon spheres of radius \(r_{o}=1 \mathrm{~mm}\), are burned in a pure oxygen atmosphere at \(1450 \mathrm{~K}\) and 1 atm. Oxygen is transferred to the particle surface by diffusion, where it is consumed in the reaction \(\mathrm{C}+\mathrm{O}_{2} \rightarrow \mathrm{CO}_{2}\). The reaction rate is first order and of the form \(\dot{N}_{\mathrm{O}_{2}}^{\prime \prime}=\) \(-k_{1}^{\prime \prime} C_{\mathrm{O}_{2}}\left(r_{o}\right)\), where \(k_{1}^{\prime \prime}=0.1 \mathrm{~m} / \mathrm{s}\). Neglecting changes in \(r_{o}\), determine the steady-state \(\mathrm{O}_{2}\) molar consumption rate in \(\mathrm{kmol} / \mathrm{s}\). At \(1450 \mathrm{~K}\), the binary diffusion coefficient for \(\mathrm{O}_{2}\) and \(\mathrm{CO}_{2}\) is \(1.71 \times 10^{-4} \mathrm{~m}^{2} / \mathrm{s}\).

Referring to Problem 14.34, a more representative model of respiration in a spherical organism is one for which oxygen consumption is governed by a firstorder reaction of the form \(\dot{N}_{\mathrm{A}}=-k_{1} C_{\mathrm{A}}\). (a) If a molar concentration of \(C_{\mathrm{A}}\left(r_{o}\right)=C_{\mathrm{A}, o}\) is maintained at the surface of the organism, obtain an expression for the radial distribution of oxygen, \(C_{\mathrm{A}}(r)\), within the organism. Hint: To simplify solution of the species diffusion equation, invoke the transformation \(y \equiv r C_{\mathrm{A}}\). (b) Obtain an expression for the rate of oxygen consumption within the organism. (c) Consider an organism of radius \(r_{o}=0.10 \mathrm{~mm}\) and a diffusion coefficient of \(D_{\mathrm{AB}}=10^{-8} \mathrm{~m}^{2} / \mathrm{s}\). If \(C_{\mathrm{A}, o}=5 \times 10^{-5} \mathrm{kmol} / \mathrm{m}^{3}\) and \(k_{1}=20 \mathrm{~s}^{-1}\), estimate the corresponding value of the molar concentration at the center of the organism. What is the rate of oxygen consumption by the organism?

Insulation degrades (experiences an increase in thermal conductivity) if it is subjected to water vapor condensation. The problem may occur in home insulation during cold periods, when vapor in a humidified room diffuses through the drywall (plaster board) and condenses in the adjoining insulation. Estimate the mass diffusion rate for a \(3 \mathrm{~m} \times 5 \mathrm{~m}\) wall, under conditions for which the vapor pressure is \(0.03\) bar in the room air and \(0.0\) bar in the insulation. The drywall is \(10 \mathrm{~mm}\) thick, and the solubility of water vapor in the wall material is approximately \(5 \times 10^{-3} \mathrm{kmol} / \mathrm{m}^{3}\) - bar. The binary diffusion coefficient for water vapor in the drywall is approximately \(10^{-9} \mathrm{~m}^{2} / \mathrm{s}\).
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