91Ó°ÊÓ

The oxidation of nitric oxide $$\mathrm{NO}+\frac{1}{2} \mathrm{O}_{2} \rightleftharpoons \mathrm{NO}_{2}$$ takes place in an isothermal batch reactor. The reactor is charged with a mixture containing 20.0 volume percent NO and the balance air at an initial pressure of \(380 \mathrm{kPa}\) (absolute). (a) Assuming ideal-gas behavior, determine the composition of the mixture (component mole fractions) and the final pressure (kPa) if the conversion of NO is 90\%. (b) Suppose the pressure in the reactor eventually equilibrates (levels out) at \(360 \mathrm{kPa}\). What is the equilibrium percent conversion of NO? Calculate the reaction equilibrium constant at the prevailing temperature, \(K_{p}\left[(\mathrm{atm})^{-0.5}\right]\), defined as $$K_{p}=\frac{\left(p_{\mathrm{NO}_{2}}\right)}{\left(p_{\mathrm{NO}}\right)\left(p_{\mathrm{O}_{2}}\right)^{0.5}}$$ where \(p_{i}(\mathrm{atm})\) is the partial pressure of species \(i\left(\mathrm{NO}_{2}, \mathrm{NO}, \mathrm{O}_{2}\right)\) at equilibrium. (c) Assuming that \(K_{\mathrm{p}}\) depends only on temperature, estimate the final pressure and composition in the reactor if the feed ratio of NO to \(\mathrm{O}_{2}\) and the initial pressure are the same as in \(\operatorname{Part}(\) a), but the feed to the reactor is pure \(\mathrm{O}_{2}\) instead of air. (d) Replace the partial pressures in the expression for \(K_{\mathrm{p}}\), and use the result to explain how reactor pressure influences the conversion of NO to \(\mathrm{NO}_{2}\)

Step by step solution

01

- Calculating composition and final pressure

For part (a), start by calculating the mole fractions of NO and O2. Given that air is roughly 21% O2 and 79% N2 by volume, the mole fraction \(X_{NO}\) is 0.2 and \(X_{O_2}\) is 0.21*0.8 = 0.168. The volume ratio between N2 and the reactants is 0.79/0.2=3.95, hence \(X_{N2}\) is 3.95*0.168=0.664. To get the final pressure, apply the ideal gas law. In this case, the number of moles decreases so pressure also decreases. If the conversion of NO is 90%, final pressure will be \(P_{final}\) = \(P_{initial} \times (1-0.9X_{NO}) = 380 \times (1-0.9 \times 0.2) = 344 kPa\).

02

- Calculating equilibrium percent conversion

For part (b), use Pfinal to find the equilibrium conversion. From the final pressure in the reactor, calculate the conversion of NO, which is \(X_{NO}\) = \(1-P_{final}/P_{initial}\) = \(1-360/380 = 0.053\). This is equivalent to a 5.3% conversion. Then, calculate the reaction equilibrium constant using the expression provided, which is \(K_{p} = (p_{NO_2})/(p_{NO}(p_{O_2})^{0.5})\). You get \(K_{p} = X_{NO_2}/(X_{NO}(X_{O_2})^{0.5})\). Substitute the expression above into Xs in the formula \(K_p\) obtain the value of \(K_p\).

03

- Calculating final pressure and composition

For part (c), maintain the same ratio of NO to O2 from part (a), but change the feed to the reactor to pure O2. This means the mole fraction of O2 will change to 0.8. Substitute the expression \(X_O2\) into the formulas provided above for the final pressure and the equilibrium percentage conversion of NO to get the results.

04

- Evaluating the influence of reactor pressure

For part (d), analyze how reactor pressure influences the conversion of NO to NO2 by looking at the expression for \(K_{p}\). This equation can be written in terms of pressure and mole fractions. Analzes the relationship between the reaction equilibrium constant (\(K_{p}\)) and the reactor pressure in terms of affecting the conversion of NO to NO2.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Ideal Gas Law

The Ideal Gas Law is a fundamental principle in Chemical Reaction Engineering used to relate pressure, volume, temperature, and the number of moles of gases. The law is expressed as \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the universal gas constant, and \( T \) is temperature.
Understanding this relationship is crucial when dealing with gases in reactors, as it allows predictions of how the system will behave under different conditions.

• It assumes gases behave ideally, which means their particles occupy negligible space and have no interactions except during collisions.
• This assumption is often valid under high temperature and low-pressure conditions.

When applied, this law helps to calculate parameters like the final pressure or composition of a gas mixture after a reaction, as demonstrated in the nitric oxide oxidation problem.
By understanding the Ideal Gas Law, you'll be able to predict changes in system variables, allowing for better control and optimization of reactions.

Equilibrium Constant

The Equilibrium Constant, \( K_p \), is a vital concept that quantifies the state of a reaction at equilibrium, especially concerning pressure in gas-phase reactions.
For the reaction involving nitric oxide oxidation, \( K_p \) is defined as:\[K_{p} = \frac{(p_{NO_{2}})}{(p_{NO})(p_{O_{2}})^{0.5}}\]
At equilibrium, the forward and reverse reaction rates are equal, and the concentrations of reactants and products become constant.

• If \( K_p \) is large, the reaction favors the formation of products at equilibrium.
• If \( K_p \) is small, the reaction favors the reactants.

In the exercise, understanding \( K_p \) allows for the calculation of the equilibrium conversion of nitric oxide into nitrogen dioxide. It also helps demonstrate how changes in pressure or composition affect the equilibrium state.
Remember, \( K_p \) depends on temperature and can be used to predict how a reaction will shift when subjected to temperature changes.

Isothermal Batch Reactor

An Isothermal Batch Reactor is a type of chemical reactor where reactions occur at a constant temperature over a period of time.
In such reactors, the composition of the reaction mixture changes as the reaction progresses, but there is no inflow or outflow of material.

• Maintaining constant temperature is vital because reaction rates and equilibria can be highly temperature dependent.
• The reactants are initially charged into the reactor, and products are removed only after the reaction is concluded.

For the oxidation of nitric oxide given in the exercise, using an isothermal batch reactor ensures that the reaction dynamics, such as pressure changes and conversion rates, depend only on the initial conditions and the reaction's kinetics.
This type of reactor is ideal for reactions where a single, controlled environment is necessary to simplify analysis and optimize reaction conditions, as sharply varying external conditions could complicate the reaction outcome.

Nitric Oxide Oxidation

Nitric Oxide Oxidation is a chemical reaction where nitric oxide \( NO \) reacts with oxygen \( O_2 \) to form nitrogen dioxide \( NO_2 \).
This process is demonstrated as:\[\mathrm{NO} + \frac{1}{2} \mathrm{O}_2 \rightleftharpoons \mathrm{NO}_2\]
This reaction plays a significant role in atmospheric chemistry and industrial processes.

• Nitric oxide is a significant pollutant emitted from vehicles and industrial activities.
• Its oxidation to nitrogen dioxide is a critical step in forming photochemical smog.

In chemical engineering, understanding this oxidation process is crucial for pollution control and the design of reactors aimed at mitigating NO emissions.
In the context of the exercise, calculating conversions and equilibrium conditions helps illustrate how to manage chemical reactions to achieve desirable environmental outcomes.