91Ó°ÊÓ

Oxygen consumed by a living organism in aerobic reactions is used in adding mass to the organism and/or the production of chemicals and carbon dioxide. since we may not know the molecular compositions of all species in such a reaction, it is common to define the ratio of moles of \(\mathrm{CO}_{2}\) produced per mole of \(\mathrm{O}_{2}\) consumed as the respiratory quotient, \(R Q,\) where $$R Q=\frac{n_{\mathrm{CO}_{2}}}{n_{\mathrm{O}_{2}}}\left(\text { or } \frac{\dot{n}_{\mathrm{CO}_{2}}}{\dot{n}_{\mathrm{O}_{2}}}\right)$$ since it generally is impossible to predict values of \(R Q\), they must be determined from operating data. Mammalian cells are used in a bioreactor to convert glucose to glutamic acid by the reaction $$\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}+a \mathrm{NH}_{3}+b \mathrm{O}_{2} \rightarrow p \mathrm{C}_{5} \mathrm{H}_{9} \mathrm{NO}_{4}+q \mathrm{CO}_{2}+r \mathrm{H}_{2} \mathrm{O}$$ The feed to the bioreactor comprises \(1.00 \times 10^{2} \mathrm{mol} \mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6} / \mathrm{day}, 1.20 \times 10^{2} \mathrm{mol} \mathrm{NH}_{3} / \mathrm{day},\) and \(1.10 \times\) \(10^{2}\) mol \(\mathrm{O}_{2} /\) day. Data on the system show that \(R Q=0.45 \mathrm{mol} \mathrm{CO}_{2}\) produced/mol \(\mathrm{O}_{2}\) consumed. (a) Determine the five stoichiometric coefficients and the limiting reactant. (b) Assuming that the limiting reactant is consumed completely, calculate the molar and mass flow rates of all species leaving the reactor and the fractional conversions of the non-limiting reactants.

Step by step solution

01

Finding stoichiometric coefficients

Firstly calculate stoichiometric coefficients 'p' and 'q'. Start out by using the given Respiratory Quotient (RQ). Rearrange the RQ formula to solve for \(q\): \(RQ = n_{CO2} / n_{O2}\) can be written as \(q = RQ * n_{O2}\). Substituting known values gives: \(q = 0.45 * 110 = 49.5 mol/day\). Second, the reaction formula also equates carbon atoms on both sides. This gives \(6 = p*5 + q\). Substituting the known value for 'q' in this equation yields 'p' as \(56 = p*5 + 49.5 -> p = (56 - 49.5) / 5 = 1.3\).

02

Identifying the limiting reactant

The limiting reactant is the one which is consumed the most in relation to its stoichiometric coefficient. Here, the coefficient relates to 'a' and 'b'. Using the equation for reaction, we get \(C6H12O6 + aNH3 + bO2 -> 1.3C5H9NO4 + 49.5CO2 + rH2O\). By comparing the ratios of reactants to their stoichiometric coefficients, it can be seen that the O2 is limiting since it has the lowest amount of 110/0.45 = 244.44. So, \(a = 49.5 / 1.3 = 38.08\) and \(b = 1 / RQ = 1 / 0.45 = 2.22\). 'r' can be calculated by balancing Hydrogen and Oxygen on both sides of the reaction.

03

Molar and Mass Flow Rates, Fractional Conversions

Assuming that the limiting reactant (O2) is completely consumed, the mass flow rates and fractional conversion of the non-limiting reactants (NH3 and Glucose) can be calculated. Firstly, the flow rates leaving the reactor can be calculated by the reaction stoichiometry. For CO2, it would be \(0.45 * 110\). For C5H9NO4, it would be \(1.3 * 110\) and for NH3, \(38.08 * 110\). For glucose, \(110 - (1*110) = 0\) since it is completely consumed. Lastly, the fractional conversion of NH3 can be calculated as \(1 - (-consuming flows / original feeding flows)\).

04

Calculating 'r'

Balance the number of Hydrogen and atoms to get \(r\). On the left-hand side of the equation, we have \(12 + 3a\) Hydrogen atoms and \(6 + 3b\) Oxygen atoms. On the right-hand side of the equation, we have \(9*1.3 + 2q + 2r\) Hydrogen atoms and \(4*1.3 + 2q + r\) Oxygen atoms. Equating both sides for Hydrogen and Oxygen balances, we obtain ordinary equations which we can solve to get \(r = 135.37\).

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Key Concepts

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

Aerobic Reactions

In the realm of biological and chemical processes, aerobic reactions are essential for sustaining life. These reactions occur in the presence of oxygen and are crucial for the conversion of substrates like glucose into energy, carbon dioxide, and water.
When organisms, particularly aerobic organisms, metabolize glucose, they undergo a series of reactions where oxygen is used to break down glucose, often producing carbon dioxide and water as byproducts. This process of aerobic respiration is pivotal as it provides energy that cells need to function effectively.
In the exercise, the focus is on the transformation of glucose into compounds like glutamic acid using oxygen as a reactant.
This understanding is key because it underlines why monitoring and optimizing oxygen consumption is critical in bioreactor operations, ensuring the desired product is maximized while maintaining metabolic balance.

Stoichiometry

Stoichiometry is a concept that depicts the quantitative nature of chemical reactions, defined by the balancing of molar amounts of reactants and products.
In this exercise, we use stoichiometry to establish the relationships between the quantities of glucose, ammonia, and oxygen and the resulting products such as carbon dioxide and glutamic acid.
By calculating stoichiometric coefficients, students can understand how much of each reactant is needed or produced to maintain balance within a chemical equation. For instance, knowing the stoichiometric coefficient of a reactant helps determine how it proportionally affects the products formed.
The stoichiometric coefficients allow us to interpret how efficiently the reaction occurs and if the reactants are fully utilized, which is directly applied to practical settings like managing reactants within a bioreactor.

Limiting Reactant

The concept of a limiting reactant is critical in chemical reactions as it dictates the extent to which a reaction can proceed. It is the substance that gets completely consumed first, thus limiting the formation of products.
In our exercise, identifying oxygen as the limiting reactant was paramount. Why? Because it allows us to calculate the maximum amount of products that can be formed.
Without determining the limiting reactant, any calculation regarding yields and efficiency would be skewed, potentially leading to a misunderstanding of the system's behavior.
This concept not only helps in academic exercises but also in industrial processes, where ensuring that limiting reactants are adequately supplied can influence the cost and efficiency of chemical production.

Bioreactor

Bioreactors are vessels or devices used to carry out a chemical process involving organisms or biochemically active substances derived from such organisms. They are integral in various biotechnological applications, especially in the production of pharmaceuticals, biofuels, and food products.
In the given exercise, the bioreactor functions as the setting where mammalian cells convert glucose into glutamic acid under controlled aerobic conditions.
Understanding how to control the parameters within a bioreactor, such as oxygen levels, is crucial. This is particularly significant when managing metabolic rates and ensuring the efficient production of desired substances with minimal waste.
Therefore, bioreactors exemplify the practical application of aerobic reactions and chemical balancing through stoichiometry to maximize output and improve process efficiency.