Question

The growth of \(S\). cerevisiae on glucose under anaerobic conditions can be described by the following overall reaction: $$ \begin{aligned} \mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}+\beta \mathrm{NH}_{3} \longrightarrow & 0.59 \mathrm{CH}_{1.74} \mathrm{~N}_{0.2} \mathrm{O}_{0.45} \text { (biomass) }+0.43 \mathrm{C}_{3} \mathrm{H}_{8} \mathrm{O}_{3}+1.54 \\ & \mathrm{CO}_{2}+1.3 \mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}+0.036 \mathrm{H}_{2} \mathrm{O} \end{aligned} $$ a. Determine the biomass yield coefficient \(Y_{X / S}\). b. Determine the product yield coefficients \(Y_{\mathrm{EtOHS}}, Y_{\mathrm{CO}_{2} / \mathrm{S}}, Y_{\mathrm{C}_{3} \mathrm{H}_{2} \mathrm{O} / \mathrm{S}}\). c. Determine the coefficient \(\beta\).

Short answer

Biomass yield coefficient: 0.59. Product yield coefficients: \(Y_{\mathrm{EtOH}/S} = 1.3\), \(Y_{\mathrm{CO_2}/S} = 1.54\), \(Y_{\mathrm{C_3H_8O_3}/S} = 0.43\). Coefficient \(\beta = 0.118\).

Step-by-step solution

Write the given overall reaction

Let's transcribe the given reaction equation:\[\begin{aligned}\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6} + \beta \mathrm{NH}_{3} \longrightarrow & 0.59 \mathrm{CH}_{1.74} \mathrm{~N}_{0.2} \mathrm{O}_{0.45} \text { (biomass) } + 0.43 \mathrm{C}_{3} \mathrm{H}_{8} \mathrm{O}_{3} + 1.54 \ & \mathrm{CO}_{2} + 1.3 \mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH} + 0.036 \mathrm{H}_{2} \mathrm{O}\end{aligned}\]

Determine biomass yield coefficient (\(Y_{X / S}\))

The biomass yield coefficient, \(Y_{X/S}\), is defined as the amount of biomass (\(X\)) produced per unit of substrate (\(S\)) consumed. Here, glucose (\(\mathrm{C}_6\mathrm{H}_{12}\mathrm{O}_6\)) is the substrate.From the equation, the coefficient for biomass (\(X\)) is 0.59 and the coefficient for glucose (\(S\)) is 1.Thus, \(Y_{X/S} = \frac{0.59}{1} = 0.59\).

Determine product yield coefficient (\(Y_{\mathrm{EtOH}/S}\))

The product yield coefficient for ethanol (\(\mathrm{EtOH}\)) is defined as the amount of ethanol produced per unit of substrate consumed.From the equation, the coefficient for ethanol (\(\mathrm{EtOH}\)) is 1.3 and the coefficient for glucose (\(S\)) is 1.Thus, \(Y_{\mathrm{EtOH}/S} = \frac{1.3}{1} = 1.3\).

Determine product yield coefficient (\(Y_{\mathrm{CO_2}/S}\))

The product yield coefficient for carbon dioxide (\(\mathrm{CO}_2\)) is defined as the amount of carbon dioxide produced per unit of substrate consumed.From the equation, the coefficient for carbon dioxide (\(\mathrm{CO}_2\)) is 1.54 and the coefficient for glucose (\(S\)) is 1.Thus, \(Y_{\mathrm{CO_2}/S} = \frac{1.54}{1} = 1.54\).

Determine product yield coefficient (\(Y_{\mathrm{C_3H_8O_3}/S}\))

The product yield coefficient for glycerol (\(\mathrm{C}_3\mathrm{H}_8\mathrm{O}_3\)) is defined as the amount of glycerol produced per unit of substrate consumed.From the equation, the coefficient for glycerol is 0.43 and the coefficient for glucose (\(S\)) is 1.Thus, \(Y_{\mathrm{C_3H_8O_3}/S} = \frac{0.43}{1} = 0.43\).

Balance nitrogen to determine coefficient \(\beta\)

To determine the coefficient \(\beta\), balance the nitrogen atoms in the equation. The nitrogen in ammonia (\(\mathrm{NH}_3\)) should equal the nitrogen in the biomass.From the equation, biomass is represented as \(0.59 \mathrm{CH}_{1.74} \mathrm{N}_{0.2} \mathrm{O}_{0.45}\). The amount of nitrogen in biomass is \(0.59 \times 0.2 = 0.118\).Ammonia (\(\mathrm{NH}_3\)) has 1 nitrogen atom per molecule. Therefore, for nitrogen balance:\(\beta = 0.118\)

Key concepts

Anaerobic Fermentation

Anaerobic fermentation is a biological process in which microorganisms convert carbohydrates into other compounds without the use of oxygen. This process commonly produces gases like carbon dioxide and compounds such as ethanol and organic acids. It is a staple process in industries like brewing, waste treatment, and biofuel production. Unlike aerobic fermentation which requires oxygen, anaerobic fermentation is carried out in oxygen-free environments. The yeast species Saccharomyces cerevisiae is a common organism used in anaerobic fermentation due to its efficiency. It can convert glucose into ethanol through glycolysis followed by alcoholic fermentation. During this process, glucose is broken down to form pyruvate, which is then converted into ethanol and carbon dioxide.

Product Yield Coefficient

The product yield coefficient is a measure used to describe the efficiency of the conversion of substrate into a particular product in a biochemical reaction. It is defined as the amount of product formed per unit of substrate consumed. This is especially useful in industrial processes where optimizing product yield is crucial.
For example, in the anaerobic fermentation of glucose by Saccharomyces cerevisiae, we calculate the product yield coefficients for various products like ethanol, carbon dioxide, and glycerol. From the chemical equation of the reaction, we can derive these coefficients by dividing the amount of product formed by the amount of substrate consumed. For ethanol (EtOH), carbon dioxide (CO2), and glycerol (C3H8O3), these coefficients help to understand the efficiency and balance of the fermentation process:

• Ethyl alcohol yield coefficient is calculated as: \(Y_{EtOH/S} = \frac{1.3}{1} = 1.3\).
• Carbon dioxide yield coefficient is: \(Y_{CO_2/S} = \frac{1.54}{1} = 1.54\).
• Glycerol yield coefficient is: \(Y_{C_3H_8O_3/S} = \frac{0.43}{1} = 0.43\).

These yield coefficients offer detailed insights into how effectively the substrate (glucose) is being converted into the products during the fermentation process.

Stoichiometry

Stoichiometry is the calculation of reactants and products in chemical reactions. It is based on the principle of conservation of mass where the total mass of reactants is equal to the total mass of products. This concept is crucial in understanding and balancing chemical equations to ensure that atoms are conserved during the reaction.
In the context of biochemical reactions like the anaerobic fermentation process of Saccharomyces cerevisiae, stoichiometry helps to determine the coefficients for each component in the reaction equation. For example, using the given equation:
\begin{aligned}\text{C}_{6}\text{H}_{12}\text{O}_{6} + \beta \text{NH}_{3} \rightarrow & 0.59 \text{CH}_{1.74} \text{N}_{0.2} \text{O}_{0.45} \text{ (biomass)} + 0.43 \text{C}_{3}\text{H}_{8}\text{O}_{3} + 1.54 \text{CO}_{2} + 1.3 \text{C}_{2}\text{H}_{5}\text{OH} + 0.036 \text{H}_{2} \text{O}ewline\begin{aligned}by balancing the elements (especially nitrogen in this case), we determine the coefficient \(\beta\). Knowing that \(\beta\) is the amount of ammonia needed to balance the nitrogen atoms, we found it as \(0.118\):
\begin{aligned}\beta = 0.59 \times 0.2 = 0.118ewlineewline Armed with stoichiometric calculations, chemical engineers can optimize manufacturing and biological processes for maximum efficiency.