Problem 156 - Chemistry The Molecular Science

Question

Ethanol, \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH},\) is a gasoline additive that can be produced by fermentation of glucose. $$\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6} \longrightarrow 2 \mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}+2 \mathrm{CO}_{2}$$ (a) Calculate the mass (g) of ethanol produced by the fermentation of 1.000 lb glucose. (b) Gasohol is a mixture of \(10.00 \mathrm{~mL}\) ethanol per \(90.00 \mathrm{~mL}\) gasoline. Calculate the mass (in g) of glucose required to produce the ethanol in 1.00 gal gasohol. Density of ethanol \(=0.785 \mathrm{~g} / \mathrm{mL}\) (c) By 2022, the U. S. Energy Independence and Security Act calls for annual production of \(3.6 \times 10^{10}\) gal of ethanol, no more than \(40 \%\) of it produced by fermentation of corn. Fermentation of 1 ton \(\left(2.2 \times 10^{3} \mathrm{lb}\right)\) of corn yields approximately 106 gal of ethanol. The average corn yield in the United States is about \(2.1 \times 10^{5} \mathrm{lb}\) per \(1.0 \times 10^{5} \mathrm{~m}^{2} .\) Calculate the acreage (in \(\mathrm{m}^{2}\) ) required to raise corn solely for ethanol production in 2022 in the United States.

Step-by-Step Solution

Answer

231.8 g of ethanol from 1 lb of glucose, 581.3 g of glucose for 1 gal gasohol, and 8.81 x 10¹¹ m² of land needed for 2022 U.S. ethanol production.

Step 1: Calculate moles of glucose

First, we need to convert 1 lb of glucose to grams. Since 1 lb = 453.592 g, 1.000 lb of glucose is equivalent to 453.592 g.Next, calculate the molar mass of glucose (\(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}\)).- Carbon (C): 6 atoms \(\times 12.011\, \mathrm{g/mol} = 72.066 \, \mathrm{g/mol}\)- Hydrogen (H): 12 atoms \(\times 1.008\, \mathrm{g/mol} = 12.096 \, \mathrm{g/mol}\)- Oxygen (O): 6 atoms \(\times 16.00\, \mathrm{g/mol} = 96.00 \, \mathrm{g/mol}\)Total molar mass of glucose: \(180.162\, \mathrm{g/mol}\).Calculate the moles of glucose:\[\text{Moles of glucose} = \frac{453.592 \, \mathrm{g}}{180.162 \, \mathrm{g/mol}} \approx 2.517 \, \text{mol}\]

Step 2: Calculate mass of ethanol produced

From the reaction, 1 mole of glucose produces 2 moles of ethanol. Therefore, 2.517 moles of glucose produce:\[2.517 \, \text{mol of glucose} \times 2 \, \frac{\text{mol ethanol}}{\text{mol glucose}} = 5.034 \, \text{mol of ethanol}\]Calculate the molar mass of ethanol (\(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\)):- Carbon (C): 2 atoms \(\times 12.011 \, \mathrm{g/mol} = 24.022 \, \mathrm{g/mol}\)- Hydrogen (H): 6 atoms \(\times 1.008 \, \mathrm{g/mol} = 6.048 \, \mathrm{g/mol}\)- Oxygen (O): 1 atom \(\times 16.00 \, \mathrm{g/mol} = 16.00 \, \mathrm{g/mol}\)Total molar mass of ethanol: \(46.068 \, \mathrm{g/mol}\).Calculate the mass of ethanol produced:\[\text{Mass of ethanol} = 5.034 \, \text{mol} \times 46.068 \, \mathrm{g/mol} \approx 231.8 \, \mathrm{g}\]

Step 3: Calculate volume and mass of ethanol in gasohol

1 gal is 3.78541 L, and since gasohol is 10% ethanol by volume, the volume of ethanol in 1.00 gal of gasohol:\[0.10 \times 3.78541 \, \mathrm{L} = 0.378541 \, \mathrm{L}\]Convert to mL:\[0.378541 \, \mathrm{L} \times 1000 \, \mathrm{mL/L} = 378.541 \, \mathrm{mL}\]Using the density of ethanol (0.785 g/mL), calculate the mass:\[\text{Mass of ethanol} = 378.541 \, \mathrm{mL} \times 0.785 \, \mathrm{g/mL} \approx 297.2 \, \mathrm{g}\]

Step 4: Calculate glucose needed for ethanol in gasohol

Using the stoichiometry from earlier, where 2 moles of ethanol come from 1 mole of glucose:Calculate moles of ethanol:\[\text{Moles of ethanol} = \frac{297.2 \, \mathrm{g}}{46.068 \, \mathrm{g/mol}} \approx 6.453 \, \text{mol}\]Moles of glucose required:\[\frac{6.453 \, \text{mol ethanol}}{2} = 3.2265 \, \text{mol glucose}\]Convert moles of glucose to mass:\[\text{Mass of glucose} = 3.2265 \, \text{mol} \times 180.162 \, \mathrm{g/mol} \approx 581.3 \, \mathrm{g}\]

Step 5: Calculate moles of ethanol from annual production target

Calculate moles of ethanol based on the 2022 production target of \(3.6 \times 10^{10}\) gal with at most 40% from corn fermentation:First, calculate gallons of ethanol from corn:\[0.40 \times 3.6 \times 10^{10} \, \text{gal} = 1.44 \times 10^{10} \, \text{gal}\]Convert to liters (1 gal = 3.78541 L):\[1.44 \times 10^{10} \, \text{gal} \times 3.78541 \, \text{L/gal} = 5.4534 \times 10^{10} \, \text{L}\]Convert to mL:\[5.4534 \times 10^{10} \, \text{L} \times 1000 \, \text{mL/L} = 5.4534 \times 10^{13} \, \text{mL}\]Calculate the mass of ethanol (density = 0.785 g/mL):\[\text{Mass of ethanol} = 5.4534 \times 10^{13} \, \text{mL} \times 0.785 \, \text{g/mL} = 4.28 \times 10^{13} \, \text{g}\]

Step 6: Calculate glucose needed for annual ethanol production

Using the stoichiometry earlier, convert the mass of ethanol to moles:\[\text{Moles of ethanol} = \frac{4.28 \times 10^{13} \, \mathrm{g}}{46.068 \, \mathrm{g/mol}} \approx 9.29 \times 10^{11} \, \text{mol}\]Find the moles of glucose needed:\[\frac{9.29 \times 10^{11} \, \text{mol ethanol}}{2} = 4.645 \times 10^{11} \, \text{mol glucose}\]Convert the moles of glucose to mass:\[\text{Mass of glucose} = 4.645 \times 10^{11} \, \text{mol} \times 180.162 \, \mathrm{g/mol} = 8.37 \times 10^{13} \, \text{g}\]

Step 7: Calculate acreage required for corn production

Given that 1 ton (2.2 × 10³ lb) of corn yields 106 gal of ethanol, calculate lb of corn needed for the ethanol:Convert mass of glucose to lb:\[8.37 \times 10^{13} \, \text{g} \times \frac{1 \, \text{lb}}{453.592 \, \text{g}} = 1.85 \times 10^{11} \, \text{lb}\]Calculate tons of corn needed (2.2 × 10³ lb/ton):\[\frac{1.85 \times 10^{11} \, \text{lb}}{2.2 \times 10^{3} \, \text{lb/ton}} = 8.41 \times 10^{7} \, \text{ton}\]Given 106 gal of ethanol per ton of corn, calculate area of cornfields needed:Corn yield: 2.1 × 10⁵ lb focused per 1.0 × 10⁵ m². Calculate total corn yield needed:\[\text{Total corn lb required} = 8.41 \times 10^{7} \, \text{ton} \times 2.2 \times 10^{3} \, \text{lb/ton} = 1.85 \times 10^{11} \, \text{lb}\]Calculate field area:\[\frac{1.85 \times 10^{11} \, \text{lb}}{2.1 \times 10^{5} \, \text{lb/} \times 10^{5} \, \text{m²}} \equiv 8.81 \times 10^{8} \, \text{m²}\]

Step 8: Conclusion: Completion of calculations

Through the calculations, we determined the mass of ethanol from 1 lb glucose, required glucose for 1 gal of gasohol, and the amount of land needed for 2022 ethanol production requirement. Steps 1 through 7 guide through stoichiometry, conversions, and acreage evaluation.

Key Concepts

Fermentation of glucose

Fermentation of glucose is a natural process where microorganisms, such as yeast, convert carbohydrates like glucose into ethanol and carbon dioxide. This is a crucial step in ethanol production for use as a fuel additive or in beverages.

The chemical reaction for glucose fermentation can be represented as follows:

• Glucose (\[\mathrm{C}_6\mathrm{H}_{12}\mathrm{O}_6\]) breaks down into ethanol (\[2\,\mathrm{C}_2\mathrm{H}_5\mathrm{OH}\]) and carbon dioxide (\[2\,\mathrm{CO}_2\]).
• This transformation is facilitated by microorganisms, usually yeast, that act on the sugar molecules in the absence of oxygen.

The process is anaerobic and takes place in environments where oxygen levels are low, making it appealing for applications where energy is harvested in the form of ethanol. Because two molecules of ethanol are formed from one molecule of glucose, a simple stoichiometric calculation can be used to predict output based on the input of glucose.

Gasohol

Gasohol is an alternative fuel composed of 90% gasoline and 10% ethanol by volume. The addition of ethanol to gasoline improves combustion efficiency, reduces emissions, and utilizes renewable biomass resources.

Here’s how gasohol is made:

• For every gallon of gasohol, 0.10 gallons are ethanol.
• Ethanol is typically sourced from agricultural products like corn due to its high starch content.

Ethanol's lower energy content compared to gasoline is offset by the benefits of reduced carbon monoxide emissions and reliance on fossil fuels. The production of ethanol for gasohol has led to increased interest in sustainable agriculture and energy independence. This blend is commonly used in vehicles and holds a significant place in efforts to promote cleaner burning fuels.

Corn yield for ethanol

Corn is a primary crop used in the production of ethanol, particularly in places like the United States where it's both abundant and economically viable. The yield of corn directly impacts ethanol production capabilities.

Here's how corn is used to produce ethanol:

• A specified weight of corn is fermented to produce ethanol.
• For instance, one ton of corn can generate approximately 106 gallons of ethanol.

In 2022, the U.S. regulations stipulate that a significant portion of ethanol should come from corn. Therefore, understanding the corn yield, which is around 210,000 pounds per 100,000 square meters, is critical for planning and fulfilling production quotas.
Calculations of required acreage take into account the total ethanol output, percentage derived from corn, and the corn yield efficiency per land unit, resulting in a sizeable area dedicated to this activity.

Chemical stoichiometry

Chemical stoichiometry is a branch of chemistry that deals with the calculation of reactants and products in chemical reactions. It's essential in determining the proportions of substances required or produced, like glucose and ethanol in fermentation.

Here's a quick guide to understanding stoichiometry in the context of ethanol production:

• Begin with balanced chemical reactions, such as the fermentation equation where 1 mole of glucose produces 2 moles of ethanol.
• Using molar mass, convert masses of reactants to moles and back to grams for the final product.

This process allows for precise calculations of product yields and reactant requirements. By understanding the stoichiometric coefficients from balanced equations, one can easily determine the conversion rates and efficiencies of various industrial processes such as ethanol production.