Problem 60 - Chemistry Matter and Change

Question

Ethanol \(\left(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\right),\) also known as grain alcohol, can be made from the fermentation of sugar \(\left(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}\right) .\)The unbalanced chemical equation for the reaction is shown below. \begin{equation} -\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6} \rightarrow_{-} \mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}+_{-} \mathrm{CO}_{2} \end{equation} \begin{equation} \begin{array}{l}{\text { Balance the chemical equation and determine the mass }} \\ {\text { of } \mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH} \text { produced from } 750 \mathrm{g} \text { of } \mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6} \text { . }}\end{array} \end{equation}

Step-by-Step Solution

Answer

The mass of \( \mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH} \) produced is approximately 383.4 g.

Step 1: Identify the Chemical Equation

The unbalanced reaction provided is: \( \mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6} \rightarrow \mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH} + \mathrm{CO}_{2} \). This needs to be balanced to conserve mass and atoms.

Step 2: Count the Atoms

In the unbalanced equation, count the atoms of each element on both sides of the equation. Initially: \( 6\, \text{C}, \ 12\, \text{H}, \ 6\, \text{O} \) on the left, \( 2\, \text{C}, \ 6\, \text{H}, \ 3\, \text{O} \) from ethanol and \( 1\, \text{C}, \ 2\, \text{O} \) from \( \mathrm{CO}_{2} \) on the right.

Step 3: Balancing Carbon Atoms

To balance carbon atoms, place a coefficient of 2 in front of \( \mathrm{CO}_{2} \) to have 2 carbon atoms in \( \mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH} \) and 2 in \( 2 \mathrm{CO}_{2} \), resulting in 4 carbon atoms on the right-hand side.

Step 4: Balancing Hydrogen Atoms

Originally, there are 12 hydrogen atoms in \( \mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6} \). Since \( \mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH} \) has 6 hydrogen atoms, set a coefficient of 2 in front of \( \mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH} \) so that there are 12 hydrogen atoms on the right.

Step 5: Balancing Oxygen Atoms

Now check the oxygen atoms. There are 6 oxygen atoms in \( \mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6} \), and with the current setup, 2 \( \mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH} \) introduces 2 oxygen atoms and 2 \( \mathrm{CO}_{2} \) introduces 4 more, making a total of 6 oxygen atoms, thus balancing the equation: \( \mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6} \rightarrow 2 \mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH} + 2 \mathrm{CO}_{2} \).

Step 6: Calculate Ethanol Produced

First, find the molar mass of \( \mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6} \) which is 180 g/mol. Then, the molar mass of ethanol \( \mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH} \) is 46 g/mol. The balanced equation shows that 1 mole of glucose produces 2 moles of ethanol.

Step 7: Convert Mass to Moles

Convert 750 g of \( \mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6} \) to moles: \( \frac{750\, \text{g}}{180\, \text{g/mol}} = 4.167 \times 10 = 4.167 \) mol.

Step 8: Calculate Moles of Ethanol

Since 1 mole of \( \mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6} \) produces 2 moles of \( \mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH} \), the number of moles of ethanol is \( 2 \cdot 4.167 = 8.334 \) mol.

Step 9: Convert Moles of Ethanol to Mass

The mass of ethanol produced is \( 8.334 \text{ moles} \times 46 \text{ g/mol} = 383.364 \text{ g} \).

Key Concepts

Balancing Chemical Equations

Chemistry involves many chemical reactions, and balancing chemical equations is a crucial skill for understanding these reactions. Every chemical reaction must adhere to the law of conservation of mass, meaning the number of atoms for each element must be equal on both sides of the equation. Balancing a chemical equation ensures that the same amount of mass is present before and after the reaction.

To successfully balance a chemical equation, start by identifying the chemical species involved and writing the unbalanced equation. Count the number of atoms for each element on both sides of the reaction. If they are not equal, add coefficients (whole numbers placed in front of chemical formulas) to balance the atoms.

In this exercise, the reaction of glucose converting to ethanol and carbon dioxide is provided. Initial counting showed that the carbon, hydrogen, and oxygen atoms were unbalanced. By placing coefficients of 2 in front of ethyl alcohol ( C_2H_5OH ) and carbon dioxide ( CO_2 ), we balanced the equation, which is crucial for accurately predicting product yields and interpreting chemical behavior.

Mass Calculation in Chemistry

Once a chemical equation is balanced, we can perform mass calculations to determine the amount of reactants or products. Mass calculations allow chemists to predict the quantities of substances consumed and produced in chemical reactions.

In this scenario, the goal was to calculate the mass of ethanol produced from a given mass of glucose. Firstly, calculate the molar mass for the substances. Glucose has a molar mass of 180 g/mol, and ethanol has a molar mass of 46 g/mol. Knowing that one mole of glucose produces two moles of ethanol helps in understanding the reaction's stoichiometry where molar ratios between substances are used.

To find the mass of ethanol:

• Convert the mass of glucose into moles by dividing the given mass by glucose's molar mass.
• Multiply the moles of glucose by the stoichiometric coefficient of ethanol (2 in this case) to find moles of ethanol produced.
• Lastly, multiply the moles of ethanol by its molar mass to find the mass of ethanol produced.

Through these calculations, a balanced equation provides a direct connection between reactants and products, crucial for quantitative chemical analysis.

Ethanol Production

Ethanol, commonly known as alcohol, is used extensively in industries and everyday products, including beverages, fuels, and sanitizers. It can be produced biologically through the fermentation process where microorganisms break down sugar into ethanol and carbon dioxide in anaerobic conditions.

In this exercise, glucose ( C_6H_{12}O_6 ) is fermented. It undergoes a biochemical transformation where yeast enzymes convert glucose to ethanol ( C_2H_5OH ) and carbon dioxide ( CO_2 ). The overall reaction involves breaking down a six-carbon sugar into smaller molecules.

Understanding ethanol production involves both the chemical and biological processes. Fermentation is optimized by controlling factors such as temperature, pH, and nutrient levels to maximize ethanol yield. For chemists and industries, knowing how to balance these processes and accurately calculate product outputs is crucial for efficient production.