Question

Yeast and other organisms can convert glucose \(\left(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}\right)\) to ethanol \(\left(\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{OH}\right)\) by a process called alchoholic fermentation. The net reaction is \\[ \mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}(s) \rightarrow 2 \mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}(l)+2 \mathrm{CO}_{2}(g) \\] Calculate the mass of glucose required to produce 2.25 L of \(\mathrm{CO}_{2}\) measured at \(P=1.00\) atm and \(T=295 \mathrm{K}\).

Short answer

Approximately 8.85 grams of glucose is required to produce 2.25 L of CO₂ gas at 1.00 atm and 295 K.

Step-by-step solution

Calculate moles of CO₂

First, we need to find the number of moles of CO₂ produced, using the ideal gas law equation: \[PV = nRT\] Where: - P = pressure (1.00 atm) - V = volume (2.25 L) - n = moles of CO₂ - R = ideal gas constant (0.0821 L·atm/mol·K) - T = temperature (295 K) We can solve for the number of moles (n) by rearranging the equation: \[n = \frac{PV}{RT}\] Now plug in the given values: \[n = \frac{(1.00 \ \text{atm})(2.25 \ \text{L})}{(0.0821 \ \text{L·atm/mol·K})(295 \ \text{K})}\] Calculate: \[n = 0.0982 \ \text{mol}\] So, there are 0.0982 moles of CO₂ produced.

Determine moles of glucose

Next, we use the stoichiometry of the balanced chemical equation to determine the moles of glucose needed to produce 0.0982 moles of CO₂. The balanced equation is: \[\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}(s) \rightarrow 2 \mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}(l)+2 \mathrm{CO}_{2}(g)\] From the balanced equation, we see that one mole of glucose produces two moles of CO₂. Therefore: Moles of glucose = \(\frac{\text{moles of CO₂}}{2}\) Substitute the value of moles of CO₂: Moles of glucose = \(\frac{0.0982 \ \text{mol}}{2}\) Calculate: Moles of glucose = 0.0491 mol

Calculate the mass of glucose

Finally, we need to calculate the mass of glucose. The molar mass of glucose is: \[C_{6}H_{12}O_{6} \Rightarrow (6 \times 12.01) + (12 \times 1.01) + (6 \times 16.00) = 180.18 \ \text{g/mol}\] Now multiply the moles of glucose by the molar mass to find the mass of glucose: Mass of glucose = Moles of glucose × Molar mass of glucose Substitute the values: Mass of glucose = \(0.0491 \ \text{mol} \times 180.18 \ \text{g/mol}\) Calculate: \[ Mass of glucose = 8.8454 \ \text{g}\] Approximately 8.85 grams of glucose is required to produce 2.25 L of CO₂ gas at 1.00 atm and 295 K.

Key concepts

Ideal Gas Law

The ideal gas law is a fundamental equation that describes the behavior of gases. This equation relates the pressure (P), volume (V), temperature (T), and number of moles (n) of a gas. The formula is expressed as:\[PV = nRT\]Here, \(R\) stands for the ideal gas constant, and its value is often \(0.0821\ L\cdot atm/mol\cdot K\).
In problems related to gases, the ideal gas law can be used to calculate any one of the variables if the others are known.

• If you're given the pressure, volume, and temperature of a gas, you can find the number of moles by rearranging the equation to \(n = \frac{PV}{RT}\).
• This calculation helps find the amount of a gas involved in a chemical reaction, which is crucial for further stoichiometric analysis.

Remember that conditions such as temperature must be in Kelvin for the equation to apply accurately.

Stoichiometry

Stoichiometry is the calculation of reactants and products in chemical reactions. It is based on the balanced equation of the reaction, which provides a quantitative relationship between the substances involved. For the given fermentation reaction:\[\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}(s) \rightarrow 2 \mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}(l)+2 \mathrm{CO}_{2}(g)\]

• Stoichiometry tells us that one mole of glucose produces two moles of carbon dioxide (CO₂).
• It involves using the coefficients from the balanced equation to convert between moles of different substances.
• In this exercise, we calculated that 0.0982 moles of CO₂ corresponds to half that number of moles in glucose, using the ratio from the balanced equation.

This concept is essential for determining how much of a reactant is necessary to produce a desired amount of product.

Molar Mass

Molar mass is the mass of one mole of a substance, usually measured in grams per mole (g/mol). It links the mass of a substance to the number of atoms or molecules it contains. Understanding molar mass is critical when converting between the mass of a compound and moles, which is often needed in stoichiometric calculations.

• To find the molar mass of glucose (\( \mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6} \)), we sum the masses of all atoms in the molecule. This includes carbon, hydrogen, and oxygen, resulting in a molar mass of 180.18 g/mol.
• This molar mass allows us to convert from moles of glucose, which we calculated earlier, to the mass in grams.
• Such conversions are crucial for practical applications, such as determining the amounts needed for a reaction.

By multiplying the number of moles by the molar mass, we deduced the mass of glucose required in the fermentation process.

Balanced Chemical Equation

A balanced chemical equation accurately represents a chemical reaction by ensuring that the number of each type of atom is the same on both sides of the equation. Balancing equations is a fundamental skill in chemistry because it upholds the law of conservation of mass.

• In the given reaction, balancing is achieved by ensuring that six carbon, twelve hydrogen, and six oxygen atoms from glucose result in the correct number of atoms in the ethanol and carbon dioxide produced.
• The balanced equation is crucial for stoichiometric calculations, as it shows the molar relationships between reactants and products.
• Understanding the equation allows for the calculation of required reactants or resulting products for any given tray in a chemical reaction.

This principle forms the basis for determining the theoretical yield of reactions and is an essential concept for solving problems like those found in the alcoholic fermentation of glucose.