Question

A sample of propane \(\left(\mathrm{C}_{3} \mathrm{H}_{8}\right)\) is placed in a closed ves sel together with an amount of \(\mathrm{O}_{2}\) that is 2.15 times the amount needed to completely oxidize the propane to \(\mathrm{CO}_{2}\) and \(\mathrm{H}_{2} \mathrm{O}\) at constant temperature. Calculate the mole fraction of each component in the resulting mixture after oxidation, assuming that the \(\mathrm{H}_{2} \mathrm{O}\) is present as a gas.

Short answer

The mole fractions of each component in the resulting mixture after oxidation are: Propane (C3H8): 0; Oxygen (O2): 0.20; Carbon dioxide (CO2): 0.52; Water (H2O): 0.28.

Step-by-step solution

Write a balanced chemical equation

The balanced chemical equation for the complete oxidation of propane is: \(C_3H_8 + 5O_2 \rightarrow 3CO_2 + 4H_2O\)

Determine the limiting reactant

Since O2 is present in 2.15 times the necessary amount for complete oxidation, propane (C3H8) will be the limiting reactant. Once all the propane has reacted, there will still be some oxygen left.

Calculate the moles of each substance after the reaction

Let's assume we have x moles of propane (C3H8) initially. According to the balanced equation, we have 5x moles of O2 in the vessel, 3x moles of CO2, and 4x moles of H2O after the reaction. From the given condition, we have the following equation: \(5x = 2.15 \times 5x\) Solve for x: \((2.15 - 1)5x = 5x\) Therefore, x = moles of C3H8. Now we need to determine the moles of the other components in the mixture after the reaction. The moles of each component are: - Propane: 0 moles (as all propane has reacted) - Oxygen: (2.15 - 1)5x moles (remaining after reaction) - Carbon dioxide: 3x moles (formed after the reaction) - Water: 4x moles (formed after the reaction)

Calculate the mole fraction of each component after the reaction

The total moles in the final mixture are: \(0 + (2.15 - 1)5x + 3x + 4x = (2.15 - 1)5x + 7x = (1.15)5x + 7x = 5.75x\) To determine the mole fraction of each component in the mixture, we divide the moles of each component by the total moles in the mixture: - Mole fraction of propane (C3H8): \(\frac{0}{5.75x} = 0\) - Mole fraction of oxygen (O2): \(\frac{(2.15 - 1)5x}{5.75x} = \frac{1.15}{5.75} = 0.20\) - Mole fraction of carbon dioxide (CO2): \(\frac{3x}{5.75x} = \frac{3}{5.75} = 0.52\) - Mole fraction of water (H2O): \(\frac{4x}{5.75x} = \frac{4}{5.75} = 0.28\) So, the mole fractions of each component in the resulting mixture after oxidation are: - Propane (C3H8): 0 - Oxygen (O2): 0.20 - Carbon dioxide (CO2): 0.52 - Water (H2O): 0.28

Key concepts

Mole Fraction Calculation

Understanding how to calculate mole fractions is critical in chemistry, especially when dealing with mixtures and reactions. The mole fraction is a way of expressing the concentration of a component in a mixture by using moles, which are a standard unit for the amount of substance in chemistry. It's defined as the number of moles of a particular component divided by the total number of moles in the mixture.

To calculate mole fractions, you should follow these steps: Identify the number of moles of each component in the mixture. This can be done either by measuring or by using stoichiometry if a chemical reaction has occurred. Next, add up all the moles to find the total moles in the mixture. Divide the moles of the component of interest by the total moles to find its mole fraction.

For example, if a vessel starts with x moles of a gas and ends up with a mixture of several gases after a reaction, the mole fraction of any one gas is the number of moles of that gas divided by the total moles in the mixture after the reaction. This approach is crucial for predicting the composition of the final gas mixture.

Limiting Reactant Determination

The concept of a limiting reactant is fundamental in stoichiometry. It helps to determine which reactant will be entirely used up first in a chemical reaction, thereby stopping or limiting the reaction. To identify the limiting reactant, you must first start with a balanced chemical equation. This gives you the proportions in which reactants combine to form products.

Once you have the balanced equation, calculate the number of moles of each reactant that you have. Compare these quantities with the ratios from the balanced equation. The reactant that has the smallest stoichiometric amount relative to what is needed is the limiting reactant. In our example, we determined that propane (C3H8) was the limiting reactant because oxygen (O2) was present in excess (2.15 times the amount needed).

Identifying the limiting reactant is crucial for calculating the theoretical yield of a reaction and for understanding how much of each product will form.

Balanced Chemical Equation

A balanced chemical equation accurately reflects the law of conservation of mass, meaning the number of atoms of each element is the same on both sides of the reaction. To balance a chemical equation, you must ensure that the quantity of each type of atom on the reactant side is equal to that on the product side.

In practice, start with the most complex molecule or the one that contains the most elements. Adjust coefficients—the numbers in front of the chemical formulas—until you have the same number of atoms for each element on both sides. For our example involving propane (C3H8) and oxygen (O2), the balanced equation is: \(C_3H_8 + 5O_2 \rightarrow 3CO_2 + 4H_2O\).

This balanced equation sets the stage for determining the amount of products produced and the amount of reactants needed, which students use for mole fraction calculations and limiting reactant determinations.