Question

Analysis of the exhaust composition of the supersonic transport for one hour of flight yielded the following information: \begin{tabular}{|l|l|l|} \hline Compound & Molecular Weight (g/mole) & \(\underline{\text { Mass }}(\mathrm{g})\) \\ \hline \(\mathrm{H}_{2} \mathrm{O}\) & 18 & \(8.0 \times 10^{7}\) \\ \hline \(\mathrm{CO}_{2}\) & 44 & \(6.6 \times 10^{7}\) \\ \hline \(\mathrm{CO}\) & 28 & \(3.6 \times 10^{6}\) \\ \hline \(\mathrm{NO}\) & 30 & \(3.6 \times 10^{6}\) \\ \hline \end{tabular} Determine the mole fraction of \(\mathrm{CO}_{2}\) in this mixture.

Short answer

The mole fraction of CO2 in the exhaust mixture is calculated as follows: Step 1: Moles of H2O = \(\frac{(8.0\times{10}^7\,g)}{18\,g/mole}\), Moles of CO2 = \(\frac{(6.6\times{10}^7\,g)}{44\,g/mole}\), Moles of CO = \(\frac{(3.6\times{10}^6\,g)}{28\,g/mole}\), and Moles of NO = \(\frac{(3.6\times{10}^6\,g)}{30\,g/mole}\). Step 2: Total moles = Moles of H2O + Moles of CO2 + Moles of CO + Moles of NO. Step 3: Mole fraction of CO2 = Moles of CO2 / Total moles of the mixture.

Step-by-step solution

Calculate the moles of each compound

To calculate the moles of each compound, we will use the following formula: Moles = (Mass of the compound)/ (Molecular weight of the compound) Let's calculate the moles of each compound in the mixture: Moles of H2O: \(\frac{(8.0\times{10}^7\,g)}{18\,g/mole}\) Moles of CO2: \(\frac{(6.6\times{10}^7\,g)}{44\,g/mole}\) Moles of CO: \(\frac{(3.6\times{10}^6\,g)}{28\,g/mole}\) Moles of NO: \(\frac{(3.6\times{10}^6\,g)}{30\,g/mole}\)

Calculate the total moles of the mixture

Add the moles of all the compounds to find the total moles in the mixture: Total moles = Moles of H2O + Moles of CO2 + Moles of CO + Moles of NO

Compute the mole fraction of CO2

To calculate the mole fraction of CO2, divide the moles of CO2 by the total moles of the mixture: Mole fraction of CO2 = Moles of CO2 / Total moles of the mixture Calculate the mole fraction of CO2 using the respective moles calculated in Step 1 and Step 2.

Key concepts

Molecular Weight

Understanding molecular weight is essential for those delving into chemistry. Molecular weight, often referred to as molecular mass or molar mass, is the weight of one mole of a given substance. It is usually expressed in units of grams per mole (g/mol) and is calculated by summing the atomic weights of the atoms that make up the molecule, as found on the periodic table. This concept is critical when converting between mass and moles of a substance, as seen in our exercise with compounds like H2O and CO2, where molecular weights are given as 18 g/mol and 44 g/mol, respectively.

Grasping the concept of molecular weight is the stepping stone to understanding more complex chemical calculations because you can't delve into stoichiometry or determining mixture composition without it. Always ensure to refer to a reliable periodic table for accurate atomic masses when calculating molecular weights for different compounds.

Moles Calculation

The moles calculation is an integral part of understanding chemical reactions and compositions. A mole is a unit of measurement used in chemistry to express amounts of a chemical substance. One mole contains exactly 6.02214076×10²³ (the Avogadro constant) of some chemical unit, be it atoms, molecules, ions, or others. The calculation for moles is a simple ratio: Moles = Mass of the Compound / Molecular Weight of the Compound.

In the provided exercise, to calculate the moles of each gas in the exhaust mixture, the mass of each substance is divided by their respective molecular weights. For example, the moles of H2O are calculated by taking its given mass and dividing it by its molecular weight (18 g/mol). This forms the basis of stoichiometry and allows for the further analysis of the mixture, such as determining the mole fraction.

Stoichiometry

Stoichiometry can be seen as the 'recipe' of chemistry. It's the calculation of the quantities of reactants and products involved in a chemical reaction. This discipline of chemistry centers around the law of conservation of mass where matter is neither created nor destroyed, meaning the mass of the reactants equals the mass of the products. In practical terms, stoichiometry allows chemists to predict the amount of product that can be produced from a given amount of reactants, or conversely, the amount of reactants needed to create a desired quantity of product.

In stoichiometry, performing calculations based on mole ratios from a balanced chemical equation is standard practice. In the context of our exhaust mixture problem, understanding stoichiometry is helpful in determining the total moles of the mixture and in calculating the mole fraction.

Gas Analysis

Gas analysis is a crucial concept when dealing with mixtures involving gases. It entails determining the composition of a gas mixture by evaluating the amount of each component. This information is vital in many fields, including environmental science, chemical engineering, and materials science. Gas analysis often involves identifying the molecular weight and the mass of each component, and converting these masses to moles, which can then be used to determine the mole fraction.

In the exercise, gas analysis is achieved through quantifying the mass of each gas present in the exhaust and calculating its respective moles. It allows us to further dissect the mixture's composition and calculate each gas's mole fraction, granting insight into the exhaust's overall makeup over a certain period, like the one-hour flight considered in the problem.

Mixture Composition

Mixture composition is all about understanding the proportions of the different components in a mixture. In chemistry, this is often articulated in terms of mole fractions, which provide a way to express the concentration of a particular component relative to the entire mixture. The mole fraction is dimensionless and is calculated by dividing the number of moles of one particular component by the total number of moles in the mixture. It's like finding out your slice of the pie in a total pie chart.

To find the mole fraction of CO2 in the given exercise, you first calculate the moles of CO2 and the total moles of all gases combined. Then, by dividing these two values, you determine the mole fraction of CO2. Mole fraction is a paramount concept for the characterization of mixture composition and can be applied to understand different mixtures in both gaseous and liquid states.