Problem 63 - Physics

Question

Consider a mixture of three different gases: \(1.20 \mathrm{~g}\) of argon (molecular mass \(=39.948 \mathrm{~g} / \mathrm{mol}), 2.60 \mathrm{~g}\) of neon \((\) molecular mass \(=20.180 \mathrm{~g} / \mathrm{mol})\), and \(3.20 \mathrm{~g}\) of helium (molecular mass \(=4.0026 \mathrm{~g} / \mathrm{mol})\). (a) Explain how to calculate the number of moles of each species in terms of its mass and its molecular mass. (b) Of the total number of molecules in the mixture, each component is a certain percentage. Explain how to calculate that percentage. (c) Without doing any detailed calculations, determine which component has the greatest percentage. Give your reasoning. (d) Which component has the smallest percentage? Account for your answer. For the mixture described in the Concept Questions, determine the percentage of the total number of molecules that corresponds to each of the components. Verify that

Step-by-Step Solution

Answer

Helium has the greatest percentage; it has the largest number of moles. Argon has the smallest percentage; it has the fewest moles.

Step 1: Calculate Moles of Each Gas

To find the number of moles for each gas, we use the formula \( n = \frac{m}{M} \), where \( n \) is the number of moles, \( m \) is the mass, and \( M \) is the molecular mass. For Argon: \[ n_{Ar} = \frac{1.20 \text{ g}}{39.948 \text{ g/mol}} = 0.0301 \text{ mol} \]For Neon: \[ n_{Ne} = \frac{2.60 \text{ g}}{20.180 \text{ g/mol}} = 0.1288 \text{ mol} \]For Helium: \[ n_{He} = \frac{3.20 \text{ g}}{4.0026 \text{ g/mol}} = 0.7999 \text{ mol} \]

Step 2: Find Total Number of Moles

Add up the moles of all the gases to find the total number of moles in the mixture: \[ n_{total} = n_{Ar} + n_{Ne} + n_{He} \]\[ n_{total} = 0.0301 + 0.1288 + 0.7999 = 0.9588 \text{ mol} \]

Step 3: Calculate Mole Percentage of Each Gas

The mole percentage can be calculated using the formula: \[ \text{Percentage} = \left( \frac{n_{component}}{n_{total}} \right) \times 100 \%\]For Argon: \[ \text{Percentage of Argon} = \left( \frac{0.0301}{0.9588} \right) \times 100 \% \approx 3.14\% \]For Neon: \[ \text{Percentage of Neon} = \left( \frac{0.1288}{0.9588} \right) \times 100 \% \approx 13.43\% \]For Helium: \[ \text{Percentage of Helium} = \left( \frac{0.7999}{0.9588} \right) \times 100 \% \approx 83.43\% \]

Step 4: Determine Gas Component with Greatest Percentage

Without performing calculations, helium will have the greatest percentage because it has the highest number of moles. The number of moles is highest for helium compared to argon and neon from initial conversion calculations.

Step 5: Identify Gas Component with Smallest Percentage

Argon has the smallest percentage in the mixture because it has the lowest number of moles calculated.

Key Concepts

Molecular Mass

Molecular mass is a fundamental concept in chemistry that refers to the mass of a given molecule. Understanding how to calculate this mass is crucial for further calculations and assessments. It is typically expressed in atomic mass units (amu) or grams per mole (g/mol). To find the molecular mass, you need to sum up the average atomic masses of each element present in the molecule. For instance, in the gas mixture from our exercise, we have argon, neon, and helium, each with known molecular masses:

• Argon: 39.948 g/mol
• Neon: 20.180 g/mol
• Helium: 4.0026 g/mol

Using molecular mass, we can convert the mass of a substance to the number of moles, by using the formula: \( = \frac{m}{M} \), where \( n \) stands for the number of moles, \( m \) is the mass, and \( M \) represents the molecular mass.

Gas Mixture

A gas mixture, in its simplest form, is a combination of different gases. These gases may have varying properties like molecular mass, density, and reactivity. In our example, the mixture is composed of argon, neon, and helium. To understand or optimize the behavior of a gas mixture, we break it down into easier-to-handle tasks by calculating specific properties, such as the number of moles of each gas. Each component contributes to the overall properties, like pressure and volume, of the mixture. This contribution can be quantified through calculations based on mole fractions and percentages. It's key to recognize in gas mixtures that while the components are mixed, each gas's behavior conforms to its ideal gas law separately, following the concept of partial pressures in Dalton's Law.

Percentage Composition

Percentage composition in a chemical context refers to the proportion of a component within a mixture or compound expressed as a percentage. In the context of our gas mixture, we are interested in how much each gas contributes to the total number of moles in the mixture. To find this, we calculate the mole percentage. The formula is:\[\text{Percentage} = \left( \frac{n_{component}}{n_{total}} \right) \times 100 \%\]For each gas, the mole percentage gives us insight into the composition:

• Argon, with only 3.14%
• Neon, with about 13.43%
• Helium, which dominates with around 83.43%

By understanding these percentages, we know which gas predominates, which helps in predicting the mixture's behavior under different conditions.

Avogadro's Number

Avogadro's number, approximately \(6.022 \times 10^{23}\), is a constant that tells us the number of units (atoms, molecules, ions, etc.) in one mole of a substance. This fundamental constant is instrumental in conversions and calculations involving moles, linking the microscopic scale to the macroscopic. When applied to our gas mixture, knowing Avogadro's number allows us to move from the abstract concept of moles to tangible values, giving a real-world sense of how many molecules are present in each quantity of gas. It provides the bridge needed to calculate the total number of individual molecules in our gas mix. This conversion from number of moles through Avogadro's number is crucial for many practical applications in chemistry and beyond.