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(i) A mixture of noble gases contains equal mole fractions of helium, argon, and xenon. What is the composition in terms of mass fractions? (ii) If the mixture contains equal mass fractions of \(\mathrm{He}, \mathrm{Ar}\), and \(\mathrm{Xe}\), what are the corresponding mole fractions?

Short Answer

Expert verified
(i) He: 2.3%, Ar: 22.8%, Xe: 74.9%. (ii) He: ~0.926, Ar: ~0.046, Xe: ~0.028.

Step by step solution

01

Understanding Mole Fractions

Mole fractions represent the ratio of moles of one component to the total moles in a mixture. Given equal mole fractions of He, Ar, and Xe, each element contributes \(\frac{1}{3}\) to the total moles since there are three different gases.
02

Calculating Mass Fractions from Mole Fractions

The mass fraction for an element is calculated using its molar mass and mole fraction. For each gas, calculate mass as \(m_i = X_i \times M_i\), where \(X_i = \frac{1}{3}\) and \(M_i\) is the molar mass: - Helium (\(M_{\text{He}} = 4.00 \text{g/mol}\)) - Argon (\(M_{\text{Ar}} = 39.95 \text{g/mol}\)) - Xenon (\(M_{\text{Xe}} = 131.29 \text{g/mol}\)).Calculate the weighted contributions and divide each by the total mass to find mass fractions.
03

Calculation Example

Calculate masses for unit moles: For helium (He): \(m_{\text{He}} = (\frac{1}{3}) \times 4.00 = 1.33 \text{g}\) For argon (Ar): \(m_{\text{Ar}} = (\frac{1}{3}) \times 39.95 = 13.32 \text{g}\)For xenon (Xe): \(m_{\text{Xe}} = (\frac{1}{3}) \times 131.29 = 43.76 \text{g}\)Total mass = \(1.33 + 13.32 + 43.76 = 58.41\text{ g}\).
04

Determining Mass Fractions

Using the masses determined:- Mass Fraction of He = \(\frac{1.33}{58.41} = 0.023\) or 2.3%- Mass Fraction of Ar = \(\frac{13.32}{58.41} = 0.228\) or 22.8%- Mass Fraction of Xe = \(\frac{43.76}{58.41} = 0.749\) or 74.9%\.
05

Understanding Equal Mass Fractions

With equal mass fractions for He, Ar, and Xe each being \(\frac{1}{3}\), we need to convert these to mole fractions. Set each mass fraction equal and solve for moles using the equations: \(X_{\text{He}}M_{\text{He}} = X_{\text{Ar}}M_{\text{Ar}} = X_{\text{Xe}}M_{\text{Xe}}\).
06

Calculating Mole Fractions from Mass Fractions

Given equal mass fractions:\(\frac{X_{\text{He}}}{4.00} = \frac{X_{\text{Ar}}}{39.95} = \frac{X_{\text{Xe}}}{131.29}\). Use algebra to express each \(X_i\), with total mole fraction \(\sum X_i = 1\).
07

Calculation Example for Mole Fractions

First solve for \(X_{\text{He}}\): \[ X_{\text{He}} = \frac{1}{\frac{1}{4} + \frac{1}{39.95} + \frac{1}{131.29}} \]Substituting values: \[ X_{\text{He}} \approx \frac{1}{0.25 + 0.025 + 0.0076} \approx 0.974 \]Repeat for Ar and Xe.
08

Final Mole Fractions

After similar calculations for Ar and Xe, the approximate mole fractions are:- Mole Fraction of He \( ≈ 0.926 \)- Mole Fraction of Ar \( ≈ 0.046 \)- Mole Fraction of Xe \( ≈ 0.028 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Noble Gases
Noble gases are a group of chemical elements that are very stable and unreactive. This stability arises from their complete outer electron shell, meaning they do not easily form chemical compounds with other elements. This group includes elements like helium (He), neon (Ne), argon (Ar), krypton (Kr), xenon (Xe), and radon (Rn). Because of their low reactivity, they are often found in the atmosphere in trace amounts.
Noble gases are colorless, odorless, and tasteless, making them ideal for use in various applications where reactivity could pose a problem. They are used in neon signs, light bulbs, and as coolant materials in some advanced technologies. Helium, being lighter than air, is popularly used in balloons while argon is often used in welding to protect heated metals from atmospheric gases.
Despite being largely non-reactive, xenon can form some compounds under certain extreme conditions, showcasing that while noble gases are stable, they can still participate in chemical reactions given the right circumstances.
Mass Fractions
Mass fractions represent how much of a total mass is made up by each component in a mixture. It is calculated by dividing the mass of an individual component by the total mass of all components combined. This gives a clearer understanding of the composition and concentration of substances in a mixture.
For example, if you know the individual masses of helium, argon, and xenon in a mixture, you can compute their mass fractions. Each fraction indicates the portion of the total mass that the specific gas contributes. The sum of all mass fractions in a mixture must equal 1, reflecting 100% of the sample.
Mass fractions are crucial for practical applications, such as formulating mixtures in industry and research. They allow chemists to quantify and control the presence of components, ensuring consistency and accuracy in chemical processes and product formulations.
Molar Mass
Molar mass is a key concept in chemistry, representing the mass of one mole of a substance. It is expressed in grams per mole ( ext{g/mol}) and is calculated by summing the atomic masses of all atoms in a chemical formula based on the periodic table.
The molar mass assists in converting between the mass of a substance and the number of moles of that substance. For instance, helium has a molar mass of approximately 4.00 g/mol because each helium atom weighs about 4 atomic mass units. This means that one mole of helium, roughly equivalent to Avogadro's number of atoms, would weigh 4 grams.
Understanding molar mass is pivotal in chemical stoichiometry, where it's used to determine the proportions of reactants and products in chemical reactions. It also supports calculations in processes like mass-to-mole and mole-to-mass conversions, essential for accurately preparing chemical solutions and mixtures.
Chemical Composition Analysis
Chemical composition analysis involves breaking down a substance into its constituent parts to understand its makeup. This analysis can involve various techniques, from qualitative methods, which identify what elements are present, to quantitative methods, which measure the exact amount of each component in a mixture.
A popular method is using mole fractions and mass fractions to determine the proportion of each component. In mixtures like gases, it's often necessary to convert mole fractions to mass fractions and vice versa to fully understand the chemical distribution.
Chemical composition analysis plays a critical role in quality control, environmental monitoring, pharmaceuticals, and materials science, allowing for precise control over the product's characteristics and performance. It can also help identify impurities and ensure compliance with regulatory standards, ensuring safety and efficacy in various applications.

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Most popular questions from this chapter

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