Problem 72 - Chemistry: The Central Science

Question

(a) What are the mole fractions of each component in a mixture of \(15.08 \mathrm{~g}\) of \(\mathrm{O}_{2}, 8.17 \mathrm{~g}\) of \(\mathrm{N}_{2}\), and \(2.64 \mathrm{~g}\) of \(\mathrm{H}_{2}\) ? (b) What is the partial pressure in atm of each component of this mixture if it is held in a 15.50- \(\mathrm{L}\) vessel at \(15^{\circ} \mathrm{C}\) ?

Step-by-Step Solution

Answer

(a) Mole fractions: O₂: 0.226, N₂: 0.140, H₂: 0.634. (b) Partial pressures: O₂: \(0.696 \: atm\), N₂: \(0.431 \: atm\), H₂: \(1.948 \: atm\).

Step 1: Calculate the number of moles for each component

Using the given masses of each component and their respective molar masses, we can find the number of moles for each component. The molar mass of O₂ is 32 g/mol, N₂ is 28 g/mol, and H₂ is 2 g/mol. n(O₂) = mass(O₂) / molar mass(O₂) = 15.08 g / 32 g/mol = 0.471 mol n(N₂) = mass(N₂) / molar mass(N₂) = 8.17 g / 28 g/mol = 0.292 mol n(H₂) = mass(H₂) / molar mass(H₂) = 2.64 g / 2 g/mol = 1.32 mol

Step 2: Calculate the mole fractions of each component

Next, we'll find the total number of moles in the mixture and calculate the mole fraction of each component. Total moles = n(O₂) + n(N₂) + n(H₂) = 0.471 + 0.292 + 1.32 = 2.083 mol Now let's calculate the mole fractions: Mole fraction(O₂) = n(O₂) / total moles = 0.471 / 2.083 = 0.226 Mole fraction(N₂) = n(N₂) / total moles = 0.292 / 2.083 = 0.140 Mole fraction(H₂) = n(H₂) / total moles = 1.32 / 2.083 = 0.634

Step 3: Use the ideal gas law to find the total pressure of the mixture

We're given the volume (V = 15.50 L) and temperature (T = 15°C = 288.15 K). Applying the ideal gas law (PV = nRT) with total moles and ideal gas constant R = 0.0821 L atm/(mol K): Total pressure (P) = n_total * R * T / V = (2.083 mol) * (0.0821 L atm/(mol K)) * (288.15 K) / (15.50 L) Total pressure (P) = 3.074 atm

Step 4: Calculate the partial pressures of each component

Using the mole fractions and the total pressure, we can find the partial pressures of each component: Partial pressure(O₂) = Mole fraction(O₂) * Total pressure = 0.226 * 3.074 atm ≈ 0.696 atm Partial pressure(N₂) = Mole fraction(N₂) * Total pressure = 0.140 * 3.074 atm ≈ 0.431 atm Partial pressure(H₂) = Mole fraction(H₂) * Total pressure = 0.634 * 3.074 atm ≈ 1.948 atm The partial pressures of O₂, N₂, and H₂ in the mixture are approximately 0.696 atm, 0.431 atm, and 1.948 atm, respectively.

Key Concepts

Ideal Gas Law

The ideal gas law is a fundamental equation for describing the state of a gas in terms of its pressure, volume, temperature, and number of moles. It integrates several key concepts in chemistry and physics to give a comprehensive understanding of gas behavior. The law is expressed as:\[ PV = nRT \]Here:

• \( P \) represents the pressure of the gas in atmosphere (atm).
• \( V \) is the volume of the gas in liters (L).
• \( n \) stands for the number of moles of the gas.
• \( R \) is the ideal gas constant, which is approximately 0.0821 L atm/(mol K).
• \( T \) is the temperature in Kelvin (K).

It's important to remember that temperature has to be in Kelvin because the ideal gas law is derived from absolute temperature scales. To convert degrees Celsius to Kelvin, simply add 273.15.This law assumes that the gas particles do not interact with each other and occupy no volume themselves, which is approximately true for many gases under standard conditions. Ideal gas law is widely applied to determine one of these parameters if the others are known, making it a powerful tool in calculations involving gases.

Partial Pressure

Partial pressure is the pressure exerted by a single gas component in a mixture of gases. Each gas in a mixture exerts its own pressure as if the other gases were not present, which we call its partial pressure. The relationship between mole fractions and partial pressures is given by Dalton's Law of Partial Pressures.The law states:"In a mixture of non-reacting gases, the total pressure exerted is equal to the sum of the partial pressures of individual gases."To find the partial pressure of a gas, multiply its mole fraction by the total pressure of the gas mixture:\[ P_i = X_i \times P_{\text{total}} \]Where:

• \( P_i \) is the partial pressure of the gas.
• \( X_i \) is the mole fraction of the gas in the mixture.
• \( P_{\text{total}} \) is the total pressure of the gas mixture.

Understanding partial pressures is crucial in fields like chemistry and environmental science, where gas mixtures are common. It allows us to predict and account for the behavior of individual gas components within a mixture.

Mole Concept

The mole concept is fundamental to chemistry, linking the microscopic world of atoms and molecules to quantities we can measure in the laboratory. A mole is defined as exactly 6.02214076 × 10²³ elementary entities (such as atoms, molecules, or ions). This number is known as Avogadro's number.In practical terms, the mole concept allows us to convert between mass and number of moles, using the substance's molar mass (the mass of one mole of a substance). The formula to find the number of moles \( n \) is:\[ n = \frac{\text{mass}}{\text{molar mass}} \]This concept helps us understand relationships in chemical reactions and stoichiometry, enabling us to calculate how much product will form from given reactants, or how much reactant we need to form a desired amount of product. The mole concept is also instrumental in calculating concentrations, pressures in gases, and understanding the proportions in which elements combine to form compounds.By grasping the mole concept, students are better equipped to solve problems involving chemical quantities and predict outcomes in chemical experimentation and real-world applications.