Question

At \(0^{\circ} \mathrm{C}\) a \(1.0-\mathrm{L}\) flask contains \(5.0 \times 10^{-2}\) mole of \(\mathrm{N}_{2}, 1.5 \times\) \(10^{2} \mathrm{mg} \mathrm{O}_{2},\) and \(5.0 \times 10^{21}\) molecules of \(\mathrm{NH}_{3} .\) What is the partial pressure of each gas, and what is the total pressure in the flask?

Short answer

The partial pressures of the gases N₂, O₂, and NH₃ in the flask are 1.1240 atm, 0.1045 atm, and 0.1854 atm, respectively. The total pressure inside the flask is 1.4139 atm.

Step-by-step solution

Convert given quantities to moles

First, we need to find the moles of each gas in the flask. We're given the amount of N₂ in moles already, but we need to convert the mass of O₂ and the number of NH₃ molecules into moles. For O₂, we are given 1.5 * 10² mg. To convert this to moles, we can use the molar mass of O₂ (32 g/mol): moles of O₂ = (1.5 * 10² mg) * (1 g/1000 mg) * (1 mol/32 g) = 0.0046875 mol For NH₃, we are given 5.0 * 10²¹ molecules. To convert this to moles, we can use Avogadro's number (6.022 * 10²³ molecules/mol): moles of NH₃ = (5.0 * 10²¹ molecules) * (1 mol/6.022 * 10²³ molecules) = 0.0083045 mol Now we have the moles of each gas: - N₂: 5.0 * 10⁻² mol - O₂: 0.0046875 mol - NH₃: 0.0083045 mol

Calculate partial pressure for each gas using Ideal Gas Law equation

Now we need to calculate the partial pressure of each gas using the Ideal Gas Law equation: PV = nRT. In this case, we know the volume (V = 1.0 L), and the temperature (T = 273.15 K, because we need to convert from 0°C to Kelvin). The universal gas constant R is 0.0821 (L * atm) / (mol * K). Next, we will solve for the partial pressure (P) for each gas. Partial pressure of N₂ (P_N₂) = (5.0 * 10⁻² mol) * (0.0821 L⋅atm/mol⋅K) * (273.15 K) / (1.0 L) = 1.1240 atm Partial pressure of O₂ (P_O₂) = (0.0046875 mol) * (0.0821 L⋅atm/mol⋅K) * (273.15 K) / (1.0 L) = 0.1045 atm Partial pressure of NH₃ (P_NH₃) = (0.0083045 mol) * (0.0821 L⋅atm/mol⋅K) * (273.15 K) / (1.0 L) = 0.1854 atm

Calculate total pressure inside the flask

To find the total pressure inside the flask, we can simply add the partial pressures of each gas: Total pressure (P_total) = P_N₂ + P_O₂ + P_NH₃ = 1.1240 atm + 0.1045 atm + 0.1854 atm = 1.4139 atm. Therefore, the partial pressures of the gases are: - N₂: 1.1240 atm - O₂: 0.1045 atm - NH₃: 0.1854 atm And the total pressure inside the flask is 1.4139 atm.

Key concepts

Partial Pressure

The concept of partial pressure is crucial in understanding gas mixtures. In any mixture of gases, each gas exerts a pressure as if it occupied the entire volume alone. This pressure is known as the partial pressure. It contributes to the total pressure within a container, which is simply the sum of the partial pressures of all gases present.

To calculate the partial pressure of a gas, we use the Ideal Gas Law in the form:\[P = \frac{{nRT}}{V}\]where:

• \(P\) is the partial pressure of the gas.
• \(n\) represents the moles of the gas.
• \(R\) is the universal gas constant.
• \(T\) is the temperature in Kelvin.
• \(V\) is the volume in liters.

Understanding these calculations helps us predict how gases will behave in mixtures and allows us to determine pressures in chemical reactions or various environmental conditions.

Mole Conversion

Mole conversion is an essential step in solving many chemistry problems involving gases, as it links the macroscopic and molecular scales. Moles provide a bridge between atomic and molecular scales and practical quantities we can measure, such as mass or volume.

To convert different quantities to moles, you follow these steps:

• For mass to moles: use the formula \[\text{moles} = \frac{\text{mass (g)}}{\text{molar mass (g/mol)}}\].
• For molecules to moles: apply Avogadro's number \(6.022 \times 10^{23}\) molecules/mol with \[\text{moles} = \frac{\text{number of molecules}}{6.022 \times 10^{23}}\].

By converting measurements like grams or molecules to moles, we can easily work with the Ideal Gas Law, as it requires quantities in terms of moles, making this skill invaluable for understanding chemical equations and reactions.

Avogadro's Number

Avogadro's number is a fundamental constant in chemistry, precisely defined as \(6.022 \times 10^{23}\). It represents the number of atoms, ions, or molecules in one mole of a substance, connecting the micro world of atoms and molecules to the macro world we observe.

This constant is crucial for converting between the amount of substance in moles and the number of particles, allowing chemists to:

• Calculate the number of molecules needed for reactions.
• Understand how gases and other molecules interact in practical quantities.
• Work with molecular weights and determine molar mass.

Utilizing Avogadro's number enables achievements such as calculating the moles from a specific number of molecules, crucial for lab work and theoretical chemistry.

Universal Gas Constant

The universal gas constant \(R\) plays a central role in equations involving gases, particularly within the Ideal Gas Law. Its value ties together pressure, volume, temperature, and the amount of substance into a cohesive framework for understanding gas behaviors under various conditions. The value of \(R\) used in these calculations is typically \(0.0821\) L·atm/(mol·K).

This constant is used in the Ideal Gas Law formula:\[PV = nRT\]where \(P\) is pressure, \(V\) is volume, \(n\) is moles, \(R\) is the universal gas constant, and \(T\) is temperature in Kelvin.

The universal gas constant allows chemists and students alike to coordinate their calculations with total consistency. It serves as an anchor in theoretical and empirical approaches, helping to create standard solutions that apply broadly across many types of chemical problems and conditions.