Understanding the Ideal Gas Law with Mixtures of Gases

The Ideal Gas Law, represented by the equation PV = nRT, is a fundamental relationship in chemistry that connects pressure (P), volume (V), number of moles (n), gas constant (R), and temperature (T). When dealing with mixtures of gases, the law can be applied to find properties of the mixture as a whole. Below are three diverse examples illustrating the application of the Ideal Gas Law to mixtures of gases.

Example 1: Calculating the Pressure of a Gas Mixture in a Container

In many industrial processes, gas mixtures are common, and understanding their behavior is crucial. Consider a container holding a mixture of nitrogen (N₂) and oxygen (O₂) at a specific temperature. The mixture’s pressure can be calculated using the Ideal Gas Law.

Suppose we have a 10.0 L container with 2.0 moles of nitrogen and 1.0 mole of oxygen at a temperature of 300 K. To find the total pressure:

1. Calculate the total number of moles (n):

   • n = 2.0 (N₂) + 1.0 (O₂) = 3.0 moles

2. Use the Ideal Gas Law (PV = nRT):

   • P = nRT / V
   • R (Ideal Gas Constant) = 0.0821 L·atm/(K·mol)
   • P = (3.0 moles) × (0.0821 L·atm/(K·mol)) × (300 K) / (10.0 L)
   • P = 7.39 atm

Thus, the pressure of the gas mixture in the container is approximately 7.39 atm.

Notes: The total pressure of a gas mixture can also be determined by Dalton’s Law of Partial Pressures, where each gas contributes to the total pressure based on its mole fraction.

Example 2: Determining the Composition of a Gas Mixture from Pressure Measurements

In environmental science, monitoring the composition of air pollutants is essential. By capturing a gas sample and measuring its pressure and temperature, one can deduce the composition of the mixture. For instance, let’s analyze a gas mixture containing carbon dioxide (CO₂) and methane (CH₄).

Assume a 5.0 L container with a total pressure of 1.5 atm at 298 K. If the mole fraction of CO₂ is known to be 0.4, we can find the number of moles of each gas present:

1. Use the Ideal Gas Law to find total moles (n):

   • P = nRT / V
   • n = PV / RT
   • n = (1.5 atm) × (5.0 L) / (0.0821 L·atm/(K·mol) × 298 K)
   • n = 0.307 moles

2. Calculate the moles of each gas:

   • Moles of CO₂ = 0.4 × 0.307 = 0.123 moles
   • Moles of CH₄ = 0.6 × 0.307 = 0.184 moles

Notes: This method can help in understanding air quality and compliance with environmental regulations. Variations can include measuring other gases or changing the temperature and pressure conditions.

Example 3: Using the Ideal Gas Law to Analyze a Reaction Involving Gaseous Mixtures

Chemical reactions involving gaseous reactants can be analyzed using the Ideal Gas Law to predict changes in pressure or volume. For example, consider a reaction between hydrogen (H₂) and nitrogen (N₂) to form ammonia (NH₃) in a closed system. If we start with a 2.0 L reaction vessel with 1.0 mole of hydrogen and 0.5 moles of nitrogen at a temperature of 350 K:

1. Calculate the initial pressure of the system:

   • Total moles (n) = 1.0 (H₂) + 0.5 (N₂) = 1.5 moles
   • P = nRT / V
   • P = (1.5 moles) × (0.0821 L·atm/(K·mol)) × (350 K) / (2.0 L)
   • P = 19.55 atm

2. Consider the reaction stoichiometry (N₂ + 3H₂ → 2NH₃) and determine how pressure changes if 0.5 moles of H₂ reacts completely:

   • Remaining moles = 1.0 (H₂) - 0.5 (H₂) + 0.25 (N₂) = 1.25 moles
   • New P = (1.25 moles) × (0.0821 L·atm/(K·mol)) × (350 K) / (2.0 L)
   • New P = 18.64 atm

Notes: This illustrates how reactions can affect gas behavior in terms of pressure and volume. It is essential for process design in chemical engineering.

By exploring these examples, we see practical applications of the Ideal Gas Law with mixtures of gases, which is vital in numerous scientific and industrial fields.