Gases With Equal Molecules Determining Pairs Of CO2 And N2

Introduction

In the realm of chemistry, understanding the molecular composition of gases is crucial for various applications, from industrial processes to environmental studies. A fundamental concept in this understanding is Avogadro's Law, which states that equal volumes of all gases, at the same temperature and pressure, contain the same number of molecules. This principle allows us to compare the number of molecules in different gas samples by examining their molar quantities. In this article, we will delve into a specific problem that requires us to identify pairs of gases containing the same number of molecules. We will dissect the problem statement, explore the underlying concepts, and systematically evaluate each option to arrive at the correct answer. This exploration will not only provide a solution to the given problem but also reinforce our understanding of key chemical principles.

Avogadro's Law serves as the cornerstone for comparing gas samples. It elegantly connects the macroscopic properties of gases, such as volume, with the microscopic realm of molecules. This law underscores that the number of molecules in a gas sample is directly proportional to the volume it occupies, provided the temperature and pressure remain constant. This proportionality constant is Avogadro's number, approximately 6.022 x 10²³, which represents the number of molecules in one mole of any substance. Understanding this fundamental relationship allows us to quantitatively compare different gas samples, even if they are composed of different chemical species. For instance, one mole of carbon dioxide (CO₂) will contain the same number of molecules as one mole of nitrogen gas (N₂), regardless of their differing molecular weights and chemical properties. This equivalence forms the basis for many stoichiometric calculations and gas-phase reactions.

To effectively apply Avogadro's Law, it is essential to grasp the concept of molar mass. Molar mass is defined as the mass of one mole of a substance, typically expressed in grams per mole (g/mol). It serves as a bridge between the mass of a substance and the number of moles present. The molar mass of a compound can be calculated by summing the atomic masses of all the atoms in its chemical formula. For example, the molar mass of carbon dioxide (CO₂) can be calculated by adding the atomic mass of carbon (approximately 12 g/mol) to twice the atomic mass of oxygen (approximately 16 g/mol), resulting in a molar mass of 44 g/mol. Similarly, the molar mass of nitrogen gas (N₂) is approximately 28 g/mol, as it consists of two nitrogen atoms, each with an atomic mass of approximately 14 g/mol. Having a clear understanding of how to calculate and utilize molar mass is crucial for accurately determining the number of moles in a given sample and, consequently, the number of molecules present.

Problem Statement

Our task is to identify a pair of gases from the given options that contain the same number of molecules. The options provide different masses of carbon dioxide (CO₂) and nitrogen gas (N₂). To solve this problem, we need to convert the given masses into moles and then compare the number of moles for each gas in each option. The option where the number of moles of CO₂ and N₂ are equal will be the correct answer. This approach directly applies Avogadro's Law, as equal moles of any gases contain the same number of molecules. The problem highlights the importance of understanding the relationship between mass, moles, and the number of molecules, a cornerstone of stoichiometry.

The problem statement provides us with four options, each presenting a pair of masses for CO₂ and N₂. These masses are given in grams, a unit that is directly measurable in a laboratory setting. However, grams do not directly tell us the number of molecules present. To bridge this gap, we need to employ the concept of molar mass, which, as previously discussed, allows us to convert mass into moles. Moles, in turn, are directly proportional to the number of molecules, as dictated by Avogadro's number. Therefore, our strategy involves a two-step process for each option: first, convert the given mass of each gas into moles using its molar mass; and second, compare the number of moles calculated for CO₂ and N₂. If the number of moles is equal, then that option represents a pair of gases with the same number of molecules.

Before we dive into the calculations, let's recap the key concepts and values we will need. We know that Avogadro's Law states that equal moles of gases contain equal numbers of molecules. We also know the approximate molar masses of CO₂ and N₂: 44 g/mol and 28 g/mol, respectively. These molar masses will serve as our conversion factors between grams and moles. With these tools in hand, we are now ready to systematically analyze each option and determine which one satisfies the condition of having the same number of molecules for both gases. The problem is a practical application of fundamental chemical principles, emphasizing the quantitative relationships between mass, moles, and molecular count. By carefully working through each option, we will reinforce our understanding of these principles and develop our problem-solving skills in chemistry.

Detailed Analysis of Options

Option 1: 22 gm of CO₂ and 72 gm of N₂

To determine if this pair of gases contains the same number of molecules, we need to convert the given masses to moles using the molar masses of CO₂ and N₂. The molar mass of CO₂ is approximately 44 g/mol, and the molar mass of N₂ is approximately 28 g/mol.

• Moles of CO₂: (22 gm CO₂) / (44 g/mol CO₂) = 0.5 moles CO₂
• Moles of N₂: (72 gm N₂) / (28 g/mol N₂) ≈ 2.57 moles N₂

Comparing the number of moles, we see that 0.5 moles of CO₂ is not equal to 2.57 moles of N₂. Therefore, this option does not represent a pair of gases with the same number of molecules.

Option 2: 11 gm of CO₂ and 28 gm of N₂

Following the same procedure, we convert the masses to moles:

• Moles of CO₂: (11 gm CO₂) / (44 g/mol CO₂) = 0.25 moles CO₂
• Moles of N₂: (28 gm N₂) / (28 g/mol N₂) = 1 mole N₂

In this case, 0.25 moles of CO₂ is not equal to 1 mole of N₂. Hence, this option is also incorrect.

Option 3: 44 gm of CO₂ and 7 gm of N₂

Converting the masses to moles:

• Moles of CO₂: (44 gm CO₂) / (44 g/mol CO₂) = 1 mole CO₂
• Moles of N₂: (7 gm N₂) / (28 g/mol N₂) = 0.25 moles N₂

Here, 1 mole of CO₂ is not equal to 0.25 moles of N₂. This option is incorrect as well.

Option 4: 11 gm of CO₂ and 7 gm of N₂

Finally, let's analyze the last option:

• Moles of CO₂: (11 gm CO₂) / (44 g/mol CO₂) = 0.25 moles CO₂
• Moles of N₂: (7 gm N₂) / (28 g/mol N₂) = 0.25 moles N₂

In this case, we find that 0.25 moles of CO₂ is equal to 0.25 moles of N₂. Therefore, this option represents a pair of gases with the same number of molecules.

Conclusion

After a systematic analysis of all the options, we have determined that Option 4, 11 gm of CO₂ and 7 gm of N₂, is the correct answer. This pair of gases contains the same number of molecules because the number of moles of CO₂ and N₂ are equal (0.25 moles each). This conclusion is based on the fundamental principle of Avogadro's Law, which states that equal volumes (or moles) of gases at the same temperature and pressure contain the same number of molecules.

This problem highlights the importance of understanding the relationship between mass, moles, and the number of molecules. By converting the given masses to moles using the molar masses of the respective gases, we were able to directly compare the molecular quantities. This approach is a cornerstone of stoichiometry, the quantitative study of chemical reactions.

Moreover, this exercise reinforces the significance of Avogadro's Law in understanding gas behavior. Avogadro's Law provides a powerful tool for relating macroscopic properties, such as mass and volume, to the microscopic world of molecules. This connection is essential for many applications in chemistry, including gas-phase reactions, volumetric analysis, and the determination of gas densities.

In summary, the problem-solving process involved a clear understanding of the question, a methodical conversion of masses to moles, and a direct application of Avogadro's Law. This approach not only led us to the correct answer but also strengthened our comprehension of fundamental chemical principles. The ability to solve such problems is crucial for further studies in chemistry and related fields. This exercise demonstrates how a seemingly simple question can encapsulate profound scientific concepts and principles. It underscores the beauty of chemistry in its ability to relate the macroscopic world we observe to the microscopic world of atoms and molecules.