Problem 48 - Chemistry: The Central Science

Question

Which of the following statements best explains why nitrogen gas at STP is less dense than Xe gas at STP? (a) Because Xe is a noble gas, there is less tendency for the Xe atoms to repel one another, so they pack more densely in the gas state. (b) Xe atoms have a higher mass than \(\mathrm{N}_{2}\) molecules. Because both gases at STP have the same number of molecules per unit volume, the Xe gas must be denser. (c) The Xe atoms are larger than \(\mathrm{N}_{2}\) molecules and thus take up a larger fraction of the space occupied by the gas. (d) Because the Xe atoms are much more massive than the \(\mathrm{N}_{2}\) molecules, they move more slowly and thus exert less upward force on the gas container and make the gas appear denser.

Step-by-Step Solution

Answer

The best explanation for why nitrogen gas at STP is less dense than Xe gas at STP is statement (b): Xe atoms have a higher mass than \(\mathrm{N}_{2}\) molecules. Since both gases at STP have the same number of molecules per unit volume, the Xe gas is denser.

Step 1: At standard temperature and pressure (0°C and 1 atm), all gases have the same number of molecules per unit volume, which is essentially the Avogadro's law. Hence, the density difference between two gases at STP depends mainly on the mass difference between their molecules or atoms. #Step 2: Evaluate statement (a)#

Statement (a) states that Xe is a noble gas, and there is less tendency for the Xe atoms to repel one another so they pack more densely in the gas state. However, this statement does not explain the actual density difference, as both nitrogen and xenon gases are non-polar and will not have significant attraction or repulsion between their particles. #Step 3: Evaluate statement (b)#

Step 2: Statement (b) says that Xe atoms have a higher mass than \(\mathrm{N}_{2}\) molecules. As both gases at STP have the same number of molecules per unit volume, the Xe gas must be denser. This statement seems to be correct because density directly depends on the mass of particles, and xenon atoms are indeed heavier than nitrogen molecules. #Step 4: Evaluate statement (c)#

Statement (c) claims that the Xe atoms are larger than the \(\mathrm{N}_{2}\) molecules, taking up a larger fraction of the space occupied by the gas. While it is true that Xe atoms are larger than \(\mathrm{N}_{2}\) molecules, this statement does not explain the density difference since both gases have the same number of particles per unit volume at STP. #Step 5: Evaluate statement (d)#

Step 3: Statement (d) says that the Xe atoms are much more massive than the \(\mathrm{N}_{2}\) molecules, so they move more slowly and exert less upward force on the gas container. However, this statement is not relevant to the density difference. Moreover, the upward force on the container is related to the pressure, not the density. #Step 6: Conclusion#

Based on the analysis of the given statements, the best explanation for the difference in densities between nitrogen gas and xenon gas at STP is statement (b). Xe atoms have a higher mass than \(\mathrm{N}_{2}\) molecules. Since both gases at STP have the same number of molecules per unit volume, the Xe gas is denser.

Key Concepts

Avogadro's Law

A fundamental principle of chemistry is Avogadro's law, which proposes that equal volumes of all gases, at the same temperature and pressure, contain the same number of molecules. Applied to the scenario of comparing gas densities at standard temperature and pressure (STP), which is 0°C and 1 atmosphere, Avogadro's law becomes particularly useful. It means that regardless of the gas type, be it nitrogen or xenon, there will be an identical number of particles in a given volume.

Understanding this law helps us determine that the density differences between gases at STP must stem from other factors, since the number of particles per volume is constant. This constant is known as Avogadro's number, which is approximately \(6.022 \times 10^{23}\) entities per mole. It exemplifies that at STP, one mole of any gas occupies \(22.4 L\), often referred to as the molar volume.

Implications of Avogadro's Law

For students, Avogadro's law implies that if you're measuring two different gases at STP and they exhibit varying densities, the difference is due to the mass of individual particles and not the amount of substance or its volume. This concept becomes the foundation for solving problems related to gas densities and comparing the physical properties of different gases under standard conditions.

Molar Mass and Density

Molar mass and density are intrinsically linked concepts in chemistry, especially when discussing gases. Density of a substance is defined as its mass per unit volume. For gases, density can be calculated using the equation \(\rho = \frac{M}{V}\), where \(\rho\) represents density, \(M\) is the mass, and \(V\) is the volume. By considering the molar mass, which is the mass of one mole of a substance, and the molar volume at STP, we can align our understanding of density with the mole concept.

Since Avogadro's law tells us that one mole of any gas has the same volume at STP, we infer that the density of a gas is directly proportional to its molar mass. Therefore, a gas with a higher molar mass will be denser, as more mass is concentrated in the same volume. This relationship allows us to solve the textbook problem by concluding that xenon, which has a much higher molar mass than nitrogen, is denser at STP even though both gases have the same number of molecules per unit volume.

Calculating Density Using Molar Mass

To calculate the density of a gas at STP, you can divide its molar mass by the molar volume of gas at STP (\(\frac{molar mass}{22.4 L/mol}\)). It's a straightforward way for students to compare the densities of different gases, given their molar masses.

Properties of Noble Gases

The noble gases, which include helium, neon, argon, krypton, xenon, and radon, are notable for their lack of reactivity due to a full valence shell of electrons. This stability is a defining property of the noble gases, making them unique among the elements. Additionally, many noble gases have high molar masses compared to other gases, such as hydrogen or nitrogen, which directly impacts their densities.

Noble gases at STP behave ideally, meaning they adhere perfectly to the assumptions made in the ideal gas law. These assumptions include no intermolecular forces and that the volume occupied by the gas particles themselves is negligible. The textbook exercise insinuates this by dismissing the relevance of repulsion between xenon atoms as the cause for its higher density compared to nitrogen.

Noble Gases and Density Predictions

When discussing noble gases and their properties, it's clear that their larger atomic sizes and higher molar masses contribute to their densities. Xenon, being one of the heaviest noble gases, is denser than nitrogen gas at STP, not because of intermolecular forces or the size of its atoms, but due to its significantly greater molar mass. This information empowers students to understand why certain gases are heavier than others and highlights the practical implications of the periodic table's organization in real-world scenarios.