Problem 105 - Chemistry The Molecular Science

Question

Consider a sample of \(\mathrm{N}_{2}\) gas under conditions in which it obeys the ideal gas law exactly. Which of these statements is/are true? (a) A sample of \(\mathrm{Ne}(\mathrm{g})\) under the same conditions must obey the ideal gas law exactly. (b) The speed at which one particular \(\mathrm{N}_{2}\) molecule is moving changes from time to time. (c) Some \(\mathrm{N}_{2}\) molecules are moving more slowly than some of the molecules in a sample of \(\mathrm{O}_{2}(\mathrm{~g})\) under the same conditions. (d) Some \(\mathrm{N}_{2}\) molecules are moving more slowly than some of the molecules in a sample of \(\mathrm{Ne}(\mathrm{g})\) under the same conditions. (e) When two \(\mathrm{N}_{2}\) molecules collide, it is possible that both may be moving faster after the collision than they were before.

Step-by-Step Solution

Answer

Statements (a), (b), (d), and (e) are true.

Step 1: Analyze Statement (a)

The ideal gas law applies universally to ideal gases, meaning all gases under conditions where they behave ideally follow the same law: \( PV = nRT \). Since \( \mathrm{N}_{2} \) obeys this law under given conditions, \( \mathrm{Ne} \) will also obey the ideal gas law under the same conditions. Thus, Statement (a) is true.

Step 2: Analyze Statement (b)

For an individual molecule, its speed varies due to random thermal motion. The speed of one \( \mathrm{N}_{2} \) molecule fluctuates over time as the molecules collide and transfer kinetic energy. Thus, Statement (b) is true.

Step 3: Analyze Statement (c)

At the same temperature, all gases have the same average kinetic energy. However, \( \mathrm{O}_{2} \) molecules are heavier than \( \mathrm{N}_{2} \) molecules. Therefore, some \( \mathrm{N}_{2} \) molecules can be slower than some \( \mathrm{O}_{2} \) molecules due to the distribution of individual molecule speeds, known as the Maxwell-Boltzmann distribution. Thus, Statement (c) is false.

Step 4: Analyze Statement (d)

Given the masses, \( \mathrm{Ne} \) (neon) atoms are lighter than \( \mathrm{N}_{2} \). Therefore, at the same temperature, \( \mathrm{Ne} \) atoms will generally move faster. Consequently, some \( \mathrm{N}_{2} \) molecules will indeed be moving more slowly than some \( \mathrm{Ne} \) molecules. Thus, Statement (d) is true.

Step 5: Analyze Statement (e)

In elastic collisions, which are typical for molecules in an ideal gas, kinetic energy is conserved. It is possible for molecules to gain speed after a collision due to the distribution of energy. Thus, some \( \mathrm{N}_{2} \) molecules can be faster after a collision. Hence, Statement (e) is true.

Key Concepts

Kinetic Molecular Theory

Kinetic Molecular Theory (KMT) is a fundamental concept for understanding the behavior of gases. It provides a picture of how gas molecules move and interact with each other. According to KMT, gases are composed of large numbers of small particles moving in constant, random motion. This movement causes them to collide with each other and the walls of their container.

Key principles of Kinetic Molecular Theory include:

• Gas molecules are in perpetual motion and move in straight lines.
• The motion of these molecules is random, leading to collisions with other molecules and the container walls.
• Molecular collisions are perfectly elastic, meaning no kinetic energy is lost in the process (more on this later).
• The average kinetic energy of gas molecules is directly proportional to the temperature of the gas in Kelvin.
• Gas molecules are considered to have no volume and exert no forces on each other except during collisions.

Understanding KMT helps in explaining why gases behave differently under various temperature and volume conditions, as well as providing a basis for the Ideal Gas Law.

Maxwell-Boltzmann Distribution

The Maxwell-Boltzmann Distribution describes how the speeds of particles within a gas are spread over a range of values at a given temperature. It is a statistical distribution that helps in understanding the behavior of gas molecules.

Key points about the Maxwell-Boltzmann Distribution:

• The distribution shows that not all molecules are moving at the same speed; there is a range of speeds.
• Most molecules have speeds around a certain average, but some move much faster or much slower.
• Higher temperatures result in a broader distribution, with more molecules moving at higher speeds.
• The peak of the distribution shifts to higher speeds as the temperature increases.

This concept is crucial for explaining why different gases may behave differently even under the same temperature and pressure conditions, as the speed distribution can affect reaction rates and diffusion.

Elastic Collisions

Elastic collisions are an important concept when discussing ideal gases. In an ideal gas, the collisions between molecules are considered elastic. This means that the total kinetic energy of the system is conserved during and after a collision.

In the context of gases:

• An elastic collision means that two colliding molecules can exchange energy but no energy is lost to sound, heat, or deformation.
• These collisions allow for the redistribution of kinetic energy among molecules.
• Because energy is conserved, the speed of individual molecules can change, but the overall energy of the system remains the same.

Elastic collisions help to maintain the properties of the gas as they allow for constant molecular motion and energy exchange without loss. This is a key aspect of ideal gas behavior, making the kinetic molecular theory applicable.

Gas Laws

Gas laws are equations that describe the behavior of gases in response to changes in temperature, pressure, volume, and the number of moles of gas. The most widely known is the Ideal Gas Law, represented by the equation: \[ PV = nRT \] Where:

• \( P \) is the pressure of the gas,
• \( V \) is the volume of the gas,
• \( n \) is the number of moles,
• \( R \) is the ideal gas constant,
• \( T \) is the temperature in Kelvin.

Ideal Gas Law provides a simple and effective model to predict the behavior of gases under many conditions, assuming that the gas behaves ideally, which means that the gas molecules do not exert forces on each other except during collisions and occupies no volume.

Other significant gas laws include Boyle's Law, Charles's Law, and Avogadro's Law, each addressing a specific aspect of gas behavior and allowing us to solve various real-world problems involving gases.