Problem 32 - Chemistry An Atoms First Approach

Question

Which of the following statements is(are) true? For the false statements, correct them. a. At constant temperature, the lighter the gas molecules, the faster the average velocity of the gas molecules. b. At constant temperature, the heavier the gas molecules, the larger the average kinetic energy of the gas molecules. c. A real gas behaves most ideally when the container volume is relatively large and the gas molecules are moving relatively quickly. d. As temperature increases, the effect of interparticle interactions on gas behavior is increased. e. At constant \(V\) and \(T,\) as gas molecules are added into a container, the number of collisions per unit area increases resulting in a higher pressure. f. The kinetic molecular theory predicts that pressure is inversely proportional to temperature at constant volume and moles of gas.

Step-by-Step Solution

Answer

The correct statements are a, c, and e. The false statements and their corrections are: b. At constant temperature, the heavier the gas molecules, the smaller the average velocity of the gas molecules, but the average kinetic energy remains the same. d. As temperature increases, the effect of interparticle interactions on gas behavior is generally decreased. f. The kinetic molecular theory predicts that pressure is directly proportional to temperature at constant volume and moles of gas.

Step 1: a. Average velocity and molecular weight

The statement is true. At constant temperature, lighter gas molecules have higher average velocity than heavier gas molecules. This is due to the relationship between molecular weight and root-mean-square speed: \(v_{rms} = \sqrt{\dfrac{3kT}{m}}\) where \(v_{rms}\) is the root-mean-square speed, \(k\) is the Boltzmann constant, \(T\) is the temperature, and \(m\) is the mass of the gas molecule. As the mass of the gas molecule decreases, the root-mean-square speed (and thus the average velocity) increases.

Step 2: b. Average kinetic energy and molecular weight

The statement is false. At constant temperature, all gas molecules (regardless of their mass) have the same average kinetic energy. The correct statement is: At constant temperature, the heavier the gas molecules, the smaller the average velocity of the gas molecules, but the average kinetic energy remains the same. The average kinetic energy (\(KE\)) of gas molecules is given by: \(KE = \dfrac{3}{2}kT\) where \(k\) is the Boltzmann constant and \(T\) is the temperature. As we can see, the average kinetic energy is independent of the molecular mass.

Step 3: c. Real gas behavior at large volume and high speed

The statement is true. A real gas behaves most ideally (i.e., more like an ideal gas) when the container volume is relatively large and the gas molecules are moving relatively quickly. In these conditions, the effects of intermolecular forces and the volume of the gas molecules become negligible compared to the volume of the container and the kinetic energy of the gas molecules.

Step 4: d. Interparticle interactions with increasing temperature

The statement is false. As temperature increases, the effect of interparticle interactions on gas behavior is generally decreased. Higher temperatures result in gas molecules moving faster, which weakens the influence of intermolecular forces, causing the gas to behave more like an ideal gas.

Step 5: e. Pressure, collisions, and adding gas molecules

The statement is true. At constant volume and temperature, as gas molecules are added into a container, the number of collisions per unit area (with the walls of the container) increases. This leads to an increase in pressure, as described by the ideal gas law: \(PV = nRT\).

Step 6: f. Pressure and temperature relationship from kinetic molecular theory

The statement is false. The kinetic molecular theory predicts that pressure is directly proportional to temperature at constant volume and moles of gas, not inversely proportional as stated. The correct relationship is given by: \(P \propto T\) at constant volume and moles of gas, according to the ideal gas law: \(PV = nRT\).

Key Concepts

Root-Mean-Square Speed

The root-mean-square (rms) speed is a way to define the average speed of gas molecules in a container. It's derived from the kinetic molecular theory, which helps us understand gas behavior. The rms speed is calculated using the formula:
\(v_{rms} = \textstyle\sqrt{\dfrac{3kT}{m}}\),
where \(k\) is the Boltzmann constant, \(T\) is the absolute temperature, and \(m\) is the mass of an individual gas molecule. This equation implies that at a given temperature, lighter gas molecules will move faster than heavier ones, as they have less mass \(m\). It's a vital concept when analyzing how different gases behave under the same conditions.

Average Kinetic Energy of Gases

The average kinetic energy of a gas molecule is directly related to the temperature of the gas and is consistent across different gases at the same temperature. It's expressed as:
\(KE = \textstyle\dfrac{3}{2}kT\),
where \(k\) represents the Boltzmann constant and \(T\) is the temperature. Despite the differences in mass or size of gas molecules, their average kinetic energy will be the same if they are at an equal temperature. This uniformity in kinetic energy is fundamental to understanding why gases despite having different molecular weights, can exert the same pressure under the same conditions if their temperatures are identical.

Ideal Gas Behavior

Gases tend to exhibit ideal behavior under certain conditions, particularly when they are at high temperature and low pressure. This means that the interactions between molecules are minimal, and their actual volume is small compared to the space they occupy. Ideal gas behavior is modeled by the ideal gas law and assumes no intermolecular forces and that the molecules occupy no volume. Real gases approach ideal behavior when the gas molecules move quickly and the container volume is relatively large, reducing the impact of these assumptions. Understanding this can help predict and calculate gas behavior in a variety of scientific and engineering contexts.

Intermolecular Forces in Gases

Intermolecular forces, such as Van der Waals forces, play a crucial role in the behavior of real gases. They affect properties such as boiling point, melting point, and viscosity. Generally, the effect of these forces becomes less significant at higher temperatures, as the increased kinetic energy of the molecules allows them to overcome the attractive forces between them. Hence, gases are less likely to liquefy or solidify at high temperatures because the molecules are moving too fast to stick together. This reduction in the influence of intermolecular forces is why gases can behave more 'ideally' in such conditions.

Ideal Gas Law

The ideal gas law is a cornerstone in understanding gas behavior and is represented by the equation:
\(PV = nRT\),
where \(P\) is pressure, \(V\) is volume, \(n\) is the number of moles of gas, \(R\) is the ideal gas constant, and \(T\) is temperature. This law combines several gas laws, including Boyle's, Charles's, and Avogadro's laws. It suggests that the pressure exerted by a gas is directly proportional to the temperature and the number of gas molecules and inversely proportional to the gas's volume. Ideal gas law is an exceptional starting point for chemistry and physics students to predict how changes in one state variable will affect another, assuming ideal gas behavior.