Problem 78 - Chemistry The Central Science

Question

Indicate which of the following statements regarding the kineticmolecular theory of gases are correct. For those that are false, formulate a correct version of the statement. (a) The average kinetic energy of a collection of gas molecules at a given temperature is proportional to \(\mathrm{m}^{1 / 2}\). (b) The gas molecules are assumed to exert no forces on each other. (c) All the molecules of a gas at a given temperature have the same kinetic energy. (d) The volume of the gas molecules is negligible in comparison to the total volume in which the gas is contained. (e) All gas molecules move with the same speed if they are at the same temperature.

Step-by-Step Solution

Answer

(a) False. Correct statement: "The average kinetic energy of a collection of gas molecules at a given temperature is proportional to the temperature." (b) True. (c) False. Correct statement: "The gas molecules at a given temperature exhibit a distribution of kinetic energies that follow the Maxwell-Boltzmann distribution." (d) True. (e) False. Correct statement: "The gas molecules at a given temperature exhibit a distribution of speeds that follow the Maxwell-Boltzmann distribution."

Step 1: Statement (a)

The statement says that the average kinetic energy of a collection of gas molecules at a given temperature is proportional to \(m^{1/2}\), where m is the mass of gas molecules. This statement is false. According to the kinetic molecular theory of gases, the average kinetic energy of a collection of gas molecules at a given temperature is proportional to the temperature: \[\overline{KE} \propto T\]A correct version of the statement would be: "The average kinetic energy of a collection of gas molecules at a given temperature is proportional to the temperature."

Step 2: Statement (b)

The statement says that gas molecules are assumed to exert no forces on each other. This statement is true. One of the assumptions of the kinetic molecular theory of gases is that there are no intermolecular forces acting between gas molecules. This simplifies the understanding of the fascinating behavior of gases.

Step 3: Statement (c)

According to the statement, all molecules of a gas at a given temperature have the same kinetic energy. This statement is false. The kinetic molecular theory of gases assumes that the distribution of kinetic energy of gas molecules at a given temperature follows the Maxwell-Boltzmann distribution. This means that gas molecules at a given temperature can have a variety of kinetic energies, not necessarily the same. A correct version of the statement would be: "The gas molecules at a given temperature exhibit a distribution of kinetic energies that follow the Maxwell-Boltzmann distribution."

Step 4: Statement (d)

The statement says that the volume of the gas molecules is negligible in comparison to the total volume in which the gas is contained. This statement is true. In the kinetic molecular theory of gases, it is assumed that the volume occupied by gas molecules is insignificant compared to the total volume in which the gas is contained. This allows the gas molecules to be considered as point particles.

Step 5: Statement (e)

The statement says that all gas molecules move with the same speed if they are at the same temperature. This statement is false. As mentioned earlier, gas molecules at a given temperature have a range of kinetic energies, which implies that they have a variety of speeds as well. As a consequence, the gas molecules do not have the same speed at a given temperature. A correct version of the statement would be: "The gas molecules at a given temperature exhibit a distribution of speeds that follow the Maxwell-Boltzmann distribution."

Key Concepts

Average Kinetic Energy

Let's talk about the average kinetic energy of gas molecules. Imagine gas molecules buzzing around inside a balloon. Each molecule is moving at its own pace, bouncing off the walls. Importantly, at a given temperature, the average kinetic energy depends on the temperature itself, not on the mass of the molecules as one might mistakenly think.
The kinetic molecular theory of gases states:

• The average kinetic energy of gas molecules is directly proportional to the temperature of the gas in Kelvin.
• As temperature rises, the average kinetic energy increases.

This proportionality can be represented as: \[ \overline{KE} \propto T\] This means that, for gas molecules in a container, when the temperature goes up, they buzz around faster! It’s like adding heat to a pot of popcorn kernels—the hotter the pot, the more energy each kernel has to pop.

Maxwell-Boltzmann Distribution

The Maxwell-Boltzmann distribution is like a guidebook to understanding the behavior of gas molecules at various speeds and energies. It explains how gas molecules in a sample do not all move at the same speed or have the same energy, even if they're at the same temperature. Instead, there is a distribution:

• Some molecules move fast and some move slow, while most are in between.
• This distribution changes with temperature and accommodates a range of speeds.

When we look at a graph of the Maxwell-Boltzmann distribution, we can see a peak representing the most probable speed (the speed that many molecules have). As temperature increases:

• The peak shifts towards higher speeds, illustrating that more molecules are moving faster.
• The range of speeds becomes wider showing more diversity among the speeds.

This vivid depiction helps us understand why not all molecules move alike and how temperature influences their pace.

Gas Molecules

Gas molecules are endlessly fascinating! According to the kinetic molecular theory, these tiny particles explain the behavior of gases with a few key assumptions:

• They are in constant, random motion, moving in straight lines until they either collide with other molecules or with the walls of the container.
• They exert no attractive or repulsive forces on each other, which means their collisions are perfectly elastic.
• Their volume is negligible compared to the container's volume, enabling simpler calculations and simulations.

Together, these assumptions allow us to predict the properties of gases under different conditions, like temperature and pressure. It's why balloons inflate, tires pop when overfilled, and if you open a bottle of perfume in a room, eventually everyone knows because the molecules spread out evenly across the room, freely and energetically bouncing around!