Question

Indicate which of the following statements regarding the kinetic-molecular 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.

Short answer

The correct statements regarding the kinetic-molecular theory of gases are: (b) The gas molecules are assumed to exert no forces on each other (except during perfectly elastic collisions). (d) The volume of the gas molecules is negligible in comparison to the total volume in which the gas is contained. For the incorrect statements: (a) The corrected statement is "The average kinetic energy of a collection of gas molecules at a given temperature is proportional to the temperature." (c) The corrected statement is "At a given temperature, the gas molecules have a range of kinetic energies, but their average kinetic energy is proportional to the temperature."

Step-by-step solution

Statement (a)

The average kinetic energy of a collection of gas molecules at a given temperature is proportional to \(\mathrm{m}^{1 / 2}\). This statement is incorrect. The average kinetic energy of a collection of gas molecules at a given temperature is proportional to the temperature (T), not the square root of the mass (m). The correct statement should be: "The average kinetic energy of a collection of gas molecules at a given temperature is proportional to the temperature."

Statement (b)

The gas molecules are assumed to exert no forces on each other. This statement is correct. In the kinetic-molecular theory of gases, it is assumed that the gas molecules do not exert any forces on each other, except during collisions which are considered to be perfectly elastic.

Statement (c)

All the molecules of a gas at a given temperature have the same kinetic energy. This statement is incorrect. While the average kinetic energy of the gas molecules is proportional to the temperature, individual molecules will have a range of kinetic energies due to their different velocities. The correct statement should be: "At a given temperature, the gas molecules have a range of kinetic energies, but their average kinetic energy is proportional to the temperature."

Statement (d)

The volume of the gas molecules is negligible in comparison to the total volume in which the gas is contained. This statement is correct. In the kinetic-molecular theory of gases, it is assumed that the volume of the gas molecules themselves is much smaller compared to the total volume in which the gas is contained. This allows for the ideal gas laws to be applicable in most cases.

Key concepts

average kinetic energy

The concept of average kinetic energy is central to understanding the behavior of gases. In the world of gases, kinetic energy refers to the motion of gas particles. The kinetic-molecular theory posits that the average kinetic energy of gas particles is proportional to the temperature of the gas in Kelvin.

This means that as the temperature of a gas increases, the average motion or kinetic energy of its particles also rises. This relationship can be mathematically represented by the equation:

• \( KE_{avg} = \frac{3}{2}kT \)

where \( KE_{avg} \) is the average kinetic energy, \( k \) is the Boltzmann constant, and \( T \) is the temperature in Kelvin.

Thus, even though individual molecules in a gas might have different kinetic energies due to varied speeds, collectively they follow this temperature-dependent average behavior.

ideal gas laws

The ideal gas laws provide a framework for understanding how gases behave under various conditions. They are based on a set of assumptions that simplify the behavior of gas molecules and make calculations more manageable. These laws encompass relations between pressure, volume, temperature, and the number of gas particles.

One of the fundamental assumptions is that gas molecules are always in motion and move randomly, colliding with each other and the walls of their container. The ideal gas laws include:

• Boyle's Law: At constant temperature, the pressure of a gas is inversely proportional to its volume (\( PV = k \))
• Charles's Law: At constant pressure, the volume of a gas is directly proportional to its temperature (\( V/T = k \))
• Avogadro's Law: At constant temperature and pressure, the volume of a gas is directly proportional to the number of moles (\( V/n = k \))
• Ideal Gas Law: It combines the above into \( PV = nRT \) where \( R \) is the ideal gas constant.

These laws help predict how a gas will respond to changes in temperature, pressure, or volume.

gas molecular forces

In the realm of gases, gas molecular forces are a topic of interest when discussing the ideal gas assumptions. According to the kinetic-molecular theory, for a gas to behave ideally, it is assumed that there are no attractive or repulsive forces between the molecules.

In reality, this assumption holds very well under conditions of high temperature and low pressure where the molecules are far apart and interactions are minimal. Under these conditions:

• Collisions between molecules are considered perfectly elastic, meaning there is no net loss in kinetic energy from the collisions.
• Intermolecular forces like van der Waals forces, if present, are negligible.

However, in very high pressure or low temperature situations, these forces become more pronounced, causing deviations from ideal behavior known as real gas behavior.

temperature dependence of kinetic energy

The temperature dependence of kinetic energy is a key concept in understanding how gases respond to temperature changes. As the temperature of a gas increases, so does the kinetic energy of its particles. This is because temperature is a measure of the average kinetic energy of the particles within a substance.

Therefore, when temperature rises:

• Gas particles move faster, increasing their kinetic energy.
• This increase leads to more frequent and energetic collisions between the particles and with the walls of their container.

On a practical level, this means that the pressure of a gas also increases if the volume is kept constant, as depicted by Gay-Lussac's Law which states that the pressure of a given mass of gas varies directly with the Kelvin temperature at constant volume (\( P/T = k \)).
This relationship underscores the interconnectedness of temperature, pressure, and kinetic energy in gas behavior.