Problem 76 - Chemistry: The Central Science in SI Units, Global Edition

Question

Indicate which of the following statements regarding the kinetic-molecular theory of gases are correct. (a) The average kinetic energy of a collection of gas molecules at a given temperature is proportional to \(m^{1 / 2}\). (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

Statements (b) and (d) are correct. In the kinetic-molecular theory of gases, gas molecules are assumed to exert no forces on each other (statement b), and the volume of the gas molecules is negligible in comparison to the total volume in which the gas is contained (statement d). Statements (a), (c), and (e) are incorrect.

Step 1: Statement a: Average kinetic energy

Statement a says that the average kinetic energy of a collection of gas molecules at a given temperature is proportional to \(m^{1/2}\). This is incorrect. 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, not the square root of their mass.

Step 2: Statement b: No forces between gas molecules

Statement b states that gas molecules are assumed to exert no forces on each other. This is correct. One of the main assumptions of the kinetic-molecular theory of gases is that gas particles do not experience any long-range forces, which allows gases to easily expand to fill any container they are placed in.

Step 3: Statement c: Same kinetic energy at a given temperature

Statement c says that all the molecules of a gas at a given temperature have the same kinetic energy. This is incorrect. Although the average kinetic energy of all the molecules within a gas at a given temperature is the same, individual gas particles can have a wide range of kinetic energies due to their varying velocity.

Step 4: Statement d: Negligible volume of gas molecules

Statement d states that the volume of the gas molecules is negligible in comparison to the total volume in which the gas is contained. This is correct. According to the kinetic-molecular theory of gases, the individual gas particles are assumed to have no volume in relation to the space they occupy. This allows us to approximate the behavior of gases in many situations.

Step 5: Statement e: Same speed at the same temperature

Statement e states that all gas molecules move with the same speed if they are at the same temperature. This is incorrect. While the average speed of gas particles is related to the temperature, individual gas particles can have a wide range of speeds due to their varying masses and initial speeds. The distribution of their speeds is given by the Maxwell-Boltzmann distribution.

Key Concepts

Average Kinetic Energy

The average kinetic energy of gas molecules is a core concept in the kinetic-molecular theory of gases. It is important to understand that at a given temperature, the average kinetic energy of gas particles is directly proportional to the absolute temperature (measured in Kelvin). This means that as the temperature increases, the average kinetic energy of the gas particles also increases. This relationship is mathematically represented as \( KE_{avg} = \frac{3}{2}kT \), where \( k \) is the Boltzmann constant and \( T \) is the temperature.
It is a common misconception that the average kinetic energy is dependent on other factors like molecular mass (\( m \)). However, according to the kinetic-molecular theory, the temperature alone governs the average kinetic energy, not the mass of the molecules.
This principle highlights why all gases, regardless of the type of particles they are made of, tend to behave similarly under the same conditions of temperature and pressure.

Forces Between Gas Molecules

In kinetic-molecular theory, it is assumed that gas molecules do not exert any forces on each other. This assumption allows gas particles to move freely and is why gases can fill any container they are placed in, as they spread out evenly without attracting or repelling each other.
There are no significant attractions or repulsions between gas particles within the gas, which means they can collide without any loss of kinetic energy, a condition known as perfectly elastic collisions.
This absence of intermolecular forces is one of the main differences between gases and other states of matter like liquids and solids, where such interactions are significant and dictate their structure and properties.

Volume of Gas Molecules

The kinetic-molecular theory assumes that the volume of the individual gas particles is negligible compared to the total volume of the gas. This means that, for a large number of particles, their size is so small compared to the distance between them that their actual volume doesn't affect the volume of the gas as a whole.
This assumption is practical and simplifies calculations by considering that gas particles take up no space, thus allowing the assumption that a gas's volume is equal to the volume of its container.
This concept helps explain why gases are highly compressible as the particles are assumed to be tiny with large spaces between them, allowing them to be easily compacted.

Speed of Gas Molecules

It might be intuitive to think that all gas molecules move at the same speed at a given temperature. However, this is not the case. Each molecule in a collection of gas molecules moves with its own speed, which varies widely from one molecule to another.
The average speed of gas molecules is related to the temperature, but individual particle speeds vary due to differences in mass and initial energy. As temperature affects kinetic energy, it does influence average speed, but not all molecules travel at this average speed.
This disparity in speeds is better explained by looking at the Maxwell-Boltzmann distribution, which shows the spread of speeds within a gas sample at a specific temperature.

Maxwell-Boltzmann Distribution

The Maxwell-Boltzmann distribution provides a statistical means of describing the range of speeds within a gas at a certain temperature. While temperature gives us the average kinetic energy, this distribution tells us about the variability in the speeds of gas molecules.
It shows that most molecules have speeds around an average (or most probable) speed, but there are also molecules with much higher or lower speeds.
The shape of the Maxwell-Boltzmann curve changes with temperature: at higher temperatures, the distribution widens, indicating a greater range of particle speeds and a higher average speed. This is why even at the same temperature, not all molecules have the same speed, emphasizing the diversity of kinetic energies within any sample of gas.