Question

According to the kinetic theory of gases (a) the pressure exerted by a gas is proportional to mean square velocity of the molecules (b) the pressure exerted by the gas is proportional to the root mean square velocity of the molecules (c) the root mean square velocity is inversely proportional to the temperature (d) the mean translational K.E. of the molecule is directly proportional to the absolute temperature.

Short answer

Option (a) and (d) are correct: pressure is proportional to mean square velocity and K.E. is proportional to temperature.

Step-by-step solution

Understanding Kinetic Theory Basics

The kinetic theory of gases relates the microscopic motion of gas particles to macroscopic properties like pressure and temperature. It states that gas pressure results from collisions of gas molecules with container walls.

Analyzing Pressure and Mean Square Velocity

The kinetic theory shows that pressure is proportional to the mean square velocity of gas molecules. Mathematically, pressure \( P \) is expressed as \( P \propto \langle v^2 \rangle \), where \( \langle v^2 \rangle \) is the mean square velocity of the molecules.

Examining Pressure and Root Mean Square Velocity

The pressure of a gas is not directly proportional to the root mean square velocity (rms velocity) but to the mean square velocity. Therefore, option (b) is incorrect as it confuses rms velocity with mean square velocity.

Relationship between Root Mean Square Velocity and Temperature

The root mean square velocity \( v_{rms} \) is given by \( v_{rms} = \sqrt{\frac{3kT}{m}} \), where \( k \) is the Boltzmann constant, \( T \) is the temperature, and \( m \) is the mass of a molecule. Root mean square velocity is directly, not inversely, proportional to the square root of temperature.

Mean Translational K.E. and Temperature Relationship

The mean translational kinetic energy (K.E.) of gas molecules is given by \( \langle KE \rangle = \frac{3}{2}kT \). This expression shows that the mean translational K.E. is directly proportional to the absolute temperature \( T \).

Key concepts

Pressure and Mean Square Velocity

In the kinetic theory of gases, pressure is a key concept that is carefully linked to the behavior of gas molecules moving in a container. When we think about how gas molecules move, we can imagine tiny particles zooming around in random directions. These molecules are responsible for exerting pressure on the walls of their container through constant collisions.

• The concept of pressure being related to molecular motion is based on the mean square velocity of these molecules.
• Mathematically, pressure, denoted by \( P \), is proportional to the mean square velocity \( \langle v^2 \rangle \).

This relationship can be expressed as \( P \propto \langle v^2 \rangle \), and it helps us understand that when molecules move faster, with greater velocity, they hit the walls of their container more frequently and with more force. This increased activity results in higher pressure.
To understand this better, keep in mind that the mean square velocity is different from the regular mean velocity. It makes use of squaring the velocities before averaging them, which gives more weight to higher speeds. This is why it's critical in analyzing gas pressure and provides a foundation for further understanding the kinetic theory of gases.

Root Mean Square Velocity and Temperature

The root mean square velocity, often abbreviated as rms velocity, is crucial in connecting molecular motion to temperature. The rms velocity \( v_{rms} \) describes the average velocity of gas molecules, taking into account their speed variety and effect.

• The mathematical expression for root mean square velocity is \( v_{rms} = \sqrt{\frac{3kT}{m}} \).
• Here, \( k \) is the Boltzmann constant, \( T \) represents the absolute temperature, and \( m \) is the mass of a single molecule.

This formula demonstrates that rms velocity is directly proportional to the square root of the absolute temperature \( T \). Simply put, as the temperature rises, the molecules gain more thermal energy, which translates into higher velocities. It's crucial to note that the rms velocity is not inversely proportional to temperature. This understanding underlines how temperature directly influences the speed of gas molecules.
The root mean square velocity helps explain phenomena like how quickly a gas spreads in a room. As the temperature increases, the molecules spread out faster due to their increased velocities. Recognizing this link is essential, especially in predicting and analyzing gas behavior under different temperature conditions.

Mean Translational Kinetic Energy

Another vital concept in the kinetic theory of gases is the mean translational kinetic energy, which helps us understand the energy dynamics at the molecular level. This energy is associated with the motion of the molecules and is deeply linked to temperature.

• The formula that expresses mean translational kinetic energy is \( \langle KE \rangle = \frac{3}{2}kT \).
• This equation highlights that the mean translational kinetic energy is directly proportional to the absolute temperature \( T \).

Here, \( \langle KE \rangle \) is the average kinetic energy of molecules in the gas, \( k \) is the Boltzmann constant, and \( T \) is the absolute temperature in Kelvin.
The significance of this relationship is that it directly links the thermal energy possessed by a gas to its kinetic energy. An increase in temperature results in an increase in the kinetic energy of molecules, reflecting increased molecular motion. Similarly, a decrease in temperature indicates less molecular motion, hence, a reduction in kinetic energy.
Understanding mean translational kinetic energy helps grasp how gases store energy and how this affects their behavior under various thermal conditions. It provides a foundational insight into why gases expand when heated and contract when cooled. Such knowledge is invaluable in fields ranging from meteorology to engineering, as it reveals the intricacies of gas behavior.