Problem 56 - A Complete Resource Book in Chemistry for JEE Main

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.

Step-by-Step Solution

Answer

The correct statements are (a) and (d).

Step 1: Analyzing Gas Pressure and Velocity

According to the kinetic theory of gases, the pressure \( P \) exerted by a gas is related to the molecules' behavior. The relevant formula used is \( P = \frac{1}{3} \text{density} \times \overline{v^2} \), where \( \overline{v^2} \) is the mean square velocity of gas molecules. Therefore, pressure is directly proportional to the mean square velocity, which makes option (a) correct.

Step 2: Evaluating Pressure and Root Mean Square Velocity

The root mean square velocity \( v_{rms} \) is given by \( v_{rms} = \sqrt{\overline{v^2}} \). Although related, the pressure is not directly proportional to the root mean square velocity but instead to the mean square velocity mentioned in Step 1. Thus, option (b) is incorrect.

Step 3: Considering Root Mean Square Velocity and Temperature

The root mean square velocity is given by \( v_{rms} = \sqrt{\frac{3kT}{m}} \), where \( k \) is the Boltzmann constant, \( T \) is the absolute temperature, and \( m \) is the molecule's mass. From this, we see that \( v_{rms} \) is directly proportional to the square root of temperature, contradicting option (c). Thus, option (c) is incorrect.

Step 4: Relating Kinetic Energy to Temperature

The mean translational kinetic energy per molecule is given by \( KE = \frac{3}{2}kT \), showing that it is directly proportional to the absolute temperature \( T \). Thus, option (d) is correct.

Key Concepts

Mean Square Velocity

In the kinetic theory of gases, mean square velocity is a fundamental concept that describes the average of the squares of the velocities of gas molecules. It is an important measure because it links microscopic particle motion to macroscopic properties like pressure. Pressure exerted by a gas in a container results from collisions of molecules with the walls, and according to the equation: \( P = \frac{1}{3} \cdot \text{density} \cdot \overline{v^2} \), where \( P \) is pressure, the density is the mass per unit volume, and \( \overline{v^2} \) is the mean square velocity. This relationship indicates that pressure is directly proportional to mean square velocity. So, if the mean square velocity increases, the pressure exerted by the gas increases. This understanding helps explain how temperature and energy are intrinsically tied to gas behavior.

Pressure and Velocity Relationship

The relationship between pressure and velocity of gas molecules is critical to understanding gas behavior. As gas molecules move and collide within a container, their velocities determine the pressure exerted on the container walls. This is captured in the equation previously mentioned: \( P = \frac{1}{3} \cdot \text{density} \cdot \overline{v^2} \). In simpler terms:

• Higher velocities lead to more forceful collisions, increasing pressure.
• Pressure is directly related to the mean square of velocities, not root mean square velocity.

By observing this, we can conclude that knowing how velocities change allows predictions about how the pressure of a gas will behave under different circumstances, such as changes in temperature or volume.

Root Mean Square Velocity

Root mean square (RMS) velocity provides another useful way to measure the speed of gas molecules. It is calculated as the square root of the mean square velocity: \( v_{rms} = \sqrt{\overline{v^2}} \). The RMS velocity gives a sense of the average velocity of gas particles and is essential for comprehending kinetic energy. One fact to note is that RMS velocity is directly related to the temperature of the gas. The formula for the RMS velocity, \( v_{rms} = \sqrt{\frac{3kT}{m}} \), shows that as absolute temperature \( T \) increases, the RMS velocity also increases. The variable \( k \) represents the Boltzmann constant, and \( m \) is the mass of a molecule. Thus, option (c) is incorrect in stating an inverse relationship between RMS velocity and temperature.

Mean Translational Kinetic Energy

Mean translational kinetic energy is a simple yet powerful concept explained by the kinetic theory of gases. Every molecule in a gas has some kinetic energy due to its motion, and this energy impacts the behavior of gases significantly. The mean translational kinetic energy for a gas molecule is expressed as: \( KE = \frac{3}{2}kT \), where \( k \) is the Boltzmann constant and \( T \) is the absolute temperature. This equality demonstrates that mean kinetic energy is directly proportional to the absolute temperature. Thus, as temperature increases, so does the kinetic energy of the gas particles. This relationship underscores the idea that gases behave more energetically as they are heated, explaining numerous everyday phenomena related to thermal expansion and temperature-dependent reactions.

Absolute Temperature

Absolute temperature is a critical element when studying the behavior of gases. It is measured on a scale beginning at absolute zero, the theoretical point where molecules have minimum kinetic energy. Absolute temperature is often measured in Kelvin (K), which is convenient for scientific calculations because it avoids negative numbers.In the kinetic theory of gases:

• Absolute temperature is directly tied to the energetic activity of gas molecules.
• It provides a scale where one unit (1 Kelvin) corresponds to the same energy difference.

Absolute temperature is central to equations like those concerning RMS velocity and kinetic energy, as seen in the equations \( v_{rms} = \sqrt{\frac{3kT}{m}} \) and \( KE = \frac{3}{2}kT \). Understanding absolute temperature helps connect theoretical concepts with practical observations, explaining how molecular activity changes with adjustments in temperature.