Question

STATEMENT-1 The total translational kinetic energy of all the molecules of a given mass of an ideal gas is \(1.5\) times the product of its pressure and its volume. because STATEMENT-2 The molecules of a gas collide with each other and the velocities of the molecules change due to the collision. (A) Statement- 1 is True, Statement- 2 is True; Statement- 2 is a correct explanation for Statement-1 (B) Statement- 1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1 (C) Statement- 1 is True, Statement- 2 is False (D) Statement-1 is False, Statement-2 is True

Short answer

(B) Statement- 1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1

Step-by-step solution

Analyzing Statement-1

Statement-1 says that the total translational kinetic energy of all the molecules of a given mass of an ideal gas is 1.5 times the product of its pressure and its volume. The kinetic theory of gases states that the average translational kinetic energy of a mole of gas is given by (3/2)kT, where k is the Boltzmann constant and T is the temperature. According to the ideal gas law, PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the gas constant, and T is the temperature. By equating (3/2)nRT to 1.5PV, it's concluded that this statement is consistent with the kinetic theory of gases.

Analyzing Statement-2

Statement-2 tells us about the collisions between gas molecules. While it is true that gas molecules collide with each other and the velocities may change due to these collisions, this statement is more related to the random and unpredictable nature of gas molecules rather than explaining the relationship between kinetic energy, pressure and volume.

Determining the Relationship Between the Statements

To establish the relationship between the two statements, one must determine whether the collision of gas molecules and the change in their velocities is directly related to the translational kinetic energy being 1.5 times the product of pressure and volume. Statement-2 does not provide a causal explanation for Statement-1, rather it describes a different aspect of gas behavior.

Key concepts

Translational Kinetic Energy

When discussing the movement of particles in a gas, we refer to translational kinetic energy. This is a type of kinetic energy that results from the motion of particles as they travel in straight lines in different directions and at various speeds within a container. As per the kinetic theory of gases, the total translational kinetic energy of the gas molecules is directly proportional to the temperature of the gas. More precisely, for an ideal gas, the average kinetic energy per molecule is given by the formula \(\frac{3}{2}kT\), where \(k\) is the Boltzmann constant and \(T\) represents the temperature in Kelvin. This energy accounts for the motion in all three spatial dimensions, reflecting the fact that gas molecules move freely in space without any specific direction or pattern.

This property has significant consequences, as it is correlated with measurable quantities such as the gas’s temperature, pressure, and volume. Knowing that the energy is proportional to the temperature, we can deduce that an increase in the energy would imply an increase in the temperature. This understanding is essential when studying gases and their behaviors in different conditions, especially when analyzing how they respond to changes in their environment.

Ideal Gas Law

The ideal gas law is a fundamental equation that describes the state of an ideal gas by relating its pressure (P), volume (V), temperature (T), and amount of substance in moles (n). The equation is given as \(PV = nRT\), where \(R\) is the universal gas constant. This equation assumes that the gas particles are point masses with no volume and that there are no intermolecular forces affecting them; hence the term 'ideal.'

In the context of translational kinetic energy, the ideal gas law integrates the microscopic view provided by kinetic theory with macroscopic observations, enabling us to describe the state of a gas in a comprehensive manner. Since the law includes the temperature of the gas and the number of moles, it indirectly speaks of the average kinetic energy of the molecules because, as we know from the theory, temperature is a measure of kinetic energy. Understanding the ideal gas law is crucial for predicting how a gas will behave under various conditions and for performing calculations related to gas processes in industrial, environmental, and biological systems.

Molecular Collisions

The concept of molecular collisions is a key component in the kinetic theory of gases and plays an instrumental role in understanding gas behavior. Gas molecules are in constant random motion, and because they are tiny and far apart, they frequently collide with each other and with the walls of their container. These collisions are perfectly elastic, meaning that the total kinetic energy is conserved—though energy is redistributed among the molecules.

It’s the collisions with the container walls that result in the observable pressure exerted by the gas. When these collisions are analyzed statistically, they explain how the macroscopic properties of gases, such as pressure and temperature, arise from microscopic events. Although these collisions are crucial to the properties of gases, they do not individually account for the relationship between kinetic energy, pressure, and volume as described in our initial statements. Rather, it is the collective effect of countless collisions that correlates with the average kinetic energy of the molecules and hence the temperature and pressure of the gas. By integrating the ideas of molecular collisions with the broader concepts of kinetic energy and gas laws, students can attain a holistic understanding of gas dynamics.