Question

The root mean square velocity of a gas molecule of mass \(\mathrm{m}\) at a given temperature is proportional to (A) \(\mathrm{m}^{0}\) (B) \(\mathrm{m}^{-1 / 2}\) (C) \(\mathrm{m}^{1 / 2}\) (D) \(\mathrm{m}\)

Short answer

The root mean square velocity of a gas molecule with mass m at a given temperature is proportional to \(m^{-1/2}\), as described by the formula \(v_{rms} = \sqrt{\frac{3kT}{m}}\), where \(v_{rms}\) is the root mean square velocity, \(k\) is the Boltzmann constant, \(T\) is the temperature, and \(m\) is the mass of the gas molecule. Comparing this expression to the given options, we find the correct answer is Option (B): \(v_{rms} \propto m^{-1/2}\).

Step-by-step solution

Root Mean Square Velocity Formula

The root mean square velocity \(v_{rms}\) of a gas molecule can be found using the kinetic theory of gases, which relates the molecular mass, temperature, and heat capacity: \[v_{rms} = \sqrt{\frac{3kT}{m}}\] Here, \(v_{rms}\) = root mean square velocity \(k\) = Boltzmann constant \(T\) = temperature \(m\) = mass of the gas molecule

Find Proportionality

Now, let's find the proportionality between the mass of the gas molecule and the root mean square velocity. From the formula, we have: \[v_{rms} \propto \sqrt{\frac{3kT}{m}}\] Since \(3kT\) is a constant term, we can write: \[v_{rms} \propto \frac{1}{\sqrt{m}}\] or equivalently \[v_{rms} \propto m^{-1/2}\] Comparing this expression to the given options, we can see that it matches Option (B): \[v_{rms} \propto m^{-1/2}\] Thus, the root mean square velocity of a gas molecule with mass m at a given temperature is proportional to \(m^{-1/2}\).

Key concepts

Kinetic Theory of Gases

The kinetic theory of gases is a fundamental part of our understanding of how gases behave. It describes a gas as a large number of small particles (atoms or molecules) in constant, random motion. This theory provides insights into the macroscopic properties of gases such as pressure, temperature, and volume, by looking at the microscopic behaviors of molecules.
The core premise of the kinetic theory is that these moving molecules collide with each other and the walls of their container. These collisions are perfectly elastic, meaning there is no loss of energy during the interactions. Because of this constant motion and collision, the gas molecules maintain pressure and distribute temperature uniformly within the system.
Some key points to keep in mind regarding the kinetic theory are:

• The pressure exerted by a gas is due to the collisions of its molecules with the walls of its container.
• Temperature is a measure of the average kinetic energy of the molecules in a gas.
• Molecules in a gas move randomly and independently of one another unless they collide.

Understanding these concepts is pivotal as it allows us to compute properties like root mean square velocity, tying together the micro and macro pictures of gas behavior.

Proportionality

Proportionality helps us understand how two quantities change in relation to each other. Specifically, when we look at root mean square velocity in relation to the mass of gas molecules, we are examining how the velocity changes as the mass changes.
From the formula derived from the kinetic theory, we have
\[v_{rms} = \sqrt{\frac{3kT}{m}}\]
In this context, we can say that the root mean square velocity is inversely proportional to the square root of mass. As mass increases, the velocity decreases according to the relationship \(v_{rms} \propto m^{-1/2}\).
This means:

• Smaller molecules (less molecular mass) tend to move faster.
• Larger molecules (more molecular mass) tend to move slower.

Proportionality simplifies and predicts how changes in one factor affect another, making it a powerful tool in physics.

Molecular Mass

Molecular mass, or molecular weight, refers to the mass of a single molecule of a substance, measured in atomic mass units (amu). It is a vital factor in determining a gas molecule's behavior under various physical conditions.
In the context of gases, molecular mass directly influences properties like diffusion, effusion, and velocity. For example, lighter molecules, due to their smaller mass, can move at higher speeds and diffuse more quickly than their heavier counterparts.
Key aspects regarding molecular mass include:

• The molecular mass of a gas affects how quickly it reacts to changes in temperature and pressure.
• Root mean square velocity is inversely proportional to the square root of molecular mass (as discussed in the proportionality section).
• Understanding molecular mass is crucial for calculations involving stoichiometry, reaction rates, and other chemical reactions.

This understanding helps us analyze and control gas behavior in both natural and industrial processes.

Temperature Dependence

Temperature plays a critical role in the behavior of gases, directly affecting their molecular motion. As temperature increases, the kinetic energy of gas molecules also increases, which in turn affects their speed and the root mean square velocity.
The root mean square velocity formula
\[v_{rms} = \sqrt{\frac{3kT}{m}}\]
shows that velocity is directly proportional to the square root of temperature. Higher temperatures result in faster-moving molecules due to the increased energy.
A few points to note about temperature dependence:

• An increase in temperature leads to an increase in kinetic energy, causing molecules to move more rapidly.
• The root mean square velocity will increase as temperature increases.
• If temperature drops, the molecules slow down, leading to a decrease in pressure as their impact against the container walls reduces.

Grasping the temperature dependence helps to predict and explain changes in gas behavior under different thermal conditions.