Question

Relation between the three types of velocities, i.e., most probable velocity : average velocity : root mean square velocity is (a) \(\sqrt{3}: \sqrt{2}: \sqrt{\frac{8}{\pi}}\) (b) \(\sqrt{3}: \sqrt{2}: \sqrt{8}\) (c) \(\sqrt{2}: \sqrt{(8 / \pi)}: \sqrt{3}\) (d) \(1: 2: 3\)

Short answer

The correct answer is option (c): \(\sqrt{2}: \sqrt{(8 / \pi)}: \sqrt{3}\).

Step-by-step solution

Understanding the Problem

We need to find the ratio of the three types of molecular velocities: the most probable velocity \(v_p\), the average velocity \(\overline{v}\), and the root mean square velocity \(v_{rms}\). In this context, these velocities are important in the study of gases and are derived from the Maxwell-Boltzmann distribution.

Identify Formulas for Each Velocity

- Most probable velocity, \(v_p = \sqrt{\frac{2kT}{m}}\).- Average velocity, \(\overline{v} = \sqrt{\frac{8kT}{\pi m}}\).- Root mean square velocity, \(v_{rms} = \sqrt{\frac{3kT}{m}}\). Here, \(k\) is the Boltzmann constant, \(T\) is the temperature, and \(m\) is the mass of a molecule.

Calculate the Ratios

To find the ratio between these velocities, compare their formulas:- \(v_p = \sqrt{2} \cdot \sqrt{\frac{kT}{m}}\)- \(\overline{v} = \sqrt{\frac{8}{\pi}} \cdot \sqrt{\frac{kT}{m}}\)- \(v_{rms} = \sqrt{3} \cdot \sqrt{\frac{kT}{m}}\)Thus, the ratio becomes \(\sqrt{2} : \sqrt{\frac{8}{\pi}} : \sqrt{3}\).

Choose the Correct Option

The problem presents different options for these ratios. From our calculations, the ratios are \(\sqrt{2}: \sqrt{(8 / \pi)}: \sqrt{3}\). Comparing with the given options, we find that option (c) \(\sqrt{2}: \sqrt{(8 / \pi)}: \sqrt{3}\) correctly matches our calculated answer.

Key concepts

Maxwell-Boltzmann distribution

The Maxwell-Boltzmann distribution is a statistical tool used to describe the distribution of speeds among molecules in a gas. It tells us how molecular speeds are spread over a range, depending on factors like temperature and mass of the molecules. Imagine a bell-shaped curve representing the speeds of gas molecules; most molecules will have moderate speeds while fewer will be much faster or slower. This distribution helps in understanding the kinetic behavior of gases and predicting how molecules will behave under different conditions, like changes in temperature. Molecular velocity in gases is essentially a random phenomenon, and the Maxwell-Boltzmann distribution provides a probability distribution for the speeds. Its shape is characterized by the temperature of the gas: as the temperature increases, the curve flattens and expands, indicating higher speeds are more probable and spread out over a broader range. This distribution is foundational for deriving expressions for different characteristic velocities of gas molecules, such as the most probable, average, and root mean square velocities.

most probable velocity

The most probable velocity is a crucial concept when analyzing the behavior of gases through the lens of the Maxwell-Boltzmann distribution. It refers to the speed at which the largest number of molecules in a gas are moving. Imagine it as the peak of the bell-shaped curve of the Maxwell-Boltzmann distribution. The formula for the most probable velocity is given by:\[v_p = \sqrt{\frac{2kT}{m}}\]where:- \(v_p\) is the most probable velocity,- \(k\) is the Boltzmann constant,- \(T\) is the temperature in Kelvin,- \(m\) is the mass of one molecule of the gas. Because it pinpoints the speed of the largest population of molecules, identifying the most probable velocity helps in predicting and understanding the overall behavior of a gaseous sample.

average velocity

The average velocity of gas molecules is another important concept derived from the Maxwell-Boltzmann distribution. It represents the mean speed of all the molecules in a gas sample. While the most probable velocity gives the peak point of speeds, the average velocity provides a more general sense of how fast, on average, the molecules are moving. The formula to calculate the average velocity is:\[\overline{v} = \sqrt{\frac{8kT}{\pi m}}\]where:- \(\overline{v}\) is the average velocity,- \(k\) is the Boltzmann constant,- \(T\) is the temperature in Kelvin,- \(m\) is the mass of one molecule of the gas,- \(\pi\) represents the mathematical constant (approx. 3.14159). The average velocity is slightly higher than the most probable velocity since it accounts for the tail of the distribution where higher speeds occur, thus giving a fuller picture of molecule movement in the gas.

root mean square velocity

The root mean square velocity is a concept that quantifies the speed of molecules in a gas. Unlike the average velocity, the root mean square velocity accounts for the distribution of molecular speeds by considering the square of each speed, providing a measure of the magnitude of their movement. The formula for root mean square velocity is:\[v_{rms} = \sqrt{\frac{3kT}{m}}\]where:- \(v_{rms}\) is the root mean square velocity,- \(k\) is the Boltzmann constant,- \(T\) is the temperature in Kelvin,- \(m\) is the mass of one molecule of the gas. Root mean square velocity is typically the highest among the three mentioned velocities. This is because it accounts for higher velocities more prominently by squaring them, thus weighting them more heavily than lower ones. It is particularly useful in kinetic energy calculations of gases because it correlates directly with the kinetic energy of the molecules. Understanding this velocity is critical for a deep comprehension of the kinetic theory of gases and molecular dynamics.