Question

The ratio among most probable velocity, mean velocity and root mean square velocity is given by (a) \(1: 2: 3\) (b) \(1: \sqrt{2}: \sqrt{3}\) (c) \(\sqrt{2}: \sqrt{3}: \sqrt[1]{8 / \pi}\) (d) \(\sqrt{2}: \sqrt{8 / \pi}: \sqrt{3}\)

Short answer

The correct ratio among most probable velocity, mean velocity, and root mean square velocity is \(\sqrt{2}: \sqrt{8 / \pi}: \sqrt{3}\), which is option (d).

Step-by-step solution

Understanding Molecular Velocities

In a sample of a gas there are three important velocities to consider: the most probable velocity (\(v_p\)), the mean velocity (\(v_m\)), and the root mean square velocity (\(v_{rms}\)). These velocities are derived from the Maxwell-Boltzmann distribution and are given by the following formulas:nmost probable velocity, \(v_p = \sqrt{(2kT/m)}\)nmean velocity, \(v_m = \sqrt{(8kT/\pi m)}\)nroot mean square velocity, \(v_{rms} = \sqrt{(3kT/m)}\)where \(k\) is the Boltzmann's constant, \(T\) is the absolute temperature, and \(m\) is the mass of a gas molecule.

Calculate Relative Velocities

To compare the velocities, we should calculate their ratios, taking the most probable velocity as a reference. Using the formulas for each velocity type:n\(v_m/v_p = \sqrt{(8/\pi)} / \sqrt{2} = \sqrt{(4/\pi)} = \sqrt{(2/(\pi/2))} = \sqrt{2/\pi)}\)n\(v_{rms}/v_p = \sqrt{3} / \sqrt{2} = \sqrt{3/2}\)nHence, the ratio of the mean velocity to the most probable velocity is \(\sqrt{2/\pi}\) and the ratio of the root mean square velocity to the most probable velocity is \(\sqrt{3/2}\).

Solve the Ratio

Expressing all particles' velocity as a ratio of the most probable velocity, we set the most probable velocity to 1:n\(v_p : v_m : v_{rms} = 1 : \sqrt{2/\pi} : \sqrt{3/2}\)nWhich upon simplification gives:n\(v_p : v_m : v_{rms} = \sqrt{2} : \sqrt{8/\pi} : \sqrt{3}\)nSo the correct ratio is given by option (d).

Key concepts

Maxwell-Boltzmann Distribution

When exploring the behavior of gases, one of the most important concepts is the Maxwell-Boltzmann distribution. It describes how gas molecules with varying amounts of kinetic energy are distributed at a certain temperature. This statistical distribution shows us that not all molecules in a gas move at the same speed—some move slowly, some at average speeds, and others very quickly.

This distribution is crucial for understanding other properties of gases like pressure and temperature, as it explains the spread of molecular speeds in a sample. The shape of the Maxwell-Boltzmann curve is skewed to the right, indicating that there are more slow-moving molecules than very fast ones, but there's still a considerable number of molecules that move faster than the average.

Most Probable Velocity

The most probable velocity is a key concept derived from the Maxwell-Boltzmann distribution. It refers to the speed that the most significant number of molecules in a gas sample is likely to have at a given temperature. It's not the average or the maximum speed, but the peak point on the Maxwell-Boltzmann curve.

Mathematically, the most probable velocity, often represented by the symbol \(v_p\), is calculated using the formula \(v_p = \sqrt{(2kT/m)}\), where \(k\) is the Boltzmann constant, \(T\) is the temperature in Kelvin, and \(m\) is the mass of the gas molecule. This velocity serves as a baseline for comparing other measures of molecular speed.

Root Mean Square Velocity

Root mean square velocity, or RMS velocity, is another term that arises from the distribution of molecular speeds in a gas. Represented by \(v_{rms}\), this value is a type of average velocity that considers the square of the velocities of all the molecules, before finding the square root of the average value.

The formula for the root mean square velocity is \(v_{rms} = \sqrt{(3kT/m)}\). RMS velocity provides important insights into the kinetic energy of the molecules in the gas, as it is directly proportional to it. In essence, \(v_{rms}\) takes into account the distribution of speeds to give a single value representing the 'average' speed in terms of energy content.

Mean Velocity

The term 'mean velocity' refers to the arithmetic mean of the speeds of all the molecules in a gas. It differs from the most probable velocity in that it is not the speed of the majority of molecules, but rather an average considering how fast each molecule is moving. The mean velocity is denoted by \(v_{m}\) and often calculated with the formula \(v_m = \sqrt{(8kT/\text{\(\pi\)} m)}\).

In contrast to the most probable velocity, the mean velocity gives us a sense of the central tendency of molecular speeds; it's akin to the 'balance point' of the system's kinetic energies.

Gas Molecule Velocities

Overall, when we refer to 'gas molecule velocities,' we are contemplating a variety of speeds at which molecules in a gas can travel. These different measures of velocity—most probable, mean, and root mean square—provide us with comprehensive insights into the kinetic behavior of gas particles.

Having an understanding of these different velocities is fundamental in fields like thermodynamics and kinetic theory. It helps us predict how gases will behave under different conditions, such as changes in temperature or volume. The exercise explained earlier showcases how these velocities are interconnected and illustrates the importance of being comfortable with these concepts when solving problems in gas kinetics.