Write the expression for the root-mean-square speed for a gas at temperature \(T\). Define each term in the equation and show the units that are used in the calculations

The Boltzmann constant (\( k \)) is a fundamental physical constant that plays a vital role in the kinetic theory of gases. It bridges the macroscopic and microscopic worlds by linking temperature, an observable physical property, to the energy of the microscopic particles that make up a gas. Specifically, the Boltzmann constant relates the average kinetic energy of particles in a gas to the temperature of the gas.

When we consider the root-mean-square (rms) speed of gas molecules, the Boltzmann constant serves as a scaling factor, converting the energy attributed to temperature into the kinetic energy of individual gas molecules. With a value of approximately \(1.38 \times 10^{-23} \, \text{J/K}\), it provides a means of calculating speeds at a molecular level using macroscopic measurements like temperature.

In formulas, the unit for the Boltzmann constant is Joules per Kelvin (J/K), illustrating how energy per degree of temperature is assigned to each molecule. The importance of this constant cannot be overstated, as it is fundamental to statistical mechanics and thermodynamics, offering deep insights into how gases behave at different temperatures.

The gas laws are a set of rules that describe the relationship between the pressure, volume, temperature, and quantity (in moles) of a gas. These laws are based on the assumptions made in the kinetic theory of gases, which state that a gas consists of a large number of small particles (atoms or molecules), all of which are in constant, random motion.

The most familiar of these laws include Boyle’s Law (pressure inversely relates to volume at constant temperature), Charles's Law (volume is directly proportional to temperature at constant pressure), and Avogadro's Law (volume is directly proportional to the number of gas molecules at constant temperature and pressure). The combined gas law, which unites these relationships into a single equation \( P \times V = n \times R \times T \), serves as a cornerstone in understanding gas behavior under various conditions.

In the context of root-mean-square speed, applying the ideal gas law, we can relate the temperature of the gas to the kinetic energy and thus the speed of gas particles. The laws governing the behavior of ideal gases are critical for understanding how temperature influences the kinetic energy and ultimately the rms speed of the gas molecules.

The kinetic theory of gases describes the fundamental particle nature and behavior of gases, explaining the observed macroscopic properties such as pressure and temperature. It posits that gas particles move rapidly in random directions and undergo elastic collisions with other particles and their container.

Through the lens of this theory, the root-mean-square (rms) speed is an essential concept that conveys the average speed of gas particles within a specified volume. The rms speed takes into account the spread of particle velocities and is thereby a more representative measure of particle motion than a simple arithmetic mean. It provides a deeper understanding of the energy distribution among particles in a gas.

To derive the rms speed, the kinetic theory applies principles of energy and momentum. Particles with greater mass or higher speed contribute more to an average measure of energy because kinetic energy depends on both mass and the square of velocity. This is why, when calculating rms speed, we square the velocity, average it, and then take the square root, hence the name 'root-mean-square'.

Understanding the kinetic theory of gases is crucial for comprehending various phenomena, from the behavior of gases in different environments to the efficiency of heat engines, and is a foundational concept in physics and chemistry that underlines the study of thermodynamics and statistical mechanics.