91Ó°ÊÓ

The Ideal Gas Law is a fundamental principle in chemistry and physics used to describe the behavior of gases. It is expressed by the equation \( PV = nRT \), where:

• \( P \) represents the pressure of the gas.
• \( V \) is the volume occupied by the gas.
• \( n \) is the number of moles of gas.
• \( R \) is the universal gas constant \( (8.314 \text{ J/mol.K}) \).
• \( T \) is the temperature in Kelvin.

This law helps us relate the pressure, volume, and temperature of an ideal gas. An ideal gas is a theoretical gas that perfectly follows these conditions without interactions between particles.

In the context of our exercise, we are tasked with finding the gas pressure in a sealed box. The problem provides information about the average kinetic energy of the gas, from which we calculate the temperature. Using the equation \( P = n k T \), where \( n \) is the number of atoms per unit volume and \( k \) is the Boltzmann constant, allows us to estimate the pressure efficiently without directly using the total volume.

The root mean square speed \( \upsilon_{rms} \) is a measure of the average speed of particles in a gas. Mathematically, it is expressed as:

• \( \upsilon_{rms} = \sqrt{\dfrac{3kT}{m}} \)

Here, \( k \) is the Boltzmann constant, \( T \) is the temperature of the gas in Kelvin, and \( m \) is the mass of a single gas atom.

Finding \( \upsilon_{rms} \) involves converting the molar mass of neon to an individual atom's mass. This is done by dividing the molar mass by Avogadro's number, giving us a per-atom mass in kilograms.

In our problem, the temperature was determined in a previous step using the average kinetic energy of gas particles. Once you have both the temperature and atomic mass, you can substitute these values into the \( \upsilon_{rms} \) formula to find the average speed of gas atoms. This concept is crucial because it ties together kinetic energy, temperature, and the velocity of particles.

The average kinetic energy of gas particles is central to understanding molecular motion in gases. Defined by the formula:

• \( KE_{avg} = \frac{3}{2} kT \)

This expression shows the direct proportionality between temperature \( T \) and kinetic energy, tied together by the Boltzmann constant \( k \).

The key takeaway is that as the temperature increases, the average kinetic energy of gas molecules also rises. For example, the given kinetic energy in the problem helps us back-calculate the temperature of the gas, which is then used to find both pressure and the root mean square speed.

Understanding this relationship enables predictions about other properties of gases such as their speed and pressure changes concerning temperature fluctuations. When linked with the Ideal Gas Law, it helps in conducting a more detailed analysis of gas behaviors in various conditions.

The Boltzmann Constant \( k \) acts as a bridge between macroscopic and microscopic physics. Its value, \( 1.38 \times 10^{-23} \text{ J/K} \), precisely relates the average kinetic energy of particles in a gas with the temperature.

• \( k = \dfrac{R}{N_A} \)

Here, \( R \) is the universal gas constant and \( N_A \) is Avogadro's number. This arises because while \( R \) is used for one mole of gas, \( k \) is applicable on a per-particle basis.

In solving the exercise, the Boltzmann constant played an essential role in both calculating the temperature from kinetic energy and determining the root mean square speed. By quantifying how much energy each degree of freedom contributes, the Boltzmann constant gives profound insights into thermodynamics and statistical mechanics.

This constant is crucial when analyzing the thermodynamic properties of gases, as seen in our problem. It provides the quantitative link needed to navigate between the microscopic world of particle behavior and the macroscopic concepts of temperature and pressure.