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The internal energy of a gas represents the total energy contained within it, originating from the motion of its molecules. This energy is substantial because it encompasses all the kinetic energy of the atoms or molecules within the gas. For ideal gases, especially monatomic gases, the internal energy can be calculated with the formula: \[ U = \frac{3}{2} nRT \]Here, \( n \) is the number of moles of gas, \( R \) is the ideal gas constant (8.31 J/(mol·K)), and \( T \) is the absolute temperature in Kelvin. The factor \( \frac{3}{2} \) is used for monatomic gases, reflecting that all the energy is in the form of translational kinetic energy. In our context, with helium being a monatomic gas at room temperature (298 K), we found the internal energy to be approximately 0.928 Joules. Understanding internal energy is crucial, as it helps explain how energy is stored in gases and how they respond to changes in conditions like temperature and pressure.
Knowing the internal energy allows us to link microscopic molecular behavior to macroscopic thermodynamic properties, essential for various applications in science and engineering.

The concept of moles is fundamental in chemistry and physics, providing a way to quantify the amount of substance present. One mole corresponds to Avogadro's number, which is approximately 6.022 x 10²³ particles, whether they be atoms, molecules, or ions. In this exercise, we use the ideal gas law, which is given by:\[ PV = nRT \]Where \( P \) is the pressure of the gas, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is the temperature. Here, solving for \( n \), the moles of gas, we substituted the known values for pressure, volume, and temperature to find that there are approximately 0.00025 moles of helium in the container. This calculation is pivotal because many other properties, including the internal energy of the gas, depend on the number of moles. By knowing how many moles are present, we can predict and calculate other important properties and behaviors of the gas.

Monatomic gases are composed of single atoms, in contrast to diatomic or polyatomic gases, which are made up of molecules consisting of two or more atoms. Common examples include noble gases like helium, neon, and argon. These gases are significant in physics because they follow the ideal gas law under a broad range of conditions, serving as models for understanding gas behavior. A key property of monatomic gases is that their internal energy is purely translational kinetic energy. This means each atom moves independently in three-dimensional space, without contributing rotational or vibrational kinetic energies, which is different for more complex molecules. The simplicity of monatomic gases results in straightforward calculations for properties such as internal energy, which depends only on temperature and moles, not on additional vibrational or rotational contributions. Such characteristics make monatomic gases ideal for studying basic thermodynamic principles and using the ideal gas law to solve a variety of real-world engineering problems.