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An ideal gas is a theoretical model that simplifies the study of gas behaviors under various conditions. In this model, gas particles are assumed to have no volume and do not exert forces on one another, except when they collide. This assumption allows us to predict the behavior of gases using mathematical equations. One crucial equation in thermodynamics for ideal gases is the Ideal Gas Law, represented as \( PV = nRT \), where:

• \( P \) is the pressure
• \( V \) is the volume
• \( n \) is the number of moles
• \( R \) is the ideal gas constant
• \( T \) is the temperature

For our exercise, it is important to understand that helium gas behaves as an ideal gas, simplifying our calculations on changes in internal energy.

Work and energy are foundational concepts in physics that describe how forces and energy interact. When you perform work on a system, you transfer energy to or from that system. This exercise considers the work done by applying a force to compress or expand a gas. Specifically, when work is done on the gas, energy is transferred to the gas. However, in this context, it leads to changes in the internal energy of the gas, without involving changes in heat due to the insulation. Understanding that work involves changes in energy helps us analyze how energy transformations affect gas properties in systems, such as our perfectly insulated helium gas case.

The first law of thermodynamics is a fundamental principle that governs the behavior of energy in physical systems. It states that energy cannot be created or destroyed, only transformed from one form to another. In mathematical terms, it's expressed as:\[\Delta U = Q - W\]Where:

• \( \Delta U \) represents the change in internal energy of the system
• \( Q \) is the heat added to the system
• \( W \) is the work done by the system

For our problem with a perfectly insulated container, no heat transfer occurs (\( Q = 0 \)). Therefore, the change in internal energy \( \Delta U \) is essentially \(-W\), the negative of the work done on the system. This principle highlights how work directly influences the internal energy of the gas in an insulated system.

The internal energy change of a gas is a crucial concept in thermodynamics, representing the total energy stored within the gas molecules. This energy comprises kinetic energy, due to the motion of the particles, and potential energy resulting from interactions between particles.In our exercise, because the helium gas is in a perfectly insulated container, no heat is exchanged with the surroundings. Therefore, the entire focus is on the work done on the gas, which changes the internal energy. Here, the work performed (\(457 \text{ J}\)) is the only factor altering the internal energy, leading to a decrease by that amount. This is reflected in the equation \( \Delta U = -457 \text{ J} \), explaining a reduction in energy, as the gas has absorbed work performed on it without a heat exchange.