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The First Law of Thermodynamics is a fundamental concept in physics that describes the conservation of energy. It states that the change in the internal energy of a system is equal to the amount of heat added to the system minus the work done by the system on its surroundings. In simpler terms, it can be written as: \[ \Delta U = Q - W \] Where:

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

In an adiabatic process, like the one described in the exercise, there is no heat exchange with the surroundings which makes \( Q = 0 \). This simplifies the equation to \( \Delta U = -W \). Hence, any change in internal energy is a result of the work done by or on the system.
Understanding this law, especially in adiabatic processes, is critical as it helps describe how energy transformations occur without external heat transfer.

A monatomic ideal gas is a theoretical gas composed of single-atom gas particles, and it assumes that the particles do not interact with each other except via elastic collisions. The ideal gas law, which applies to monatomic gases, simplifies how we understand and predict their behavior.
This ideal scenario follows the equations: - **Equation of State**: \[ PV = nRT \]

• Where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the ideal gas constant (8.314 J/mol·K), and \( T \) is temperature.

- **Degrees of Freedom**: A monatomic gas has three degrees of translational freedom, making its specific heat capacities quite specific.These equations form the basis for exploring changes in an ideal gas's state, particularly in cases of temperature fluctuation or volume alterations, such as in an adiabatic expansion.

The change in internal energy \( \Delta U \) of a gas is directly related to its temperature change and is crucial for understanding energy dynamics. In the context of a monatomic ideal gas, this change is calculated using the formula:\[ \Delta U = \frac{3}{2} nR \Delta T \]Where:

• \( n \) is the number of moles
• \( R \) is the ideal gas constant
• \( \Delta T \) is the change in temperature

Here, the factor \( \frac{3}{2} \) comes from the three translational degrees of freedom in a monatomic gas. It reflects how the energy is distributed among these degrees of freedom, affecting kinetic energy and, by extension, temperature.In the example, this calculation considers a specific drop in temperature. Therefore, identifying how internal energy changes is vital for determining associated energetic phenomena.

In thermodynamic processes, particularly notable in adiabatic expansions, the concept of work done by the gas plays a key role. Work \( W \) involves energy transfer that results when a gas expands or compresses against a force. For gases, this can be calculated by integrating the pressure with respect to volume.In adiabatic conditions where no heat is exchanged, the work done by the gas is directly responsible for changes in internal energy. Specifically, using the First Law of Thermodynamics, the relationship is: \[ W = - \Delta U \]Thus, in this exercise, calculating work becomes straightforward once the energy change is known, which was determined through temperature variations. The work done is positive if the gas expands, meaning it does work on its surroundings. Hence, achieving clarity on such terms provides comprehensive insights into the mechanics of gas behaviors under various thermodynamic processes.