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The ideal gas law is a fundamental principle in the study of gases. It relates the pressure (P), volume (V), temperature (T), and amount of gas (n) through a simple equation:

• PV = nRT

In this equation, R is the universal gas constant, a value that remains consistent across different scenarios involving ideal gases.
In the given exercise, we utilize this law to determine the initial and final volumes of the gas.
By plugging in the initial conditions (pressure, temperature, and number of moles), we can calculate the initial volume (V鈧�). Similarly, using the final temperature and pressure, we can compute the final volume (V鈧�). This understanding helps us set the stage for further calculations in thermodynamic processes like adiabatic expansion.

The first law of thermodynamics is a key concept describing the conservation of energy. It states that the total energy in a closed system is constant. In simpler terms, energy cannot be created or destroyed; it can only change forms. This is expressed mathematically as:

• 螖U = q + w

where 螖U represents the change in the internal energy of the system, q is the heat exchanged, and w is the work done.
In an adiabatic process, like the one described in the exercise, there is no heat exchange with the surroundings. This makes q = 0, simplifying the equation to 螖U = w.
Thus, any work done by or on the gas leads to a change in its internal energy, underscoring the first law's role in energy transformation during gas expansion.

Internal energy is the total energy contained within a system, due to both its molecular motion and its intermolecular forces. In the context of gases, it is primarily affected by temperature changes. For an ideal gas, the change in internal energy (螖U) is calculated using the expression:

• 螖U = nC_{V, m}螖T

where n is the number of moles, C_{V, m} is the molar heat capacity at constant volume, and 螖T is the temperature change.
Given that the process in the exercise is adiabatic, the work done by the gas reflects directly as a change in internal energy.
This relationship emphasizes how temperature variations impact the energy dynamics within a gas system as it expands or contracts adiabatically.

Enthalpy change, often symbolized as 螖H, represents the total heat content change in a system under constant pressure. Though this exercise involves adiabatic expansion with no heat exchange, we can still compute 螖H using the relationship:

• 螖H = 螖U + P螖V

Here, 螖U is the change in internal energy, P is the pressure, and 螖V is the change in volume of the gas. The relation for 螖H highlights how changes in internal energy and volume work together to determine the enthalpy change.
For ideal gases, another useful relation emerges as part of this process:

• 螖H = nC_p, m螖T

where C_p, m is the molar heat capacity at constant pressure.
This is applicable because the change in enthalpy can be directly related to the change in temperature, even in adiabatic processes, underscoring the inherent connection between heat content, energy, and pressure-volume work.