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The Ideal Gas Law is a fundamental equation in thermodynamics, expressed as \(PV = nRT\). It relates the pressure \(P\), volume \(V\), and temperature \(T\) of an ideal gas, with \(n\) representing the amount of substance (in moles), and \(R\) being the gas constant. This law is essential in determining the behavior of gases under various conditions. In the given exercise, we use the Ideal Gas Law to calculate the initial volume of the gas. By rearranging the formula as \(V = \frac{nRT}{P}\), we can input the initial conditions to find the volume. This calculation is crucial as it sets the stage for understanding how the gas will behave when it expands into the evacuated compartment.

An adiabatic process is one in which no heat transfer occurs between the system and its surroundings. This means all changes in a system's state are purely due to internal energy variations, particularly work done by or on the system. In the given problem, the tank is well-insulated, indicating an adiabatic process as no heat can enter or leave. Consequently, the gas's temperature remains constant during expansion \(T_{\text{initial}} = T_{\text{final}}\). This principle helps simplify the calculation since we only consider work done by the expanding gas without accounting for heat transfer.

Exergy represents the maximum useful work obtainable from a system. Exergy destruction occurs due to irreversibilities in real processes, signifying wasted potential. It quantifies energy that could not be converted into useful work. To evaluate exergy destruction for the solution, we use: \[X_{\text{destroyed}} = T_0 \times \text{Δ}S_{\text{universe}} \]. Here, \(T_0\) is the reference temperature provided, and \(ΔS_{\text{universe}}\) is the total entropy change. The reference temperature \(T_0 = 70^\text{°F}\) is used for this calculation, which gives a clear picture of how much potential work was lost due to the rapid, irreversible expansion of the gas.

Entropy change \( \text{Δ}S \) is a measure of disorder or randomness in a system. In thermodynamics, it helps indicate the irreversibility of processes and the amount of useful energy converted into waste. In the exercise, the gas expands into an evacuated volume leading to an increase in entropy. Since the process is adiabatic, changes in entropy are related to changes in volume and pressure. Calculating \( \text{Δ}S_{\text{universe}} \) helps in determining the exergy destruction and understanding the extent of irreversibility. Using the entropy balance and the initial and final states of the gas, we determine the increase in entropy due to expansion. This step is critical for evaluating the thermal efficiency and environmental impact of the process.