91Ó°ÊÓ

The ideal gas law is a foundational principle in thermodynamics that governs the behavior of ideal gases. It is often represented by the equation:

• \( PV = nRT \)

where:

• \( P \) is the pressure of the gas
• \( V \) is the volume
• \( n \) is the number of moles
• \( R \) is the ideal gas constant, approximately \( 8.314 \, \text{J/mol} \, \text{K} \)
• \( T \) is the absolute temperature in Kelvin

For the ideal gas law to apply, the gas should have low density and high temperature conditions. This law proves useful in determining unknown variables when you have some of the other variables' values. In the context of your exercise, it helps us determine the number of moles of the gas using the initial conditions of pressure and volume, with the process held at a constant temperature.

Entropy is a thermodynamic quantity that measures a system's thermal energy per temperature that isn't available for doing mechanical work. In simpler terms, it measures the disorder or randomness in a system. For an isothermal expansion of an ideal gas, the change in entropy \( \Delta S \) can be calculated using:

• \( \Delta S = nR\ln\left(\frac{V_f}{V_i}\right) \)

Here:

• \( \Delta S \) is the change in entropy
• \( n \) is the number of moles of the gas
• \( R \) is the ideal gas constant
• \( V_f \) and \( V_i \) are the final and initial volumes respectively

Entropy change gives insight into how much energy disperses in a process, which is particularly important in understanding the feasibility and directionality of processes. In your exercise, entropy change was evaluated by determining the final volume during the isothermal process, helping quantify how the disorder in the system increased.

An isothermal process is a type of thermodynamic process where the temperature remains constant (

• \( T = \text{constant} \)

) throughout the entire process. In such processes, while temperature remains unchanged, other properties like pressure and volume can change. Thus, the product of pressure and volume remains constant for an ideal gas:

• \( PV = \text{constant} \)

This idea comes into play in the exercise when the gas expands from a higher pressure to a lower pressure, maintaining a constant temperature. When dealing with isothermal expansions, you see a rise in volume but a drop in pressure, highlighting how the gas occupies more space without altering the thermal energy of the system. Such processes are essential when studying cycles, like those in engines, where fluctuations in pressure and volume occur without changes in heat content due to temperature.

The formula to calculate changes in entropy during processes like isothermal expansions is fundamental in thermodynamics. Entropy change is determined using the logarithmic relationship between final and initial states of the volume,

• giving: \( \Delta S = nR\ln\left(\frac{V_f}{V_i}\right) \).

This formula indicates that entropy is not solely a function of the current state of the system but is path-dependent—particularly the path of heat energy disbursement. If the final volume is larger, the entropy increases due to higher disorder. When applying this formula, ensuring accurate knowledge of the initial and final volumes, and temperature consistency is crucial, as seen in your exercise. This reveals how energy dispersal changes with different processes, central to understanding real-world thermodynamic phenomena.