91Ó°ÊÓ

The Ideal Gas Law is a fundamental equation in thermodynamics that describes the relationship between the pressure, volume, temperature, and amount of an ideal gas. It is expressed as:

• \( PV = nRT \)

Where:

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

In the given exercise, helium gas is treated as an ideal gas, allowing the use of the Ideal Gas Law to find the initial pressure and solve various parts of the problem. It helps predict how the gas will behave under different conditions and forms the basis for understanding processes like expansion and compression.

An adiabatic process is one in which no heat is exchanged between the system and its surroundings. In other words, the heat transfer \( Q = 0 \). During an adiabatic process, changes in temperature and volume occur without any heat loss or gain.

• A key feature of adiabatic processes is the relationship \( TV^{\gamma-1}= \text{constant} \)

Here, \( \gamma \) is the heat capacity ratio \( \frac{C_p}{C_v} \), which is \( \frac{5}{3} \) for helium because it is a monoatomic ideal gas.
In the exercise, after expanding at constant pressure, the helium undergoes adiabatic expansion. This means volume and temperature changes are governed by the adiabatic relationship until the temperature returns to its initial value. By leveraging the relationship between temperature and volume during adiabatic expansion, the final volume, after all changes, is determined.

Internal energy refers to the energy contained within a system that is associated with the random motion of molecules. For ideal gases, it depends only on temperature. The change in internal energy is given by:

• \( \Delta U = nC_v \Delta T \)

Where:

• \( \Delta U \) is the change in internal energy
• \( n \) is the number of moles
• \( C_v \) is the molar specific heat at constant volume, given as \( \frac{3}{2}R \) for monoatomic gases like helium
• \( \Delta T \) is the change in temperature

In this exercise, the temperature of the helium returns to its original state after the adiabatic expansion, indicating \( \Delta T = 0 \). Therefore, there is no change in internal energy, and \( \Delta U = 0 \) J. This concept helps understand energy conservation during ideal gas transformations.

The work done by a gas during a thermodynamic process can be determined by considering the pressure and change in volume. For a process at constant pressure, the work done \( W \) is given by:

• \( W = P \Delta V \)

Where:

• \( P \) is the constant pressure during the process
• \( \Delta V \) is the change in volume

In the provided exercise, the helium expands at constant pressure first, leading to the calculation of the work done as \( W = 4988.4 \) J. This work is significant because it represents the energy transferred from the gas by doing work on its surroundings. Understanding work done by gas helps in analyzing energy transformations and mechanical work conversions within thermodynamic cycles.