91Ó°ÊÓ

The Ideal Gas Law is fundamental in understanding the behavior of gases under various conditions. It is represented by the equation:

• \( PV = nRT \)

where:- \( P \) is the pressure of the gas,- \( V \) is the volume of the gas,- \( n \) is the number of moles of the gas,- \( R \) is the ideal gas constant \( (8.314 \text{ J/(mol} \cdot \text{K)}) \), and- \( T \) is the temperature in Kelvin.
In the case of an isothermal process, the temperature \( T \) remains constant, which allows us to understand how pressure and volume relate to each other. Thus, for constant temperature, the equation can be rearranged:

• \( P_1 V_1 = P_2 V_2 \)

This means that any change in the pressure of the gas will inversely affect the volume if the temperature and amount of gas remain the same. This relationship helps us solve for unknown variables, such as volume changes during gas compression.

An isothermal process is where the temperature remains constant as a gas undergoes a volume change. This concept is particularly interesting because, despite the constant temperature, other properties like pressure or volume can vary.
In our problem, the gas is compressed isothermally. This means that while the external pressure changes from 101 kPa to 121 kPa, the temperature of the gas stays at 295 K throughout. Hence, the energy added or taken from the gas during this process primarily changes its pressure and volume.
Isothermal processes are characterized by a few key points:

• Temperature \((T)\) is constant.
• Internal energy change \((\Delta U)\) is zero because temperature stays the same.
• Heat \((Q)\) added or removed equals the work \((W)\) done by or on the gas.

These characteristics are crucial when using the ideal gas law, as the equations depend on understanding how these properties interact during a process.

In thermodynamics, the work done by the gas during an isothermal process is an essential concept. When a gas changes its volume due to applied or removed heat, it performs work. The work done \((W)\) by the gas during an isothermal process can be determined using the formula:

• \( W = nRT \ln\left(\frac{V_2}{V_1}\right) \)

where \( V_1 \) and \( V_2 \) are the initial and final volumes, respectively.
This relation shows that the work done depends on the natural logarithm of the volume ratio and the constant product of moles, gas constant, and temperature.
It's essential to remember:

• The sign of the work will be positive for expansion and negative for compression.

In our exercise, since the gas is compressed, the work is done on the gas. This means that the work value will be negative, reflecting the reduction in volume.

Heat transfer in an isothermal process is closely tied to the concept of work. In thermodynamics, heat \((Q)\) is energy transferred between systems due to temperature difference. For isothermal processes, understanding this transfer is crucial.
Since the internal energy change \((\Delta U)\) is zero during an isothermal process, the first law of thermodynamics simplifies to:

• \( \Delta U = Q - W \)
• Thus, \( Q = W \)

This implies that the heat added to the system is equal to the work done by the gas when the process is isothermal. In our context, since the gas is compressed, the system loses heat proportionate to the work done, so \( Q = -W \).
This balance ensures that despite the pressure changes, the temperature remains constant due to the energy adjustments caused by heat transfer. Understanding this relationship is key to mastering isothermal processes.