Understanding Internal Energy of a Perfect Gas

The question asks about the thermodynamic process during which the internal energy of a perfect gas remains unchanged. To answer this, we need to understand how the internal energy of a perfect gas behaves.

For a perfect gas (also known as an ideal gas), the internal energy (U) depends solely on its temperature (T). It does not depend on pressure or volume. This relationship can be expressed mathematically, where the change in internal energy $\Delta U$ is proportional to the change in temperature $\Delta T$:

$$ \Delta U = n C_v \Delta T $$

where:

• $\Delta U$ is the change in internal energy.
• $n$ is the number of moles of the gas.
• $C_v$ is the molar specific heat at constant volume.
• $\Delta T$ is the change in temperature.

From this equation, it is clear that for the internal energy of a perfect gas to remain unchanged ($\Delta U = 0$), the temperature of the gas must remain unchanged ($\Delta T = 0$).

Now, let's examine each of the given thermodynamic processes:

• Isobaric process: This is a process where the pressure (P) remains constant. In an isobaric process, volume and temperature can change. If the volume changes, the temperature typically changes as well (according to the ideal gas law PV = nRT). Since temperature can change, the internal energy of a perfect gas can also change during an isobaric process.
• Isothermal process: This is a process where the temperature (T) remains constant. As we established, the internal energy of a perfect gas depends only on temperature. Therefore, if the temperature is constant, the internal energy of a perfect gas must remain constant throughout an isothermal process. $\Delta T = 0$, hence $\Delta U = 0$.
• Isochoric process: This is a process where the volume (V) remains constant. In an isochoric process, pressure and temperature can change. If the pressure changes, the temperature typically changes. Since temperature can change, the internal energy of a perfect gas can also change during an isochoric process.
• Adiabatic process: This is a process where no heat is exchanged between the system and its surroundings ($Q = 0$). According to the first law of thermodynamics, $\Delta U = Q - W$, where $W$ is the work done by the system. In an adiabatic process, $\Delta U = -W$. Work is done if the volume changes. In an adiabatic process involving a perfect gas, pressure, volume, and temperature all typically change. Since temperature usually changes (unless no work is done, e.g., free expansion into vacuum), the internal energy can change.

Based on the analysis, the only process among the options where the temperature of a perfect gas is guaranteed to remain constant is the isothermal process. Consequently, the internal energy of a perfect gas does not change during an isothermal process.

Summary of Processes and Internal Energy Change for Perfect Gas

Revision Table: Key Thermodynamic Processes

Additional Information: Perfect Gas and Internal Energy

A perfect gas is an idealized model of a gas where the particles are considered point masses with no volume and exert no forces on each other except during elastic collisions. While no real gas is perfectly ideal, this model is useful for understanding thermodynamic processes, especially at low pressures and high temperatures.

The internal energy of a substance is the total energy contained within it, including the kinetic energy of its molecules due to their motion and the potential energy due to the forces between them. For a perfect gas, the potential energy between molecules is considered negligible. Therefore, the internal energy of a perfect gas is solely due to the kinetic energy of its molecules, which in turn depends only on the temperature of the gas. This is a fundamental assumption in the kinetic theory of gases applied to perfect gases.

In summary, the unique property of a perfect gas where internal energy depends only on temperature is key to understanding its behavior during different thermodynamic processes. When the temperature is constant, the internal energy is constant, which happens specifically in an isothermal process.