Let's analyze the problem. We are given a scenario where air is compressed in a compressor. We are provided with the initial conditions (pressure, temperature, specific volume) and the final pressure. A crucial piece of information is that the process is isothermal, meaning the temperature remains constant throughout the process.

Understanding Isothermal Processes

An isothermal process is a thermodynamic process in which the temperature of the system remains constant ($T_1 = T_2$). For a closed system undergoing an isothermal process, heat transfer into or out of the system usually occurs to maintain the constant temperature.

Internal Energy of an Ideal Gas

Air can be treated as an ideal gas for most engineering purposes, especially at these conditions. The internal energy of an ideal gas is a function of temperature only. Mathematically, the internal energy $U$ of an ideal gas can be expressed as:

$\Delta U = m \cdot c_v \cdot \Delta T$

where:

• $\Delta U$ is the change in internal energy
• $m$ is the mass of the gas (here, we consider per unit mass, so $m=1$ kg)
• $c_v$ is the specific heat capacity at constant volume
• $\Delta T$ is the change in temperature ($T_2 - T_1$)

Calculating Change in Internal Energy

In this problem, the process is isothermal, which means the temperature is constant:

$T_1 = 298 \, \text{K}$

$T_2 = T_1 = 298 \, \text{K}$

Therefore, the change in temperature is:

$\Delta T = T_2 - T_1 = 298 \, \text{K} - 298 \, \text{K} = 0 \, \text{K}$

Now, using the formula for the change in internal energy for an ideal gas:

$\Delta U = m \cdot c_v \cdot \Delta T$

Since $\Delta T = 0$, the change in internal energy is:

$\Delta U = m \cdot c_v \cdot 0 = 0$

Thus, the change in internal energy during the isothermal compression process is zero.

The initial pressure (1 bar), final pressure (5 bar), and initial specific volume (1.8 m²/kg) are provided to describe the state changes of the air, but they are not required to calculate the change in internal energy for an isothermal process involving an ideal gas. Only the fact that it is an isothermal process is needed for this calculation.

Summary of the Process and Internal Energy Change

Here's a brief summary:

• Process: Isothermal compression of air
• Working substance: Air (treated as an ideal gas)
• Condition: Temperature remains constant ($T = \text{constant}$)
• Internal energy of ideal gas: Depends only on temperature ($U = U(T)$)
• Conclusion: If temperature is constant, change in internal energy is zero ($\Delta U = 0$)

Therefore, for the isothermal compression of air from 1 bar and 298 K to 5 bar, the change in internal energy is 0.