To solve the question regarding isothermal ideal gas compression, we will analyze the process step by step.
### Step 1: Understand Isothermal Process
In an isothermal process, the temperature of the system remains constant. For an ideal gas, this means that any change in volume will not affect the temperature, as the system is in thermal equilibrium with its surroundings.
**Hint:** Remember that in an isothermal process, temperature (T) is constant.
### Step 2: Work Done in Compression
When we compress an ideal gas isothermally, work is done on the gas. This means that the surroundings are applying pressure to reduce the volume of the gas. In thermodynamics, work done on the system is considered positive.
**Hint:** Work done on the system is positive, while work done by the system is negative.
### Step 3: Change in Internal Energy
For an ideal gas undergoing an isothermal process, the change in internal energy (ΔU) is given by the formula ΔU = Cv * ΔT. Since the temperature is constant (ΔT = 0), it follows that ΔU = 0. Therefore, there is no change in internal energy during isothermal compression.
**Hint:** Remember that for an ideal gas at constant temperature, the internal energy change is zero.
### Step 4: Effect on Pressure
As the gas is compressed, its volume decreases, which leads to an increase in pressure according to Boyle's Law (P1V1 = P2V2). Therefore, during isothermal compression, the pressure of the gas increases as the volume decreases.
**Hint:** Recall Boyle's Law, which relates pressure and volume for a given amount of gas at constant temperature.
### Step 5: Changes in Enthalpy and Entropy
In an isothermal process for an ideal gas, the change in enthalpy (ΔH) is also zero because ΔH is a function of temperature. However, the entropy (ΔS) of the gas increases because the gas is being compressed, leading to a greater order in the system.
**Hint:** Enthalpy change is zero in isothermal processes, but entropy can change.
### Step 6: Gibbs Free Energy
The change in Gibbs free energy (ΔG) for an isothermal process can also be analyzed. In general, for a spontaneous process at constant temperature and pressure, ΔG is negative. However, in this case, since we are compressing the gas, ΔG can be considered positive as work is done on the system.
**Hint:** Gibbs free energy is related to spontaneity; work done on the system can lead to a positive ΔG.
### Conclusion
In summary, during isothermal ideal gas compression:
- The work done on the system (W) is positive.
- The internal energy change (ΔU) is zero.
- The pressure of the gas increases.
- The enthalpy change (ΔH) is zero.
- The entropy (ΔS) of the gas increases.
- The Gibbs free energy (ΔG) can be considered positive.
**Final Answer:** The correct answer to the question is that the work done (W) is positive.
