Reversible Adiabatic Compression of an Ideal Gas

Given:
- Number of moles (n) = 0.2
- Initial volume (V1) = 1 dm^3
- Final volume (V2) = 0.25 dm^3

To calculate:
- q (heat transfer)
- w (work done)
- ΔU (change in internal energy)
- ΔH (change in enthalpy)

Solution:

1. Adiabatic Process:
An adiabatic process is a thermodynamic process in which there is no heat transfer between the system and its surroundings. In this case, the adiabatic process is a reversible adiabatic compression.

2. Work Done (w):
The work done in a reversible adiabatic process can be calculated using the formula:
w = -PΔV

Where P is the pressure and ΔV is the change in volume.

3. Pressure (P):
The pressure of an ideal gas can be calculated using the ideal gas law:
PV = nRT

Where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant (8.314 J/mol·K), and T is the temperature in Kelvin.

4. Calculation:

4.1. Initial Pressure (P1):
Using the ideal gas law, we can calculate the initial pressure:
P1 = (nRT1)/V1

4.2. Final Pressure (P2):
Using the ideal gas law, we can calculate the final pressure:
P2 = (nRT2)/V2

4.3. Change in Pressure (ΔP):
The change in pressure can be calculated as:
ΔP = P2 - P1

4.4. Change in Volume (ΔV):
The change in volume is given as:
ΔV = V2 - V1

4.5. Work Done (w):
Using the formula, we can calculate the work done:
w = -PΔV

5. Change in Internal Energy (ΔU):
For an adiabatic process, the change in internal energy is given by the formula:
ΔU = q + w

Since the process is adiabatic, q (heat transfer) is zero. Therefore, ΔU = w.

6. Change in Enthalpy (ΔH):
The change in enthalpy can be calculated using the equation:
ΔH = ΔU + PΔV

7. Summary:
- q (heat transfer) = 0 (adiabatic process)
- w (work done) = -PΔV
- ΔU (change in internal energy) = w
- ΔH (change in enthalpy) = ΔU + PΔV

Note:
Remember to convert all volume values to cubic meters (m^3) for consistency in units.