Gas Expansion and Enthalpy Change

To determine the molar enthalpy change for the given gas expansion process, we need to consider the ideal gas behavior, reversible expansion, adiabatic conditions, and initial and final volumes.

Given Data:
- Cv,m = 5/2 R (specific heat capacity at constant volume)
- Initial volume (V1) = 1 litre
- Final volume (V2) = 32 litre
- Initial temperature (T1) = 327°C

Step 1: Conversion of Temperature to Kelvin Scale
The temperature needs to be converted from Celsius to Kelvin scale since the gas law equations require temperatures in Kelvin.

T1 (Kelvin) = T1 (°C) + 273.15
T1 (Kelvin) = 327 + 273.15 = 600.15 K

Step 2: Calculation of Molar Entropy Change
Since the process is adiabatic (no heat transfer), the change in entropy is zero (ΔS = 0).

Step 3: Calculation of ΔU (Change in Internal Energy)
In an adiabatic process, the change in internal energy (ΔU) is equal to the work done (W).

ΔU = W

For an ideal gas, the work done during an expansion process can be calculated using the formula:

W = (Cv,m * ΔT)

Where ΔT is the change in temperature.

Step 4: Calculation of ΔT (Change in Temperature)
The change in temperature can be calculated using the ideal gas law equation:

ΔT = (T2 - T1)

Where T2 is the final temperature, and T1 is the initial temperature.

Step 5: Calculation of T2 (Final Temperature)
Since the process is adiabatic, there is no heat transfer. Therefore, the final temperature (T2) can be calculated using the adiabatic expansion formula:

T2 * (V2/V1)^(γ-1) = T1

Where γ is the heat capacity ratio (Cp/Cv), which is equal to 5/2 for the given gas.

T2 = T1 / (V2/V1)^(γ-1)

Step 6: Calculation of ΔU (Change in Internal Energy)
Using the values of Cv,m, ΔT, and T2, we can now calculate the change in internal energy (ΔU).

ΔU = Cv,m * ΔT

Step 7: Calculation of ΔH (Change in Enthalpy)
Since the process is adiabatic, there is no heat transfer. Therefore, the change in enthalpy (ΔH) is equal to the change in internal energy (ΔU).

ΔH = ΔU

Step 8: Calculation of Molar Enthalpy Change
Finally, to calculate the molar enthalpy change, we divide the change in enthalpy (ΔH) by the number of moles (n) of the gas.