The molar specific heat capacity at constant pressure (Cp) can be calculated using the relationship between Cp, Cv, and the ideal gas constant (R):...
Cp = Cv + R
Where:
- Cp is the molar specific heat capacity at constant pressure
- Cv is the molar specific heat capacity at constant volume (20.8 J K-1 mol-1)
- R is the ideal gas constant (8.314 J K-1 mol-1)
Substituting the values, we get:
Cp = 20.8 J K-1 mol-1 + 8.314 J K-1 mol-1 = 29.114 J K-1 mol-1
Adiabatic Constant
The adiabatic constant (γ) is the ratio of Cp to Cv:
γ = Cp / Cv
Substituting the values, we get:
γ = 29.114 J K-1 mol-1 / 20.8 J K-1 mol-1 = 1.40
Final Volume and Temperature
For an adiabatic process, the following relationship holds:
P1V1γ = P2V2γ
Where:
- P1 is the initial pressure (4.25 atm)
- V1 is the initial volume (unknown)
- P2 is the final pressure (2.50 atm)
- V2 is the final volume (unknown)
- γ is the adiabatic constant (1.40)
To find V1, we can use the ideal gas law:
P1V1 = nRT1
Where:
- n is the number of moles (1.0 mol)
- R is the ideal gas constant (8.314 J K-1 mol-1)
- T1 is the initial temperature (310 K)
Solving for V1, we get:
V1 = (nRT1) / P1 = (1.0 mol * 8.314 J K-1 mol-1 * 310 K) / (4.25 atm * 101325 Pa/atm) = 0.0596 m3
Now we can substitute V1 and the other values into the adiabatic equation to find V2:
V2 = (P1V1γ / P2)1/γ = (4.25 atm * (0.0596 m3)1.40 / 2.50 atm)1/1.40 = 0.0954 m3
To find the final temperature (T2), we can use the following relationship:
T1V1γ-1 = T2V2γ-1
Solving for T2, we get:
T2 = T1(V1/V2)γ-1 = 310 K (0.0596 m3 / 0.0954 m3)1.40-1 = 247 K
Work Done During Expansion
The work done during an adiabatic expansion is given by:
W = (P2V2 - P1V1) / (1 - γ) = (2.50 atm * 0.0954 m3 - 4.25 atm * 0.0596 m3) / (1 - 1.40) = -1.58 kJ
The negative sign indicates that work is done by the system, as the gas expands.