One mole of an ideal mono-atomic gas initially at 500
C, one atmosphere, is irreversibly
brought to 2000
C, 3.2 atm. Determine dq, dw, ΔU, ΔH, ΔS, ΔG and ΔA.
To determine dq (heat transfer), dw (work done), ΔU (change in internal energy), ΔH (change in enthalpy), ΔS (change in entropy), ΔG (change in Gibbs free energy), and ΔA (change in Helmholtz free energy) for the given process, we need to follow some steps and equations.
1. Calculate the change in temperature and pressure:
ΔT = T_final - T_initial = 2000°C - 500°C = 1500°C (convert to kelvin by adding 273.15)
ΔP = P_final - P_initial = 3.2 atm - 1 atm = 2.2 atm
2. Determine the heat transfer (dq):
Since the process is irreversible, we cannot directly calculate dq using the ideal gas equation. However, we can make use of the equation:
dq = n*Cv*(T_final - T_initial)
Where:
n = number of moles of gas (given as 1 mole)
Cv = molar specific heat capacity at constant volume
The molar specific heat capacity of an ideal monoatomic gas at constant volume is Cv = (3/2)R, where R is the gas constant.
Substituting the values, we have:
dq = (1 mole) * (3/2)*R * (1500°C + 273.15) - (500°C + 273.15)
3. Calculate the work done (dw):
Since the process is irreversible, we can't directly determine the work done. However, we can use the equation:
dw = -P_initial * ΔV
Where:
ΔV = V_final - V_initial (this can be calculated using the ideal gas law)
From the ideal gas law, we have:
V_initial = (n * R * T_initial) / P_initial
V_final = (n * R * T_final) / P_final
Substituting these values, we have:
dw = -(1 atm) * [(n * R * T_final) / P_final] - [(n * R * T_initial) / P_initial]
4. Calculate the change in internal energy (ΔU):
ΔU = dq + dw
5. Calculate the change in enthalpy (ΔH):
The monoatomic gas is an ideal gas and the process is irreversible. Therefore, for an ideal gas:
ΔH = ΔU + P_initial * ΔV
6. Calculate the change in entropy (ΔS):
ΔS = n * Cv * ln(T_final/T_initial) + n * R * ln(P_final/P_initial)
7. Calculate the change in Gibbs free energy (ΔG):
ΔG = ΔH - T_initial * ΔS
8. Calculate the change in Helmholtz free energy (ΔA):
ΔA = ΔU - T_initial * ΔS
By plugging in the values into the respective equations, you can determine dq, dw, ΔU, ΔH, ΔS, ΔG, and ΔA for the given process. Note that all temperature values need to be in Kelvin and pressure values in atm for accurate calculations.