Consider a mole of gas obeying the equation of state P⋅V = n⋅R⋅T + n⋅B(T)⋅P. Find w, q, ΔU, ΔS, ΔH, ΔA, and ΔG for a reversible isothermal expansion of this gas. Express your answers in
terms of initial and final pressures, T, B, and (dB/dT).
To find the values of work (w), heat (q), change in internal energy (ΔU), change in entropy (ΔS), change in enthalpy (ΔH), change in Helmholtz free energy (ΔA), and change in Gibbs free energy (ΔG) for a reversible isothermal expansion of the gas obeying the equation of state P⋅V = n⋅R⋅T + n⋅B(T)⋅P, we'll go step by step.
Step 1: Work (w)
The work done during an isothermal expansion can be calculated using the equation:
w = -nRT ln(Vf/Vi)
Here, n is the number of moles, R is the gas constant, T is the temperature, Vi is the initial volume, and Vf is the final volume.
Step 2: Heat (q)
Since the process is isothermal, the heat absorbed or released (q) is equal to the work done (w):
q = -w = nRT ln(Vf/Vi)
Step 3: Change in internal energy (ΔU)
For an ideal gas, the change in internal energy can be calculated using the equation:
ΔU = q + w = nRT ln(Vf/Vi) - nRT ln(Vf/Vi)
ΔU = 0 (since q = -w)
Step 4: Change in entropy (ΔS)
The change in entropy can be calculated using the equation:
ΔS = q/T
Here, T is the temperature.
ΔS = nR ln(Vf/Vi)
Step 5: Change in enthalpy (ΔH)
For an isothermal process, the change in enthalpy is equal to the heat absorbed or released:
ΔH = q = nRT ln(Vf/Vi)
Step 6: Change in Helmholtz free energy (ΔA)
The change in Helmholtz free energy can be calculated using the equation:
ΔA = ΔU - TΔS
Since ΔU = 0, ΔA = -TΔS = -nRT ln(Vf/Vi)
Step 7: Change in Gibbs free energy (ΔG)
The change in Gibbs free energy can be calculated using the equation:
ΔG = ΔH - TΔS = nRT ln(Vf/Vi) - nRT ln(Vf/Vi)
ΔG = 0 (since ΔH = TΔS)
So, the final expressions for the given quantities in terms of initial and final pressures (Pi and Pf), temperature (T), B, and (dB/dT) are:
w = -nRT ln(Vf/Vi)
q = nRT ln(Vf/Vi)
ΔU = 0
ΔS = nR ln(Vf/Vi)
ΔH = nRT ln(Vf/Vi)
ΔA = -nRT ln(Vf/Vi)
ΔG = 0