Gas expands through an adiabatic turbine. The inlet stream flows at 100 bar and 600 K, while the outlet stream is at 20 bar and 445 K. The constant-pressure molar heat capacity of the gas is $30.0 + 0.02T$, where the molar heat capacity is in J/(mol K) and the temperature is in K. The process is at steady state.

Assume the gas is non-ideal and follows the following equation:

$$p=\frac{RT}{V_\mathrm m-b}+\frac{ap^2}{T\left(V_\mathrm m-b\right)}\tag{1}$$

with $\displaystyle a=0.001\ \frac{\mathrm{m^3\ K}}{\mathrm{bar\ mol}}$

and $\displaystyle b=0.00008\ \frac{\mathrm{m^3}}{\mathrm{mol}}$

For this gas the differential molar enthalpy change is:

$$\mathrm dH_\mathrm m=C_{\mathrm m,p}\mathrm dT+\left[V_\mathrm m-T\left(\frac{\partial V_\mathrm m}{\partial T}\right)_p\right]\mathrm dp\tag{2}$$

Assume that the ideal expression for the molar heat capacity given is still valid, but that it is only true as $p$ goes to 0.

And the question is; what is the shaft work of the turbine?

This question is from a thermodynamics course and to solve it students were sort of guided through a solution. First substitute equation (1) into equation (2) and find a general expression for the differential molar enthalpy change of the non-ideal gas. Then draw a $pT$ diagram of an appropriate path to evaluate the enthalpy change of the process, and just integrate along the path to find the answer.

I still can't figure out how this was supposed to be done.

1. Hold the temperature at 600 K and lower the pressure to say 1 bar, where the gas behaves as an ideal gas. Use the equation provided with dT set to 0 to evaluate $\Delta h$ for this step
2. Hold the pressure constant at 1 atm, and use the ideal gas heat capacity to calculate $\Delta h$ for cooling the gas down from 600K to 445K.
3. Hold the temperature constant at 445K and raise the pressure from 1 bar to 20 bars. Use the equation provided with dT set to 0 to evaluate $\Delta h$ for this step.