Let's break this down step by step according the instructions and data provided. --- # **Step 1: Fugacity Coefficient Calculation at \( P_r = 1. \)** Given: \[ \ln \phi = \int_^{P_r}frac{Z-1}{P} dP \] the table for \( P_r \ and \( Z \): | \(_r \) .5 | 1. | 2. | 4. | 6. | 8. | 10. | |-----------|-----|-----|-----|-----|-----|-----|------| | \( Z \) | .92| .84| .72| .88| 1.05| 1.22| 1.40 | We want \( \phi \) at \( P_r = 1. \): ### **Numerical Integration (Trapezoidal Rule)** Use the trapezoidal rule for the integral from \( P_r = \) to \( P_r = 1. \): - At \( P_r = .5 \), \( Z = .92 \) - At \( P_r = 1. \), \( Z = .84 \) \[ \int_^{1.} \frac{Z-1}{P_r} dP_r \approx \int_{.5}^{1.} \frac{Z-1}{P_r} dP_r \] Calculate \(\frac{Z-1}{P_r}\): - At \(.5\): \((.92-1)/.5 = -.08/.5 = -.16\) - At \(1.\): \((.84-1)/1. = -.16\) Trapezoidal rule: \[ \int_{.5}^{1.} f(P_r) dP_r \approx \frac{(1.-.5)}{2} [f(.5) + f(1.)] \] \[ = \frac{.5}{2} [(-.16) + (-.16)] = .25 \times (-.32) = -.08 \] Therefore, \[ \ln \phi \approx -.08 \] \[ \phi \approx e^{-.08} \approx .923 \] **Final Answer:** \[ \boxed{\phi(P_r = 1.) \approx .92} \] --- # **Step 2: Maxwell Relations** Write the exact differential forms: - **Internal Energy:** \( dU = TdS - PdV \) - **Enthalpy:** \( dH = TdS + VdP \) - **Helmholtz Free Energy:** \( dA = -SdT - PdV \) - **Gibbs Free Energy:** \( dG = -SdT + VdP \) **Maxwell Relations:** (from second derivatives equality) 1. \( \left(\frac{\partial T}{\partial V}\right)_S = -\left(\frac{\partial P}{\partial S}\right)_V \) 2. \( \left(\frac{\partial T}{\partial P}\right)_S = \left(\frac{\partial V}{\partial S}\right)_P \) 3. \( \left(\frac{\partial S}{\partial V}\right)_T = \left(\frac{\partial P}{\partial T}\right)_V \) 4. \( \left(\frac{\partial S}{\partial P}\right)_T = -\left(\frac{\partial V}{\partial T}\right)_P \) --- # **Step 3: Relationship using Euler's Reciprocity** Given: \[ \left( \frac{\partial S}{\partial V} \right)_T = \left( \frac{\partial P}{\partial T} \right)_V \] This follows from the fundamental equation \( dA = -S dT - P dV \) and applying the reciprocity criterion. --- # **Step 4: SFEE and Specific Work for Steam Turbine** The steady-flow energy equation (SFEE): \[ \dot{Q} - \dot{W} = \dot{m}(h_{out} - h_{in}) + \frac{\dot{m}}{2} (V_{out}^2 - V_{in}^2) \] For a turbine (usually adiabatic, \( \dot{Q} \approx \)), the specific work output per kg: \[ w = h_{in} - h_{out} \] (If kinetic energy changes are negligible.) If the **inlet** is at \( P_1, T_1 \) and the **exit** is at saturated vapor, - Find \( h_{in} \) at \( (P_1, T_1) \) from steam tables. - Find \( h_{out} = h_{g}(P_{exit}) \) (saturated vapor). \[ w = h_{in} - h_{g,exit} \] --- # **Step 5: Chemical Potential and Gibbs-Duhem Equation** For multi-component systems: \[ \sum n_i d\mu_i = -S dT + V dP \] At equilibrium for a phase transition: \[ \sum n_i d\mu_i = \] --- # **Step 6: Spontaneous Processes and Equilibrium Constant** - **Total entropy change:** \( \Delta S_{univ} \geq \) for spontaneity. - **Relationship:** \( \Delta_r G^\circ = -RT \ln K_p \) (van't Hoff isotherm). --- ## **Summary Table of Key Results** | Step | Result | |------|--------| | Fugacity coefficient \(\phi\) at \(P_r = 1.\) | \(\boxed{.92}\) | | Maxwell relations | Four classic relations written above | | Euler's reciprocity | \( \left( \frac{\partial S}{\partial V} \right)_T = \left( \frac{\partial P}{\partial T} \right)_V \) | | Specific turbine work \(w\) | \( w = h_{in} - h_{g,exit} \) | | Chemical potential equilibrium | \( \sum n_i d\mu_i = \) | | Gibbs free energy and equilibrium | \( \Delta_r G^\circ = -RT \ln K_p \) | --- If you need calculations for specific values (e.g., for the steam turbine), please provide values for \( P_1, T_1 \), and the exit pressure.