# Fugacity Calculation for Propylene Gas We are to calculate the fugacity of propylene (\(C_3H_6\)) at \(275^\circ \mathrm{C}\) and \(70\) bar. ## Step 1: Gather Basic Data - **Temperature (\(T\))**: \(275^\circ \mathrm{C} = 548\, \mathrm{K}\) - **Pressure (\(P\))**: \(70\, \mathrm{bar}\) - **Propylene critical properties**: - \(T_c = 365\, \mathrm{K}\) - \(P_c = 46\, \mathrm{bar}\) - \(Z_c = .276\) (typical value) ## Step 2: Calculate Reduced Properties \[ T_r = \frac{T}{T_c} = \frac{548}{365} = 1.50 \] \[ P_r = \frac{P}{P_c} = \frac{70}{46} = 1.52 \] ## Step 3: Choose an Equation of State We will use the **compressibility factor method** (using generalized charts or correlations) to estimate the fugacity. ## Step 4: Estimate Compressibility Factor (\(Z\)) For \(T_r = 1.50\) and \(P_r = 1.52\), from generalized compressibility charts (or Lee-Kesler tables): \[ Z \approx .74 \] ## Step 5: Calculate Fugacity Coefficient (\(\phi\)) The fugacity coefficient for a real gas is: \[ \ln \phi = \int_^P \frac{Z - 1}{P} dP \] For practical purposes, for a constant \(Z\) (approximate): \[ \ln \phi \approx (Z-1) - \ln Z \] \[ \ln \phi \approx (.74-1) - \ln(.74) = -.26 - (-.301) = -.26 + .301 = .041 \] \[ \phi = e^{.041} \approx 1.042 \] ## Step 6: Calculate Fugacity \[ f = \phi \cdot P \] \[ f = 1.042 \times 70\, \mathrm{bar} = 72.94\, \mathrm{bar} \] ## Step 7: Final Answer **The fugacity of propylene at \(275^\circ \mathrm{C}\) and \(70\, \mathrm{bar}\) is:** \[ \boxed{72.9 \ \mathrm{bar}} \] --- ## Notes - More accurate results can be obtained with a more precise EOS (e.g., Peng-Robinson) and iterative calculations. - The compressibility factor \(Z\) was estimated from generalized charts. For a more precise answer, you may use EOS software or tables. - The fugacity coefficient is slightly above 1, indicating deviation from ideality, but not extremely so. --- **Summary Table:** | Parameter | Value | |-----------|-----------------| | Temperature (\(T\)) | 548 K | | Pressure (\(P\)) | 70 bar | | \(Z\) | .74 | | \(\phi\) | 1.042 | | Fugacity (\(f\)) | 72.9 bar |