# Chapter 5: Properties of Real Gases (Solution)

**Problem 5.1: **A pressure gauge connected to a cylinder containing 175 kg propane indicates 50 bar absolute pressure. Using Van der Waals equation, determine the temperature of the gas in the cylinder. The capacity of the cylinder is 1 m^{3}. The van der Waals constants are a = 0.9252 N m^{4}/mol^{2} and b = 9.05⨯10^{-5} m^{3}/mol.

**Problem 5.2: **One mole of a gas obeying Van der Waals equation is contained in a cylinder of volume 0.05 m^{3} at 100 kPa. If the van der Waals constants are a = 0.135 N m^{4}/mol^{2} and b = 3.22⨯10^{-5} m^{3}/mol, calculate the temperature of the gas.

**Problem 5.3: **One kmol CO_{2} occupies a volume of 0.381 m^{3} at 313 K. Compare the pressures given by the (a) Ideal gas equation and (b) van der Waals equation. Take the van der Waals constants to be a = 0.365 N m^{4}/mol^{2} and b = 4.28⨯10^{-5} m^{3}/mol.

**Problem 5.4: **Use the Redlich-Kwong equation to calculate the pressure of 0.5 kg gaseous ammonia contained in a vessel of 0.03 m^{3} at a constant temperature of 338 K. The critical temperature and pressure are 405.5 K and 112.8 bar respectively.

**Problem 5.5: **Calculate the pressure exerted by one mole of gaseous ammonia contained in a vessel of volume 0.03 m^{3} at a temperature of 340 K using the Redlich-Kwong equation of state. The critical constants of ammonia are 405.5 K and 112.8 bar. Compare the result with the value obtained using the ideal gas equation.

**Problem 5.6: **Ethylene at 500 bar and 375 K is contained in a cylinder of volume 0.03 m^{3}. The critical properties of ethylene are 282.4 K and 50.4 bar. How many kilograms of ethylene are there in the cylinder? Use the van der Waals equation.

**Problem 5.7: **A fire extinguisher of capacity 0.04 m^{3} contains CO_{2} at 101.3 kPa and 300 K. It is to be charged with CO_{2} to a pressure of 20 bar at the constant temperature of 300 K. Determine the kilograms of gas to be admitted assuming the van der Waals equation. The critical constants of CO_{2} are 304.2 K and 73.8 bar.

**Problem 5.8:** Calculate the molar volume and the compressibility factor of methanol vapor at 450 K and 10 bar using (a) The truncated virial equation, given the following virial coefficients: B = -2.19⨯10^{-4} m^{3}/mol and C = -1.73⨯10^{-8} m^{6}/mol^{2} (b) The Redlich-Kwong equation, given the critical constants of methanol as 512.6 K and 81.0 bar.

**Problem 5.9: **Calculate the volume of the vessel that should be provided to store 1 kmol of methane at 325 K and 600 bar using the ideal gas equation and the van der Waals equation. The critical constants of methane are 190.6 K and 46.0 bar.

**Problem 5.10:** Calculate the molar volume of a gas at 9 MPa and 373 K using the van der Waals equation given that the constants a and b of the gas are respectively, 0.231 N m^{4}/mol^{2} and 4.49⨯10^{-5} m^{3}/mol.

**Problem 5.11:** Calculate the volume occupied by one mole of oxygen at 300 K and 100 bar using (a) The ideal gas law (b) The van der Waals equation. Take a = 0.1378 N m^{4}/mol^{2}, and b = 3.18⨯10^{-5} m^{3}/mol.

**Problem 5.12: **Calculate the molar volume of ammonia at 373 K and 10 bar using (a) the van der Waals equation and (b) the Redlich-Kwong equation given that the critical temperature is 405.5 K and the critical pressure is 112.8 bar.

**Problem 5.13: **Determine the van der Waals constants and the molar volume of ethane at the critical point given that the critical temperature and pressure are respectively 305.2 K and 49.4 bar.

**Problem 5.14: **Using the Redlich-Kwong equation calculate the molal volumes of saturated liquid and saturated vapor of methyl chloride at 333 K. The saturation pressure of methyl chloride at 333 is 13.76 bar. The critical temperature and pressure are respectively 416.3 K and 66.8 bar.

**Problem 5.15: **Using the virial equation calculate the molar volume and compressibility factor of isopropanol vapor at 473 K and 10 bar. The virial coefficients are: B = -3.88⨯10^{-4} m^{3}/mol and C = -2.6⨯10^{-8} m^{6}/mol^{2}.

**Problem 5.16: **Determine the molar volume of gaseous methane at 300 K and 600 bar by the following methods: (a) Using the ideal gas equation (b) Using the van der Waals equation given that a = 0.2285 N m^{4}/mol^{2}; b = 4.27⨯10^{-5} m^{3}/mol (c) Using the Redlich-Kwong equation given that T_{c} = 191.1 K and P_{c} = 46.4 bar.

**Problem 5.17: **Using the Nelson-Obert chart calculate the pressure developed by 4.0 kg oxygen contained in a vessel of volume 0.03 m^{2} at 250 K. The critical constants of oxygen are 154.6 K and 50.5 bar.

**Problem 5.18: **A 6.0 m^{3} tank contains 25 kmol of gas at 275 bar and 882 K. Determine the critical temperature of the gas using a generalized method given that the critical pressure is 50 bar.

**Problem 5.19:** Using the Nelson-Obert chart, estimate the volume occupied by 1 kg water vapor at 17 bar and 500 K, given the critical pressure and temperature of the water as 221.2 bar and 647.3 K respectively. Compare the result with the volume calculated using the ideal gas equation.

**Problem 5.20: **Estimate the molar volume of water at 973.2 K and 200 bar using(a) The Lydersen method and (b) The Pitzer method. The critical pressure and temperature of the water are 221.2 bar and 647.3 K respectively, and the critical compressibility factor and the acentric factor are, 0.234 and 0.3448 respectively.

**Problem 5.21: **Calculate the molar volume and the compressibility factor of isopropyl alcohol at 475 K and 10 bar using (a) The van der Waals equation (b) The Lydersen method (c) The Pitzer correlation. The critical pressure and temperature of isopropyl alcohol are 47.64 bar and 508.3 K. The critical compressibility factor and the acentric factor are 0.248 and 0.6689 respectively.

**Problem 5.22: **Calculate the molar volume of isobutene at 428 K and 86.2 bar using(a) The van der Waals equation (b) The Lydersen method (c) The Pitzer correlation. The critical pressure and temperature of isobutene are 36.48 bar and 408.1 K. The critical compressibility factor and the acentric factor are 0.2627 and 0.282 respectively.

**Problem 5.23: **A gas mixture containing 20% ethanol and 80% carbon dioxide has a molar volume equal to 2.0⨯10-4 m3/mol at 500 K. The critical parameters are: ethanol (Pc = 6383 kPa, Tc = 516.3 K) and carbon dioxide (Pc = 7382 kPa, Tc = 304.1 K). What is the pressure of the gas? Use the pseudo-critical method.

**Problem 5.24: **The molar volume of a gas mixture is experimentally found to be 1.4⨯10^{-4} m^{3}/mol at 121.6⨯105 N/m2 and 298 K. The analysis of the mixture showed that it contained 57% ethylene, 40% argon, and 3% helium. By how much does the molar volume calculated using the pseudo-critical properties deviate from the experimental result? The critical properties are: ethylene (Pc = 5032 kPa, Tc = 282.4 K), argon (Pc = 4898 kPa, Tc = 150.9 K), and helium (P_{c} = 228 kPa, T_{c} = 5.2 K).

**Problem 5.25: **20 kg of a gas mixture analyzing 10% carbon dioxide, 40% methane, and 50% ethylene to be contained in a cylinder at a maximum pressure of 165 bar at 355 K. What should be the volume of the cylinder? Use Kay’s method. The critical properties are: carbon dioxide (Pc = 7382 kPa, Tc = 132.9 K), methane (Pc = 4604 kPa, Tc = 190.6 K), and ethylene (P_{c} = 5032 kPa, T_{c} = 282.4 K).

**Problem 5.26:** A gas mixture analyzed 20% methane, 30% ethylene, and 50% nitrogen. The molar volume of the mixture at 100 bar is found to be 3⨯10-4 m3/mol. The critical parameters are: methane (Pc = 4604 kPa, Tc = 190.6 K), ethylene (Pc = 5032 kPa, Tc = 282.4 K), and nitrogen (P_{c} = 3394 kPa, T_{c} = 126.1 K). Using the pseudo-critical method, determine the temperature of the gas.

**Problem 5.27: **A gas mixture analyzes 20% methane, 30% ethylene, and the rest of nitrogen. Determine the molar volume of the gas at 100 bar and 375 K using (a) The ideal gas equation and (b) The van der Waals equation. The constants a and b are: methane (a = 0.225 N m^{4}/ mol^{2}, b = 3.159⨯10^{-5} m^{3}/mol), ethylene (a = 0.448 N m^{4}/mol^{2}, b = 5.72⨯10^{-5} m^{3}/mol) and nitrogen (a = 0.1347 N m^{4}/mol^{2}, b = 3.86⨯10^{-5} m^{3}/mol). (c) The mean compressibility factor, given the critical constants as methane (P_{c} = 4604 kPa, T_{c} = 190.6 K), ethylene (P_{c} = 5032 kPa, T_{c} = 282.4 K) and nitrogen (P_{c} = 3394 kPa, T_{c} = 126.1 K) (d) The pseudo-critical properties.

**Problem 5.28: **A gas mixture consists of 70% nitrogen, 20% oxygen, 9% ammonia, and 1% water. Determine the density of the gas at 800 kPa and 925 K. (a) Using the van der Waals equation with average values of the constants. The van der Waals constants are: nitrogen (a = 0.1347 N m4/mol2, b = 3.86⨯10-5 m3/mol), oxygen (a = 0.1360 N m4/mol2, b = 3.19⨯10-5 m3/mol), ammonia (a = 0.419 N m4/mol2, b = 3.73⨯10-5 m3/mol) and water vapor (a = 0.548 N m4/mol2, b = 3.06⨯10-5 m3/mol). (b) Using the van der Waals equation with the constants evaluated using the pseudo-critical properties. The following are critical properties: nitrogen (P_{c} = 3394 kPa, T_{c} = 126.1 K), oxygen (P_{c} = 5043 kPa, T_{c} = 154.6 K), ammonia (P_{c} = 1128 kPa, T_{c} = 405.7 K) and water (P_{c} = 22090 kPa, T_{c} = 647.3 K).