# Chapter 3: PVT Behaviour and Heat Effects (Solution)

**Problem 3.1: **Sulphur dioxide is circulated as the refrigerant in a small refrigerator. SO_{2} gas at a pressure of 5 bar and temperature 340 K is to be cooled at a constant volume of 0.142 m3, to 293 K as part of the refrigeration cycle. Calculate (a) The heat liberated, (b) The work done by the gas on cooling, (c) The final pressure attained on cooling, and (d) The change in enthalpy. Sulfur dioxide may be treated as an ideal gas. The specific heat (J/mol K) is found to vary with temperature (K) according to

**Problem 3.2: **Ten kilograms of an equimolar mixture of CO and H_{2} at 1500 K and 1 bar is cooled at constant pressure to 350 K. Assume that the gases are ideal. Determine: (a) the heat given off during cooling, (b) the change in internal energy, (c) the work done on the gas. The specific heats in J/mol K are as follows:

**Problem 3.3: **CO_{2} is sold commercially in steel containers at 60 bar. The gas is leaking through the outlet valve slowly so that the temperature may be assumed constant at the room temperature of 300 K. (a) What is the work done in the expansion of 10 kg of this gas from 60 bar to 1 bar? (b) If the temperature were constant at 290 K, would there be any difference in the work done? (c) What are the changes in enthalpy in both the above cases?

**Problem 3.4: **An ideal gas is compressed adiabatically from 1.5 bar, 338 K to 9 bar. The process is reversible, and g = 1.23 is constant over the entire range of conditions. Calculate: (a) The temperature at the end of compression (b) The work of compression (c) The heat transferred (d) The change in internal energy (e) The change in enthalpy.

**Problem 3.5: **An ideal gas undergoes the following reversible processes: (a) From an initial state of 343 K and 1 bar it is compressed adiabatically to 423 K. (b) It is then cooled to 343 K at constant pressure. (c) Finally, it is expanded to its original state isothermally. Calculate dU, dH, W, and Q for each step as well as for the entire cycle. Assume C_{V} = (3/2) R.

**Problem 3.6: **Air, initially at 389 K and 8 bar is expanded reversibly and isothermally to such a pressure that when it is cooled to 278 K at constant volume, its pressure is 2 bar. Assume air to be an ideal gas with C_{P} = 29.3 J/mol K. Calculate the work, heat transferred, changes in internal energies and changes in enthalpies.

**Problem 3.7: **Methane is stored in a tank of capacity 5.7×10^{–2} m^{3} at a pressure of 15 bar and 294 K. The gas is allowed to flow from the tank through a partially opened valve to a gas holder where the pressure is constant at 1.15 bar. When the pressure in the tank dropped to 5 bar, what would be the mass of methane removed under the following conditions? (a) If the process took place slowly so that the temperature was constant. (b) If the process took place so rapidly that the heat transferred was negligible. Methane behaves as an ideal gas with g = 1.4.

**Problem 3.8: **A rigid non-conducting tank with a volume of 4 m^{3} is divided into two equal parts by a membrane. On one side of the membrane, the tank contained gas A at 5 bar and 350 K and on the other side gas B at 10 bar and 450 K. A and B are ideal gases with C_{V} values (5/2)R and (7/2)R respectively. The membrane is suddenly ruptured and the gases get mixed. What are the final temperature and pressure?

**Problem 3.9: **One cubic metre of an ideal gas at 600 K and 20 bar expands to ten times its initial volume as follows: (a) By a reversible, isothermal process (b) By a reversible, adiabatic process (c) By an irreversible, adiabatic process in which the expansion is against a restraining pressure of 1 bar. If C_{P} = 21 J/mol K, calculate the final temperature, pressure, and the work done by the gas.

**Problem 3.10: **Estimate the molar volume of CO_{2} at 500 K and 100 bar using van der Waals equation. The van der Waals constants are 0.364 m^{4} N/mol^{2} and 4.267×10^{–5} m^{3}/mol.

**Problem 3.11: **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 3.12: **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 3.13: **Calculate the pressure developed by 1 kmol gaseous ammonia contained in a vessel of 0.6 m^{3} capacity at a constant temperature of 473 K by the following methods: (a) Using the ideal gas equation (b) Using the van der Waals equation given that a = 0.4233 N m^{4}/mol^{2}; b = 3.73×10^{–5} m^{3}/mol (c) Using the Redlich–Kwong equation given that P_{C} = 112.8 bar; T_{C} = 405.5 K.

**Problem 3.14: **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 3.15: **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 K is 13.76 bar. The critical temperature and pressure are respectively 416.3 K and 66.8 bar.

**Problem 3.16: **Using the virial equation calculate the molar volume and compressibility factor of isopropanol vapour at 473 K and 10 bar. The virial coefficients are: B = –3.88×10^{–4} m^{3}/mol; C = –2.6×10^{–8} m^{6}/mol^{2}.

**Problem 3.17: **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 3.18: **An empirical equation PV^{d} = C, where C is constant, is used to relate P and V of any reversible process. Show that for an ideal gas this equation leads to

Show that in the case of an isothermal process when δ = 1, the above equation reduces to

W=RT\ln\left(\frac{P_1}{P_2}\right)**Problem 3.19: **Heat of combustion of solid carbon and gaseous CO are respectively –393.78 kJ/mol and – 283.18 kJ/mol. Determine the heat of the formation of CO.

**Problem 3.20: **Using Hess’s law evaluate the heat of formation of solid CaCO_{3}. The following data are available:

(a)\;Ca(s)+\frac12O_2(g)\rightarrow CaO(s);\;\triangle H_{298}^0=-635.77\;kJ

(b)\;C(s)+O_2(g)\rightarrow CO_2(g);\;\;\triangle H_{298}^0=-393.77\;kJ

(c)\;CaO(s)+CO_2(g)\rightarrow CaCO_3(s);\;\;\triangle H_{298}^0=-178.15\;kJ

**Problem 3.21: **Calculate the standard heat of reaction for the following reaction:

The standard heat of formation at 298 K is –178.02 kJ/mol for FeS2(s), –822.71 kJ/mol for Fe2O3(s), and –297.10 kJ/mol for SO_{2}(g).

**Problem 3.22: **The standard heat of combustion of benzene at 298 K is – 3269.5 kJ/mol when burnt completely to CO_{2} and liquid water. The standard heat of combustion of hydrogen to liquid water is –286.04 kJ/mol and that of carbon to carbon dioxide is –393.78 kJ/mol. Calculate the standard heat of the formation of liquid benzene.

**Problem 3.23: **The standard heat of reaction at 298 K for the following reaction is – 42.433 kJ.

Calculate the heat of the reaction at 400 K. The constants in the heat capacity equation C_{P} = a + bT + gT^{2} are as given below: (C_{P} is in J/mol K and T in K).

a | b | g | |

C_{2}H_{4} | 11.85 | 119.75×10^{-3} | -36.53×10^{-6} |

H_{2}O | 30.38 | 9.62×10^{-3} | +1.19×10^{-6} |

C_{2}H_{5}OH | 29.27 | 166.39×10^{-3} | -49.93×10^{-6} |

**Problem 3.24: **The standard heat of combustion of graphite at 298 K is –393.778 kJ/mol. Determine the heat of combustion at 800 K. The heat capacities in J/mol K are:

Carbon:\;11.19+1.096\times10^{-2}T-\frac{4.894\times10^5}{T^2}

Oxygen:\;34.62+1.08\times10^{-3}T-\frac{7.859\times10^5}{T^2}

CO_2:\;43.29+1.147\times10^{-2}T-\frac{8.185\times10^5}{T^2}

**Problem 3.25: **Calculate the heat of combustion of methane at 800 K given that the heat of combustion at 298 K is –802.861 kJ/mol and the mean heat capacity in the temperature range from 298 K to 800 K are 41.868 J/mol K, 30.563 J/mol K, 41.449 J/mol K, and 34.332 J/mol K for methane, oxygen, CO_{2} and water vapor respectively.

**Problem 3.26: **Ammonia is synthesized according to the following reaction:

The specific heats of the components are represented by C_{P} = a + bT + gT^{2} where C_{P} is in J/mol K and the constants a, b, and g are:

a | b | g | |

N_{2} | 27.31 | 5.2335×10^{-3} | -4.1868×10^{-9} |

H_{2} | 29.09 | -8.374×10^{-4} | +2.0139×10^{-6} |

NH_{3} | 25.48 | 36.89×10^{-3} | -6.305×10^{-6} |

Express the heat of the reaction as a function of temperature.

**Problem 3.27: **Calculate the heat of the following gas-phase reaction

if the reactants are at 473 K and the product is at 993 K. The specific heats C_{P} = a + bT + gT^{2} J/mol K may be evaluated using the data given below:

a | b | g | |

H_{2} | 29.09 | -8.374×10^{-4} | 2.0139×10^{-6} |

O_{2} | 25.74 | 12.987×10^{-3} | -3.864×10^{-6} |

H_{2}O | 30.38 | 9.62×10^{-3} | -1.185×10^{-6} |

**Problem 3.28: **Carbon monoxide (CO) at 1000 K is burned with air at 800 K in 90 % excess air. The products of combustion leave the reaction chamber at 1250 K. Calculate the heat evolved in the reaction chamber per kmol of CO burned. The standard heat of reaction at 298 K is –283.028 kJ/mol CO. The mean specific heat applicable in the temperature range of this problem are 29.38, 49.91, 33.13, and 31.43 J/mol K for CO, CO_{2}, O_{2}, and N_{2} respectively.

**Problem 3.29: **Hydrochloric acid is produced from chlorine according to

The standard heats of formation at 298 K are –241.82, –92.307 kJ/mol for water vapour and hydrochloric acid respectively. The reaction is to be carried out at 500 K with saturated steam at 500 K and chlorine gas at 500 K entering the reactor with a ratio of 3 mol steam per mol of chlorine gas. Determine the amount of heat to be added or to be removed per kmol of the acid obtained if the reaction goes to completion and the products leave at 500 K. Mean specific heats of chlorine gas, oxygen, HCl, and water vapor may be taken as 35.0, 33.0, 40.0, and 25.0 J/mol K respectively, and enthalpy of saturated water vapor at 298 K and 500 K are 2547 kJ/kg and 2801 kJ/kg respectively.

**Problem 3.30: **Calculate the theoretical flame temperature for a gas containing 25 % CO and 75 % N_{2} when burned with 80 % excess air, when both the reactants are at 298 K. The standard heat of formation are –393.70 kJ/mol for CO_{2} and –110.00 kJ/mol for CO. The mean molar specific heats in J/mol K are

T (K) | CO_{2} | O_{2} | N_{2} |

800 | 45.43 | 31.59 | 30.31 |

1000 | 47.56 | 32.37 | 30.64 |

1200 | 49.35 | 33.02 | 31.22 |

1400 | 50.82 | 33.60 | 31.77 |

1600 | 51.99 | 34.05 | 32.25 |

1800 | 53.18 | 34.40 | 32.65 |

**Problem 3.31: **Calculate the theoretical flame temperature of a gas mixture consisting of 20 % CO and 80 % N_{2} when burned with 100 % excess air, both air and gas initially being at 298 K. The standard heat of reaction at 298 K is –283.178 J/mol CO. The heat capacity in J/mol K are given by C_{P} = a + bT + gT^{2}, where

a | b | g | |

CO_{2} | 26.54 | 42.45×10^{-3} | -14.298×10^{-6} |

O_{2} | 25.61 | 13.26×10^{-3} | -4.208×10^{-6} |

N_{2} | 27.03 | 5.815×10^{-3} | -0.289×10^{-6} |

**Problem 3.32: **0.5 mol hydrogen at 300 K, 2 mol nitrogen and oxygen present in the ratio of 3:1, and 0.3 mol of CO are mixed. The nitrogen-oxygen mixture was initially present at 400 K and the initial temperature of CO was 500 K. Fifty percent of the CO present reacts. If the final temperature of the system rises to 1200 K under adiabatic conditions, and if the standard heat of the formation of CO2 is two times that of CO, calculate the standard heat of the formation of CO_{2}. The mean heat capacities in J/mol K are 32.5 for O_{2}, 31.1 for N_{2}, 29.0 for H_{2}, 31.0 for CO, and 49.0 for CO_{2}.

**Problem 3.33: **Calculate the standard heat of reaction at 800 K for the combustion of pentane gas, given that the mean heat capacities in J/mol K are 247 for C_{5}H_{12}, 33.62 for O_{2}, 52.32 for CO_{2}, and 38.49 for H_{2}O. The standard heat of combustion at 298 K is –3271.71 kJ/mol.

**Problem 3.34: **An internal combustion engine uses octane as fuel. The air and fuel vapor mixture enters the engine at 298 K. Twenty percent of excess air is supplied. Seventy-five percent of the carbon present in the fuel is converted to CO2 and the rest to CO, and the combustion products leave the engine at 800 K, calculate the energy transferred as heat to the engine per kg fuel burned. The mean heat capacities in J/mol K are 32.5 for O2, 31.1 for N2, 31.0 for CO, 50.0 for H2O, and 49.0 for CO2. The heat of formation of CO2, CO, and H2O at 298 K are –393.509, –110.525, and –241.818 kJ/mol respectively. The heat of the formation of octane is –208.75 kJ/mol.

**Problem 3.35: **Calculate the maximum flame temperature attained when methane is burned with theoretical air when both fuel and air are at 298 K initially. The mean heat capacities in J/mol K are 62.75 for CO_{2}, 52.96 for H_{2}O, 38.67 for O_{2}, 37.13 for N_{2}. The standard heat of combustion of methane at 298 K is –802.625 kJ/mol.

**Problem 3.36: **Calculate the standard heat of reaction at 298 K for the following reaction:

The standard heats of formation are –92.307 kJ/mol for HCl (g) and –241.818 kJ/mol for H_{2}O (g).

**Problem 3.37: **Hydrocarbon fuels can be produced from methanol by the following reaction:

Compare the standard heat of combustion of 6 mol CH_{3}OH (g) at 298 K with the standard heat of combustion of C_{6}H_{12}(g) at 298 K, products in both the cases being CO_{2}(g) and H_{2}O(g).

**Problem 3.38: **Methanol is synthesized according to the following reaction:

The standard heats of formation at 298 K are –110.125 kJ/mol for CO and –200.660 kJ/mol for methanol. The specific heats (J/mol K) are given by:

C_P(CH_3OH)=19.382+101.564\times10^{-3}T-28.683\times10^{-6}T^2

C_P(CO)=28.068+4.631\times10^{-3}T-2.5773\times10^4T^{-2}

C_P(H_2)=27.012+3.509\times10^{-3}T+6.9006\times10^4T^{-2}

(a) Calculate the standard heat of the reaction at 1073 K. (b) Express the heat of the reaction as a function of temperature.

**Problem 3.39: **Methane is burned with 20 % excess air; both methane and air being at 298 K. The standard heat of combustion of methane at 298 K is –802.625 kJ. The heat capacities in J/mol K are:

C_p\left(CH_4\right)=14.15+75.499\times10^{-3}T-179915\times10^{-6}T^2

C_P(O_2)=30.255+4.207\times10^{-3}T-1.8873\times10^5T^{-2}

C_P(N_2)=27.27+4.93\times10^{-3}T+3.3256\times10^4T^{-2}

C_P(CO_2)=45.369+8.688\times10^{-3}T+9.6193\times10^5T^{-2}

C_P(H_2O)=28.85+12.055\times10^{-3}T+1.006\times10^5T^{-2}

**Problem 3.40: **Carbon monoxide reacts with water vapor to form carbon dioxide and hydrogen.

The reactants are at 298 K and in stoichiometric proportions. Seventy-five percent of CO is converted in the reaction. The products leave the reaction chamber at 800 K. The mean heat capacities in J/mol K are 30.35 for CO, 45.64 for CO2, 36.00 for water vapor, and 29.30 for hydrogen. Determine the quantity of heat to be added or removed in the reaction chamber per 1000 kg of hydrogen produced.

**Problem 3.41: **Calculate the standard heat of the following reaction at 298 K:

The standard heats of formation are as follows: CO_{2}(g) = –393.509 kJ, H_{2}O(g) = –241.818 kJ, C_{5}H_{12}(g) = –146.76 kJ The latent heat of vaporization of water at 298 K = 43.967 kJ/mol.

**Problem 3.42: **A gas mixture consisting of 20 % ethane and 80 % oxygen at 298 K is burned completely after diluting it with double the volume of CO_{2} at the same temperature 298 K. The mean heat capacities in J/mol K are 54.56 for CO_{2}, 43.02 for water vapor, and 35.52 for oxygen. The standard heat of the reaction of ethane at 298 K is –1478 kJ/mol. Determine the theoretical flame temperature.