# Chapter 2: First Law of Thermodynamics (Solution)

**Problem 2.1: **Water is being heated in a closed container provided with a paddle agitator. 50 kJ of heat is supplied and 10 kJ is lost to the surroundings. The work done by the paddle wheel is 750 N m. What is the change in internal energy of the contents of the vessel?

**Problem 2.2: **An insulated rigid tank is divided into two equal parts by a partition. Initially, one part contains 5 kg of an ideal gas at 1 MPa and 400 K, and the other part is evacuated. The partition is now removed and the gas expands into the entire tank. Determine the final temperature and pressure in the tank.

**Problem 2.3: **An elevator with a mass of 2500 kg rests at a level of 7.5 m above the base of the elevator shaft. It is raised to 75 m when the cable holding it breaks. It falls freely to the base where it is brought to rest by a strong spring. Assume that the acceleration due to gravity is 9.81 m/s^{2}. Neglecting the effect of friction, Calculate (a) The potential energy of the elevator in its initial position. (b) The potential energy of the elevator in its highest position. (c) The work done in raising the elevator. (d) The kinetic energy and velocity of the elevator just before it strikes the spring. (e) The potential energy of the spring when the elevator rests on it.

**Problem 2.4: **Consider 1 kg of water in a waterfall which is flowing down from a height of 100 m. At the bottom of the fall, it joins a river the downstream velocity of which may be assumed negligible. Neglecting any energy exchange between the water and the surroundings, calculate (a) The potential energy at the top of the falls. (b) The kinetic energy just before the water strikes the bottom. (c) The change in temperature of the water when it enters the river.

**Problem 2.5: **A 10 g lead ball at 300 K is dropped from a height of 10 m. Calculate: (a) the kinetic energy and speed of the ball as it reaches the ground and (b) the temperature rise of the ball if all its kinetic energy is transformed into internal energy as the ball is suddenly stopped after 10 m, given that the specific heat of lead is 125.6 J/kg K and its volume change is negligible.

**Problem 2.6: **A mixture of saturated steam and saturated water is contained in a rigid tank of volume 0.1 m^{3} at a pressure of 2 bar. The liquid occupies 10 % of the total volume. How much heat must be added in order that the tank contain only saturated steam? What will be the pressure in the tank? The specific volumes of saturated liquid and saturated vapor are 1.061×10–3 m3/kg and 0.8857 m3/kg respectively. The internal energy of saturated liquid and saturated vapor are respectively 504.5 kJ/kg and 2529.5 kJ/kg.

**Problem 2.7: **Iron reacts with hydrochloric acid at 300 K according to

Determine the work done when 0.1 kg iron reacts with HCl in (a) a closed vessel, and (b) an open vessel.

**Problem 2.8: **A vessel of volume 1 m^{3} initially contains one percent by volume saturated liquid water and the rest saturated vapor at 1 bar. How much heat is to be supplied so that the vessel gets filled with vapor? Take data from steam tables.

**Problem 2.9: **Liquid water at 373 K and 101.3 kPa has an internal energy of 420 kJ/kg on an arbitrary basis. The specific volume at these conditions is 1.04×10^{–3} m^{3}/kg. The water is brought to the vapor state at 473 K and 700 kPa. At this condition, its specific volume is 0.3 m3/kg and its enthalpy is 2844 kJ/kg. Calculate the enthalpy of the liquid and the changes in internal energy and enthalpy accompanying the vaporization process.

**Problem 2.10: **A sample of 10 g of liquid benzene at its boiling point is vaporized using a 12 V, 0.5 A electric supply. The normal boiling point and latent heat of vaporization of benzene are 353.2 K and 30.8×103 kJ/kmol respectively. Determine: (a) the change in internal energy, (b) the time required for complete vaporization.

**Problem 2.11: **A cylinder fitted with a piston has a volume 0.1 m^{3} and contains 0.5 kg of steam at 500 kPa. How much heat is to be supplied to bring the temperature of the steam to 823 K keeping the pressure constant? What is the work done in the process?

**Problem 2.12: **A 5-kg saturated liquid water contained in a piston-cylinder arrangement at 200 kPa is heated by passing a current of 8 A through an electric resistance heater kept within the cylinder for 46 min. Water is stirred by a paddle wheel which does a work of 500 kJ. If one-half of the liquid is evaporated at constant pressure, what is the voltage of the electric source?

**Problem 2.13: **A steam boiler of volume 2.3 m^{3} initially contained 1.7 m^{3} liquid water in equilibrium with 0.6 m^{3} of vapor at 100 kPa. The boiler is heated keeping the inlet and discharge valves closed. The relief valve is set to lift when the pressure in the boiler reaches 5500 kPa. Determine the amount of heat supplied to the contents in the boiler before the relief valve lifts.

**Problem 2.14: **A gas expands from an initial volume of 0.2 m^{3} to a final volume of 0.4 m^{3} in a reversible steady flow process. During the process, the pressure varies with the volume as P = 5×10^{5} V + 7×10^{4} where P is in N/m^{2} and V is in m^{3}. The inlet line is 4 m below the outlet line and the gas enters with a negligible velocity. The internal energy of the gas decreases by 30 kJ during the process. Determine the heat transferred..

**Problem 2.15: **A trial run on a steam turbine power plant gave the following results

Entrance to boiler | Exit of turbine |

Mass flow rate = 3600 kg/h | Velocity of steam = 25 m/s |

Enthalpy of water = 850 kJ/kg | Elevation above datum = 0 |

Elevation above datum = 4.3 m | Enthalpy of steam = 2625 kJ/kg |

Velocity of water = 5 m/s |

Determine the power developed by the turbine if the heat added to the boiler is 2100 kJ/s.

**Problem 2.16: **The work required for compressing a gas from an initial condition of 100 kPa and 300 K to a final pressure of 300 kPa is found to be 280 kJ/kg of the gas. The compressed gas is admitted to a nozzle where its velocity is increased to 700 m/s. If the gas enters the compressor with negligible velocity and leaves the nozzle at 100 kPa and 300 K, what is the heat removed during compression?

**Problem 2.17: **In a test of a water-jacketed air compressor, it was found that the shaft work required to drive the compressor was 150 kJ/kg air compressed. The enthalpy of air leaving the compressor was found to be greater than that of the air entering the compressor by 80 kJ/kg and the heat removed by the cooling water was 100 kJ/kg of air. Determine the amount of energy that must have been dissipated as heat to the atmosphere from the bearings, cylinder walls, etc.

**Problem 2.18: **Water at 200 kPa and 355 K (H = 343.3 kJ/kg) enters a straight horizontal pipe at a velocity 3 m/s where it is heated by flue gases from the outside. Steam leaves the system at 100 kPa and 423 K (H = 2776.3 kJ/kg) at a velocity of 200 m/s. How much heat must have been supplied per kg of water flowing?

**Problem 2.19: **A steam-jet ejector is used to entrain saturated water vapor at 25 kPa (H = 2618.2 kJ/kg) leaving an evaporator, by high-pressure saturated steam at 1000 kPa (2778.1 kJ/kg). It is estimated that with every kg of high-pressure steam 0.75 kg of the vapour from the evaporator can be entrained. If the mixed stream leaving the ejector is at 100 kPa what is its temperature?

**Problem 2.20: **Air at 80 kPa, 300 K, and 250 m/s enters a diffuser at a rate of 2.5 kg/s and leaves at 320 K. The exit area of the diffuser is 0.04 m^{2}. The air is estimated to lose heat at a rate of 20 kJ/s during this process. Assume air to be an ideal gas with an average molecular weight of 29 and average heat capacity, CP of 1.0068×10^{3} J/kg K under the stated conditions. Determine (a) the exit velocity and (b) the exit pressure of the air.

**Problem 2.21: **Superheated steam at 2 MPa and 673 K is expanded to a pressure 15 kPa in an adiabatic turbine, the power output of which is 5 MW. The steam enters at a velocity of 50 m/s and leaves at 180 m/s as a two-phase mixture consisting of 90 % vapour. The inlet and exit of the turbine are at elevations of 10 m and 6 m from the datum. Determine the work done per unit mass of the steam flowing through the turbine and the mass flow rate of the steam.

**Problem 2.22: **A compressor receives air at 101.3 kPa at a steady rate of 0.5 kg/s and delivers at 700 kPa. At the inlet, the velocity and specific volume of air are 6 m/s and 0.83 m^{3}/kg, respectively and those at the exit are 4 m/s and 0.15 m^{3}/kg, respectively. The internal energy of compressed air is greater than that of inlet air by 90 kJ/kg. Heat is removed from the cylinder at a rate of 60 W by the circulating cooling water. Determine the power required to drive the compressor.

**Problem 2.23: **Turbines in a hydroelectric power plant receive water from a 100-m high dam, generate power, and discharge water through a 6-m diameter pipe at a velocity of 15 m/s into a river below the dam. Assume that the dam and the surroundings are at the same temperature. What is the power output from the plant?

**Problem 2.24: **A pump that supplies water at the rate of 200 cubic meter per hour is driven by a 20-kWh electric motor having an efficiency of 80 %. The pressures at the inlet and exit of the pump are 100 and 316 kPa, respectively and the potential energy change across the pump is negligible. The inlet and exit pipes are of the same diameters. Determine the mechanical efficiency of the pump.

**Problem 2.25: **The turbine in a hydroelectric power plant receives water at a rate of 120 m^{3}/s from an elevation of 150 m. If the combined efficiency of the turbine-generator system is 80 %, determine the power output from the plant in MW. Neglect frictional losses in the piping.

**Problem 2.26: **Oil flows at a rate of 1000 kg/min from an open reservoir at the top of a hill 400 m in height to another reservoir at the bottom of the hill. Heat is supplied to the oil on its way at the rate 1800 kJ/min and work is supplied by a 1 hp pump. Take the mean specific heat of oil to be 3.35 kJ/kg K. Determine the temperature change of the oil.

**Problem 2.27: **A gas initially at 100 bar is undergoing a reversible process in which its pressure gets doubled at a constant volume of 0.5 m^{3}. If the total internal energy (U) of the gas in kJ is related to the pressure P in bar and total volume (V) in m^{3} according to U = 2.0 PV, calculate the change in enthalpy and the heat supplied.

**Problem 2.28: **(a) Three kilo mole nitrogen gas (C_{P} = 29.1 kJ/kmol K and C_{V} = 20.8 kJ/kmol K) at 350 K is taken in a rigid container and is heated to 550 K. How much heat must be supplied, if the mass of the vessel is 100 kg and its specific heat is 0.5 kJ/kg K? (b) Two kilo mole nitrogen is taken in a piston/cylinder arrangement at 500 K. Determine the quantity of heat extracted from the gas to cool it to 350 K at constant pressure if the heat capacity of the container is negligible.

**Problem 2.29: **An insulated rigid container contains air at 5 bar and 375 K. The volume of the container is 1 m^{3}. Air may be assumed to behave as an ideal gas so that the P-V-T relationship is PV = nRT, where R = 8.314 kJ/kmol K. The specific heat C_{V} of air is 20.785 kJ/kmol K. The temperature of the air is increased by a rotating paddle wheel. Calculate the work done to raise the temperature of the air to 425 K.

**Problem 2.30: **Assume that the P-V-T relationship for nitrogen gas can be approximated by the ideal gas equation PV = nRT, where R = 8.314 kJ/kmol K. The heat capacities are C_{V} = 20.786 kJ/kmol K and C_{P} = 29.1 kJ/kmol K. Nitrogen which was initially at 1 bar and 280 K, is compressed to 5 bar and 280 K by two different reversible processes: (a) Cooling at constant pressure followed by heating at constant volume (b) Heating at constant volume followed by cooling at constant pressure. For each of the above paths, determine the conditions at the intermediate state and ∆U, ∆H, Q and W.

**Problem 2.31: **One kilo mole of a gas for which PV = nRT, where R = 8.314 kJ/kmol K is originally at 300 K and 1 bar. It is then heated at constant pressure to a temperature of 400 K and compressed isothermally to a volume equal to its initial volume. Assume that C_{P} = 30 kJ/kmol K. Find ∆U, ∆H, Q and W.