Chapter 19

Two moles of an ideal gas are heated at constant pressure from \(T=27^{\circ} \mathrm{C}\) to \(T=107^{\circ} \mathrm{C}\) (a) Draw a p \(V\) -diagram for this process. (b) Calculate the work done by the gas.

Six moles of an ideal gas are in a cylinder fitted at one end with a movable piston. The initial temperature of the gas is \(27.0^{\circ} \mathrm{C}\) and the pressure is constant. As part of a machine design project, calculate the final temperature of the gas after it has done \(2.40 \times 10^{3}\) J of work.

CALC Two moles of an ideal gas are compressed in a cylinder at a constant temperature of \(65.0^{\circ} \mathrm{C}\) until the original pressure has tripled. (a) Sketch a pV-diagram for this process. (b) Calculate the amount of work done.

CALC During the time 0.305 mol of an ideal gas under- goes an isothermal compression at \(22.0^{\circ} \mathrm{C}, 468 \mathrm{J}\) of work is done on it by the surroundings. (a) If the final pressure is 1.76 atm, what was the initial pressure? (b) Sketch a pV-diagram for the process.

A gas undergoes two processes. In the first, the volume remains constant at 0.200 \(\mathrm{m}^{3}\) and the pressure increases from \(2.00 \times 10^{5}\) Pa to \(5.00 \times 10^{5}\) Pa. The second process is a compression to a volume of 0.120 \(\mathrm{m}^{3}\) at a constant pressure of \(5.00 \times 10^{5} \mathrm{Pa}\) (a) In a \(p V\) -diagram, show both processes. (b) Find the total work done by the gas during both processes.

A gas in a cylinder expands from a volume of 0.110 \(\mathrm{m}^{3}\) to 0.320 \(\mathrm{m}^{3} .\) Heat flows into the gas just rapidly enough to keep the pressure constant at \(1.65 \times 10^{5}\) Pa during the expansion. The total heat added is \(1.15 \times 10^{5} \mathrm{J}\) (a) Find the work done by the gas. (b) Find the change in internal energy of the gas. (c) Does it matter whether the gas is ideal? Why or why not?

Five moles of an ideal monatomic gas with an initial temperature of \(127^{\circ} \mathrm{C}\) expand and, in the process, absorb 1200 \(\mathrm{J}\) of heat and do 2100 \(\mathrm{J}\) of work. What is the final temperature of the gas?

A gas in a cylinder is held at a constant pressure of \(1.80 \times 10^{5} \mathrm{Pa}\) and is cooled and compressed from 1.70 \(\mathrm{m}^{3}\) to 1.20 \(\mathrm{m}^{3} .\) The intermal energy of the gas decreases by \(1.40 \times 10^{5} \mathrm{J}\) . (a) Find the work done by the gas. (b) Find the absolute value \(Q\) of the heat flow into or out of the gas, and state the direction of the heat flow. (c) Does it matter whether the gas is ideal? Why or why not?

Bio Doughnuts: Breakfast of Champions! A typical doughnut contains 2.0 \(\mathrm{g}\) of protein, 17.0 \(\mathrm{g}\) of carbohydrates, and 7.0 \(\mathrm{g}\) of fat. The average food energy values of these substances are 4.0 \(\mathrm{kcal} / \mathrm{g}\) for protein and carbohydrates and 9.0 kcal/g for fat. (a) During heavy exercise, an average person uses energy at a rate of 510 kcal/h. How long would you have to exercise to "work off" one doughnut? (b) If the energy in the doughnut could somehow be converted into the kinetic energy of your body as a whole, how fast could you move after eating the doughnut? Take your mass to be \(60 \mathrm{kg},\) and express your answer in \(\mathrm{m} / \mathrm{s}\) and in \(\mathrm{km} / \mathrm{h} .\)

Boiling Water at High Pressure. When water is boiled at a pressure of 2.00 atm, the heat of vaporization is \(2.20 \times 10^{6} \mathrm{J} / \mathrm{kg}\) and the boiling point is \(120^{\circ} \mathrm{C}\) . At this pressure, 1.00 \(\mathrm{kg}\) of water has a volume of \(1.00 \times 10^{-3} \mathrm{m}^{3},\) and 1.00 \(\mathrm{kg}\) of steam has a volume of 0.824 \(\mathrm{m}^{3} .\) (a) Compute the work done when 1.00 \(\mathrm{kg}\) of steam is formed at this temperature. (b) Compute the increase in internal energy of the water.

During an isothermal compression of an ideal gas, 335 \(\mathrm{J}\) of heat must be removed from the gas to maintain constant temperature. How much work is done by the gas during the process?

A cylinder contains 0.250 mol of carbon dioxide \(\left(\mathrm{CO}_{2}\right)\) gas at a temperature of \(27.0^{\circ} \mathrm{C}\) . The cylinder is provided with a frictionless piston, which maintains a constant pressure of 1.00 Atm on the gas. The gas is heated until its temperature increases to \(127.0^{\circ} \mathrm{C}\) . Assume that the \(\mathrm{CO}_{2}\) may be treated as an ideal gas. (a) Draw a \(p V\) -diagram for this process. (b) How much work is done by the gas in this process? (c) On what is this work done? (d) What is the change in internal energy of the gas? (e) How much heat was supplied to the gas? (f) How much work would have been done if the pressure had been 0.50 atm?

Acylinder contains 0.0100 mol of helium at \(T=27.0^{\circ} \mathrm{C}\) . (a) How much heat is needed to raise the temperature to \(67.0^{\circ} \mathrm{C}\) while keeping the volume constant? Draw a \(p V\) -diagram for this process. (b) If instead the pressure of the helium is kept constant, how much heat is needed to raise the temperature from \(27.0^{\circ} \mathrm{C}\) to \(67.0^{\circ} \mathrm{C} ?\) Draw a \(p V\) -diagram for this process. (c) What accounts for the difference between your answers to parts (a) and (b)? In which case is more heat required? What becomes of the additional heat? (d) If the gas is ideal, what is the change in its internal energy in part (a)? In part (b)? How do the two answers compare? Why?

Heat \(Q\) flows into a monatomic ideal gas, and the volume increases while the pressure is kept constant. What fraction of the heat energy is used to do the expansion work of the gas?

Three moles of an ideal monatomic gas expands at a constant pressure of 2.50 atm; the volume of the gas changes from \(3.20 \times 10^{-2} \mathrm{m}^{3}\) to \(4.50 \times 10^{-2} \mathrm{m}^{3} .\) (a) Calculate the initial and final temperatures of the gas. (b) Calculate the amount of work the gas does in expanding. (c) Calculate the amount of heat added to the gas. (d) Calculate the change in internal energy of the gas.

Propane gas \(\left(\mathrm{C}_{3} \mathrm{H}_{8}\right)\) behaves like an ideal gas with \(\gamma=1.127\) . Determine the molar heat capacity at constant volume and the molar heat capacity at constant pressure.

CALC The temperature of 0.150 mol of an ideal gas is held constant at \(77.0^{\circ} \mathrm{C}\) while its volume is reduced to 25.0\(\%\) of its initial volume. The initial pressure of the gas is 1.25 atm. (a) Determine the work done by the gas. (b) What is the change in its internal energy? (c) Does the gas exchange heat with its surroundings? If so, how much? Does the gas absorb or liberate heat?

A monatomic ideal gas that is initially at a pressure of \(1.50 \times 10^{5}\) Pa and has a volume of 0.0800 \(\mathrm{m}^{3}\) is compressed adiabatically to a volume of 0.0400 \(\mathrm{m}^{3} .\) (a) What is the final pressure? (b) How much work is done by the gas? (c) What is the ratio of the final temperature of the gas to its initial temperature? Is the gas heated or cooled by this compression?

In an adiabatic process for an ideal gas, the pressure decreases. In this process does the internal energy of the gas increase or decrease? Explain your reasoning.

Two moles of carbon monoxide (CO) start at a pressure of 1.2 atm and a volume of 30 liters. The gas is then compressed adiabatically to \(\frac{1}{3}\) this volume. Assume that the gas may be treated as ideal. What is the change in the internal energy of the gas? Does the internal energy increase or decrease? Does the temperature of the gas increase or decrease during this process? Explain.

The engine of a Ferrari \(\mathrm{F} 355 \mathrm{Fl}\) sports car takes in air at \(20.0^{\circ} \mathrm{C}\) and 1.00 atm and compresses it adiabatically to 0.0900 times the original volume. The air may be treated as an ideal gas with \(\gamma=1.40\) . (a) Draw a p \(V\) -diagram for this process. (b) Find the final temperature and pressure.

A player bounces a basketball on the floor, compressing it to 80.0\(\%\) of its original volume. The air (assume it is essentially \(\mathrm{N}_{2}\) gas) inside the ball is originally at a temperature of \(20.0^{\circ} \mathrm{C}\) and a pressure of 2.00 atm. The ball's inside diameter is 23.9 \(\mathrm{cm}\) . (a) What temperature does the air in the ball reach at its maximum compression? Assume the compression is adiabatic and treat the gas as ideal. (b) By how much does the internal energy of the air change between the ball's original state and its maximum compression?

On a warm summer day, a large mass of air (atmospheric pressure \(1.01 \times 10^{5}\) Pa) is heated by the ground to a temperature of \(26.0^{\circ} \mathrm{C}\) and then begins to rise through the cooler surrounding air. (This can be treated approximately as an adiabatic process; why? Calculate the temperature of the air mass when it has risen to a level at which atmospheric pressure is only \(0.850 \times 10^{5}\) Pa. Assume that air is an ideal gas, with \(\gamma=1.40\) . (This rate of cooling for dry, rising air, corresponding to roughly \(1^{\circ} \mathrm{C}\) per 100 \(\mathrm{m}\) of altitude, is called the dry adiabatic lapse rate.)

A cylinder contains 0.100 mol of an ideal monatomic gas. Initially the gas is at a pressure of \(1.00 \times 10^{5} \mathrm{Pa}\) and occupies a volume of \(2.50 \times 10^{-3} \mathrm{m}^{3}\) (a) Find the initial temperature of the gas in kelvins. (b) If the gas is allowed to expand to twice the initial volume, find the final temperature (in kelvins) and pressure of the gas if the expansion is (i) isothermal; ( (ii) isobaric; (iii) adiabatic.

One mole of ideal gas is slowly compressed to one-third of its original volume. In this compression, the work done on the gas has magnitude 600 \(\mathrm{J}\) . For the gas, \(C_{p}=7 R / 2\) . (a) If the process is isothermal, what is the heat flow \(Q\) for the gas? Does heat flow into or out of the gas? (b) If the process is isobaric, what is the change in internal energy of the gas? Does the internal energy increase or decrease?

A quantity of air is taken from state \(a\) to state \(b\) along a path that is a straight line in the \(p V\) -diagram (Fig. Pl9.39). (a) In this process, does the temperature of the gas increase, decrease, or stay the same? Explain. (b) If \(V_{a}=0.0700 \mathrm{m}^{3}\) , \(V_{b}=0.1100 \mathrm{m}^{3}, \quad p_{a}=1.00 \mathrm{x}\) \(10^{5} \mathrm{Pa},\) and \(p_{b}=1.40 \times 10^{5}\) \(\mathrm{Pa},\) what is the work \(W\) done by that gas in this process? Assume that the gas may be treated as ideal.

Three moles of argon gas (assumed to be an ideal gas) originally at a pressure of \(1.50 \times 10^{4}\) Pa and a volume of 0.0280 \(\mathrm{m}^{3}\) are first heated and expanded at constant pressure to a volume of 0.0435 \(\mathrm{m}^{3}\) , then heated at constant volume until the pressure reaches \(3.50 \times 10^{4} \mathrm{Pa}\) , then cooled and compressed at constant pressure until the volume is again \(0.0280 \mathrm{m}^{3},\) and finally cooled at constant volume until the pressure drops to its original value of \(1.50 \times 10^{4}\) Pa. (a) Draw the \(p V\) -diagram for this cycle. (b) Calculate the total work done by (or on) the gas during the cycle. (c) Calculate the net heat exchanged with the surroundings. Does the gas gain or lose heat overall?

Two moles of helium are initially at a temperature of \(27.0^{\circ} \mathrm{C}\) and occupy a volume of 0.0300 \(\mathrm{m}^{3} .\) The helium first expands at constant pressure until its volume has doubled. Then it expands adiabatically until the temperature returns to its initial value. Assume that the helium can be treated as an ideal gas. (a) Draw a diagram of the process in the \(p V\) -plane. (b) What is the total heat supplied to the helium in the process? (c) What is the total change in internal energy of the helium? (d) What is the total work done by the helium? (e) What is the final volume of the helium?

Starting with 2.50 mol of \(\mathrm{N}_{2}\) gas (assumed to be ideal) in a cylinder at 1.00 atm and \(20.0^{\circ} \mathrm{C},\) a chemist first heats the gas at constant volume, adding \(1.52 \times 10^{4} \mathrm{J}\) of heat, then continues heating and allows the gas to expand at constant pressure to twice its original volume. (a) Calculate the final temperature of the gas. (b) Calculate the amount of work done by the gas. (c) Calculate the amount of heat added to the gas while it was expanding. (d) Calculate the change in internal energy of the gas for the whole process.

Nitrogen gas in an expandable container is cooled from \(50.0^{\circ} \mathrm{C}\) to \(10.0^{\circ} \mathrm{C}\) with the pressure held constant at \(3.00 \times 10^{3}\) Pa. The total heat liberated by the gas is \(2.50 \times 10^{4}\) . Assume that the gas may be treated as ideal. (a) Find the number of moles of gas. (b) Find the change in internal energy of the gas. (c) Find the work done by the gas. (d) How much heat would be liberated by the gas for the same temperature change if the volume were constant?

In a certain process, \(2.15 \times 10^{5} \mathrm{J}\) of heat is liberated by a system, and at the same time the system contracts under a constant external pressure of \(9.50 \times 10^{5} \mathrm{Pa}\) . The internal energy of the system is the same at the beginning and end of the process. Find the change in volume of the system. (The system is not an ideal gas.)

CALC A cylinder with a frictionless, movable piston like that shown in Fig. 19.5 contains a quantity of helium gas. Initially the gas is at a pressure of \(1.00 \times 10^{5}\) Pa, has a temperature of 300 \(\mathrm{K}\) , and occupies a volume of 1.50 \(\mathrm{L}\) . The gas then undergoes two processes. In the first, the gas is heated and the piston is allowed to move to keep the temperature equal to 300 \(\mathrm{K}\) . This continues until the pressure reaches \(2.50 \times 10^{4}\) Pa. In the second process, the gas is compressed at constant pressure until it returns to its original volume of 1.50 L. Assume that the gas may be treated as ideal. (a) In a \(p V\) -diagram, show both processes. (b) Find th volume of the gas at the end of the first process, and find the pressure and temperature at the end of the second process. (c) Fin the total work done by the gas during both processes. (d) Final would you have to do to the gas to return it to its original pres sure and temperature?

CP A Thermodynamic Process in a Liquid. A chemical engineer is studying the properties of liquid methanol \(\left(\mathrm{CH}_{3} \mathrm{OH}\right)\) . She uses a steel cylinder with a cross-sectional area of 0.0200 \(\mathrm{m}^{2}\) and containing \(1.20 \times 10^{-2} \mathrm{m}^{3}\) of methanol. The cylinder is equipped with a tightly fitting piston that supports a load of \(3.00 \times 10^{4} \mathrm{N}\) . The temperature of the system is increased from \(20.0^{\circ} \mathrm{C}\) to \(50.0^{\circ} \mathrm{C}\) . For methanol, the coefficient of volume expansion is \(1.20 \times 10^{-3} \mathrm{K}^{-1},\) the density is \(79 \mathrm{I} \mathrm{kg} / \mathrm{m}^{3},\) and the specific heat at constant pressure is \(c_{p}=2.51 \times 10^{3} \mathrm{J} / \mathrm{kg} \cdot \mathrm{K}\) .You can ignore the expansion of the steel cylinder. Find (a) the increase in volume of the methanol; (b) the mechanical work done by the methanol against the \(3.00 \times 10^{4}\) N force; (c) the amount of heat added to the methanol; (d) the change in internal energy of the methanol. (e) Based on your results, explain whether there is any substantial difference between the specific heats \(c_{p}\) (at constant pressure) and \(c_{V}\) (at constant volume) for methanol under these conditions.

A Thermodynamic Process in an Insect. The African bombardier beetle (Stenaptinus insignis) can emit a jet of defensive spray from the movable tip of its abdomen (Fig. Pl9.57). The beetle's body has reservoirs of two different chemicals; when the beetle is disturbed, these chemicals are combined in a reaction chamber, producing a compound that is warmed from \(20^{\circ} \mathrm{C}\) to \(100^{\circ} \mathrm{C}\) by the heat of reaction. The high pressure produced allows the compound to be sprayed out at speeds up to 19 \(\mathrm{m} / \mathrm{s}(68 \mathrm{km} / \mathrm{h})\) , scaring away predators of all kinds. (The beetle shown in the figure is 2 \(\mathrm{cm}\) long.) Calculate the heat of reaction of the two chemicals (in \(\mathrm{J} / \mathrm{kg} ) .\) Assume that the specific heat of the two chemicals and the spray is the same as that of water, \(4.19 \times 10^{3} \mathrm{J} / \mathrm{kg} \cdot \mathrm{K}\) , and that the initial temperature of the chemicals is \(20^{\circ} \mathrm{C}\) .

High-Altitude Research. A large research balloon containing \(2.00 \times 10^{3} \mathrm{m}^{3}\) of helium gas at 1.00 atm and a temperature of \(15.0^{\circ} \mathrm{C}\) rises rapidly from ground level to an altitude at which the atmospheric pressure is only 0.900 atm (Fig. Pl9.58. Assume the helium behaves like an ideal gas and the balloon's ascent is too rapid to permit much heat exchange with the surrounding air. (a) Calculate the volume of the gas at the higher altitude. (b) Calculate the temperature of the gas at the higher altitude. (c) What is the change in internal energy of the helium as the balloon rises to the higher altitude?

An air pump has a cylinder 0.250 \(\mathrm{m}\) long with a mov- able piston. The pump is used to compress air from the atmosphere (at absolute pressure \(1.01 \times 10^{5}\) Pa) into a very large tank at \(4.20 \times 10^{5}\) Pa gauge pressure. (For air, \(C_{V}=20.8 \mathrm{J} / \mathrm{mol} \cdot \mathrm{K}\) ) (a) The piston begins the compression stroke at the open end of the cylinder. How far down the length of the cylinder has the piston moved when air first begins to flow from the cylinder into the tank? Assume that the compression is adiabatic. (b) If the air is taken into the pump at \(27.0^{\circ} \mathrm{C}\) , what is the temperature of the compressed air? (c) How much work does the pump do in putting 20.0 mol of air into the tank?

A monatomic ideal gas expands slowly to twice its original volume, doing 300 \(\mathrm{J}\) of work in the process. Find the heat added to the gas and the change in internal energy of the gas if the process is (a) isothermal; (b) adiabatic; (c) isobaric.

CALC Acylinder with a piston contains 0.250 mol of \(0 x y-\) gen at \(2.40 \times 10^{5}\) Pa and 355 \(\mathrm{K}\) . The oxygen may be treated as an ideal gas. The gas first expands isobarically to twice its original volume. It is then compressed isothermally back to its original volume, and finally it is cooled isochorically to its original pressure. (a) Show the series of processes on a \(p V\) -diagram. (b) Compute the temperature during the isothermal compression. (c) Compute the maximum pressure. (d) Compute the total work done by the piston on the gas during the series of processes.

CALC A cylinder with a piston contains 0.150 mol of Nnitrogen at \(1.80 \times 10^{5}\) Pa and 300 \(\mathrm{K}\) . The nitrogen may be treated as an ideal gas. The gas is first compressed isobarically to half its original volume. It then expands adiabatically back to its original volume, and finally it is heated isochorically to its original pressure. (a) Show the series of processes in a \(p V\) -diagram. (b) Compute. the temperatures at the beginning and end of the adiabatic expansion. (c) Compute the minimum pressure.

Comparing Thermodynamic Processes. In a cylinder, 1.20 mol of an ideal monatomic gas, initially at \(3.60 \times 10^{5}\) Pa and \(300 \mathrm{K},\) expands until its volume triples. Compute the work done by the gas if the expansion is (a) isothermal; (b) adiabatic; (c) isobaric. (d) Show each process in a \(p V\) -diagram. In which case is the absolute value of the work done by the greatest? Least? (e) In which case is the absolute value of the heat transfer greatest? Least? (f) In which case is the absolute value of the change in internal energy of the gas greatest? Least?

CP Oscillations of a Piston. A vertical cylinder of radius \(r\) contains a quantity of ideal gas and is fitted with a piston with mass \(m\) that is free to move (Fig. P19.69). The piston and the walls of the cylinder are frictionless, and the entire cylinder is placed in a constant-temperature bath. The outside air pressure is \(p_{0} .\) In equilibrium, the piston sits at a height \(h\) above the bottom of the cylinder. (a) Find the absolute pressure of the gas trapped below the piston when in equilibrium. (b) The piston is pulled up by a small distance and released. Find the net force acting on the piston when its base is a distance \(h+y\) above the bottom of the cylinder, where \(y\) is much less than \(h\) . (c) After the piston is displaced from equilibrium and released, it oscillates up and down. Find the frequency of these small oscillations. If the displacement is not small, are the oscillations simple harmonic? How can you tell?