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.

Two moles of an ideal gas are compressed in a cylinder at a constant temperature of \(85.0^{\circ} \mathrm{C}\) until the original pressure has tripled. (a) Sketch a \(p V\) -diagram for this process. (b) Calculate the amount of work done.

During the time 0.305 mol of an ideal gas undergoes an isothermal compression at \(22.0^{\circ} \mathrm{C}, 518 \mathrm{~J}\) 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 \(p V\) -diagram for the process.

You close off the nozzle of a bicycle tire pump and very slowly depress the plunger so that the air inside is compressed to half its original volume. Assume the air behaves like an ideal gas. If you do this so slowly that the temperature of the air inside the pump never changes: (a) Is the work done by the air in the pump positive or negative? (b) Is the heat flow to the air positive or negative? (c) What can you say about the relative magnitudes of the heat flow and the work? Explain.

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 \(1.80 \times 10^{5}\) 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. 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 \(2.30 \times 10^{5} \mathrm{Pa}\) and is cooled and compressed from 1.70 \(\mathrm{m}^{3}\) to \(1.20 \mathrm{m}^{3} .\) The internal 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?

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 \(9.0 \mathrm{kcal} / \mathrm{g}\) for fat. (a) During hcavy exercise, an average person uscs cnergy at a rate of 510 keal \(/\) 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 \(60 \mathrm{~kg}_{4}\) and express your answer in \(\mathrm{m} / \mathrm{s}\) and \(\mathrm{km} / \mathrm{h}\)

A liquid is irregularly stirred in a well-insulated container and thereby undergoes a rise in temperature. Regard the liquid as the system. (a) Has heat been transferred? How can you tell? (b) Has work been done? How can you tell? Why is it important that the stirring is irregular?(c) What is the sign of \(\Delta U ?\) How can you tell?

A student performs a combustion experiment by burning a mixture of fuel and oxygen in a constant-volume metal can surrounded by a water bath. During the experiment the temperature of the water is observed to rise. Regard the mixture of fuel and oxy gen as the system. (a) Has heat been transferred? How can you tell? (b) Has work been done? How can you tell? (c) What is the sign of \(\Delta U ?\) How can you tell?

Boiling Water at High Pressure. When water is boiled at a pressure of \(2.00 \mathrm{atm},\) the heat of vaporization is \(2.20 \times 10^{5} \mathrm{J} / \mathrm{kg}\) 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 temperture. 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 \(\mathrm{atm}\) 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 \(\mathrm{atm} ?\)

A cylinder 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 pV-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?

An ideal gas expands while the pressure is kept constant. During this process, does heat flow into the gas or out of the gas? Justify your answer.

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?

A cylinder with a movable piston contains 3.00 \(\mathrm{mol}\) of \(\mathrm{N}_{2}\) gas (assumed to behave like an ideal gas). (a) The \(\mathrm{N}_{2}\) is heated at constant volume until 1557 \(\mathrm{J}\) of heat have been added. Calculate the change in temperature. (b) Suppose the same amount of heat is added to the \(\mathrm{N}_{2}\) , but this time the gas is allowed to expand while remaining at constant pressure. Calculate the temperature change. (c) In which case, (a) or \((b),\) is the final internal energy of the \(\mathbf{N}_{2}\) higher? How do you know? What accounts for the difference between the two cases?

Three moles of an ideal monatomic gas expands at a con-stant 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 encrgy of the gas.

The temperature of 0.150 mol of an ideal gas is held stant 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 \(\mathrm{atm}\) . (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?

An experimenter adds \(970 \mathrm{~J}\) of heat to \(1.75 \mathrm{~mol}\) of an ideal gas to heat it from \(10.0^{\circ} \mathrm{C}\) to \(25.0^{\circ} \mathrm{C}\) at constant pressure. The gas does \(+223 \mathrm{~J}\) of work during the expansion. (a) Calculate the change in internal energy of the gas. (b) Calculate \(\gamma\) for the gas.

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.

A monatomic ideal gas that is initially at a pressure of \(1.50 \times 10^{5} \mathrm{Pa}\) 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?

The engine of a Ferrari \(\mathrm{F} 355 \mathrm{FI}\) sports car takes in air at \(20.0^{\circ} \mathrm{C}\) and 1.00 \(\mathrm{atm}\) 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.

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.

A player bounces a basketball on the floor, connpressing it to 80.0\(\%\) of its original volume. The air (assume it is essentially N_{2} gas) inside the ball is originally at a temperature of \(20.0^{\circ} \mathrm{C}\) and a pressure of 2.00 \(\mathrm{atm}\) . The ball's diameter is \(23.9 \mathrm{cm} .\) (a) What temperature does the air in the ball reach at its maximum compression? (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 (armospheric pressure \(1.01 \times 10^{5} \mathrm{Pa}\) ) 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 \(0.850 \times 10^{5} \mathrm{Pa}\) . Assume that air is an ideal gas, with \(\gamma=1.40\) . (This rate of cool- ing for dry, rising air, corresponding to roughly \(1^{\circ} \mathrm{C}\) per 100 \(\mathrm{m}\) of attitude, is called the dry adiabatic lapse rate.)

On a warm summer day, a large mass of air (atmospheric pressure \(1.01 \times 10^{5} \mathrm{Pa}\) ) 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 \(0.850 \times 10^{5} \mathrm{Pa}\) . Assume that air is an ideal gas, with \(\gamma=1.40\) . (This rate of cool- ing for dry, rising air, corresponding to roughly \(1^{\circ} \mathrm{C}\) per 100 \(\mathrm{m}\) of altitude, is called the dry adiabatic lapse rate.)

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 \(3.00 \times 10^{5} \mathrm{Pa}\) . The total heat liberated by the gas is \(2.50 \times 10^{4} \mathrm{J}\) . 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.)

A Thermodymamic Process in a Liquid. A chemical engineer is studying the properties of liquid methanol \(\left(\mathrm{CH}_{3} \mathrm{OH}\right)\) She uses a stecl cylinder with a cross-sectional. The cylindor is 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 791 \(\mathrm{kg} / \mathrm{m}^{3}\) , and the specific heat capacity 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 stecl cylinder. Find (a) the increase in volume of the methanol; (b) the mechanical work done by the methanol against the \(3.00 \times 10^{6} \mathrm{N}\) force; (c) the amount of heat added to the methanol; (d) the change in internal cnergy of the methanol. (e) Based on your results, explain whether there is any substantial difference between the specific heat capacities \(c_{p}\) (at constant pressure) and \(c_{V}\) (at constant volume) for methanol under these conditions.

A Thermodynamic \(\mathbf{P} \mathbf{r} \mathbf{o}-\) cess In an Insect. The Africanbombardier beetle Stenaptinus insignis can emit a jet of defensive spray from the movable tip of its abdomen (Fig, 19.32\()\) . The beetle's body has reservoirs of two different chemicals; when the beetle is disturbed, these chemicals are combined in a reaction chamber, producing 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 J/kg). Assume that the specific heat capacity of the two chemicall 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 \((\text { Fig. } 19.33)\) . 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?

Chinook. During certain seasons strong winds called chinooks blow from the west across the eastern slopes of the Rockies and downhill into Denver and nearby areas. Although the mountains are cool, the wind in Denver is very hot; within a few minutes after the chinook wind arrives, the temperature can climb 20 \(\mathrm{C}^{\circ}\) ("chinook" is a Native American word meaning "snow cater"). Similar winds occur in the Alps (called fochns) and in southern California (called Santa Anas). (a) Explain why the temperature of the chinook wind rises as it descends the slopes. Why is it important that the wind be fast moving? (b) Suppose a strong wind is blowing toward Denver (elevation 1630 \(\mathrm{m} )\) from Grays Peak \((80 \mathrm{km} \text { west of Denver, at an elevation of } 4350 \mathrm{m})\) , where the air pressure is \(5.60 \times 10^{4} \mathrm{Pa}\) and the air temperature is \(-15.0 ^{\circ} \mathrm{C}\) . The temperature and pressure in Denver before the wind arrives are \(20^{\circ} \mathrm{C}\) and \(8.12 \times 10^{4} \mathrm{Pa}\) . By how many Celsius degrees will the temperature in Denver rise when the chinook arrives?

An air pump has a cylinder 0.250 m long with a movable piston. The pump is used to compress air from the atmosphere (at absolute pressure \(1.01 \times 10^{3} \mathrm{Pa}\) ) 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 trom 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 200 \(\mathrm{mol}\) of air into the tank?

Engine Thrbochargers and Intercoolers. The power output of an automobile engine is directly proportional to the mass of air that can be forced into the volume of the engine's cylinders to react chemically with gasoline. Many cars have a turbochargers which compresses the air before it enters the engine, giving a greater mass of air per volume. This rapid, essentially adiabatic compression also heats the air. To compress it further, the air then passes through an intercooler in which the air exchanges heat with its surroundings at essentially constant pressure. The air is then drawn into the cylinders. In a typical installation, air is taken into the turbocharger at atmospheric pressure \(\left(1.01 \times 10^{5} \mathrm{Pa}\right)\) , density \(p=1.23 \mathrm{kg} / \mathrm{m}^{3}\) , and temperature \(15.0^{\circ} \mathrm{C}\) . It is compressed adiabatically to \(1.45 \times 10^{5} \mathrm{Pa}\) . In the intercooler, the air is cooled to the original temperature of \(15.0^{\circ} \mathrm{C}\) at a constant pressure of \(1.45 \times 10^{5} \mathrm{Pa}\) (a) Draw a \(p V\) -diagram for this sequence of processes. b) If the volume of one of the engine's cylinders is 575 \(\mathrm{cm}^{3}\) , what mass of air exiting from the intercooler will fill the cylinder at \(1.45 \times 10^{5} \mathrm{Pa} 2\) Compared to the power output of an engine that takes in air at \(1.01 \times 10^{5} \mathrm{Pa}\) at \(15.0^{\circ} \mathrm{C}\) , what percentage increase in power is obtained by using the turbocharger and intercooler? (c) If the intercooler is not used, what mass of air exiting from the turbocharger will fill the cylinder at \(1.45 \times 10^{5} \mathrm{Pa} ?\) Compared to the power output of an engine that takes in air at obtained by using the turbocharger alone?

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.

A cylinder with a piston contains 0.250 mol of oxygen at \(2.40 \times 10^{5} \mathrm{Pa}\) 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 compresed 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. e

A cylinder with a piston contains 0.150 mol of mitrogen at \(1.80 \times 10^{5} \mathrm{Pa}\) 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.

In a cylinder sealed with a piston, you rapidly compress 3.00 \(\mathrm{L}\) of \(\mathrm{N}_{2}\) gas initially at 1.00 atm pressure and \(0.00^{\circ} \mathrm{C}\) to half its original volume. Assume the \(\mathrm{N}_{2}\) behaves like an ideal gas. (a) Calculate the final temperature and pressure of the gas. (b) If you now cool the gas back to \(0.00^{\circ} \mathrm{C}\) without changing the pressure, what is its final volume?

Oscillations of a Piston. A vertical cylinder of radins \(r\) contains a quantity of ideal gas and is fitted with a piston with mass \(m\) that is free to move (Fig. 19.34\()\) . The piston and the walls of the cylinder are frictionless and the entire cylinder is placed in aconstant-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 nthese small oscillations. If the displacement is not small, are the oscillations simple harmonic? How can you tell?