Chapter 19

A tire on a car is inflated to a gauge pressure of \(32 \mathrm{lb} / \mathrm{in}^{2}\) at a temperature of \(27^{\circ} \mathrm{C}\). After the car is driven for \(30 \mathrm{mi}\), the pressure has increased to \(34 \mathrm{lb} / \mathrm{in}^{2} .\) What is the temperature of the air inside the tire at this point? a) \(40^{\circ} \mathrm{C}\) b) \(23^{\circ} \mathrm{C}\) c) \(32^{\circ} \mathrm{C}\) d) \(54^{\circ} \mathrm{C}\)

Molar specific heat at constant pressure, \(C_{p}\), is larger than molar specific heat at constant volume, \(C_{V}\), for a) a monoatomic ideal gas. b) a diatomic atomic gas. c) all of the above. d) none of the above.

An ideal gas may expand from an initial pressure, \(p_{\mathrm{i}},\) and volume, \(V_{\mathrm{i}},\) to a final volume, \(V_{\mathrm{f}}\), isothermally, adiabatically, or isobarically. For which type of process is the heat that is added to the gas the largest? (Assume that \(p_{i}, V_{i}\) and \(V_{f}\) are the same for each process.) a) isothermal process b) adiabatic process c) isobaric process d) All the processes have the same heat flow.

Which of the following gases has the highest rootmean-square speed? a) nitrogen at \(1 \mathrm{~atm}\) and \(30^{\circ} \mathrm{C}\) b) argon at \(1 \mathrm{~atm}\) and \(30^{\circ} \mathrm{C}\) c) argon at \(2 \mathrm{~atm}\) and \(30^{\circ} \mathrm{C}\) d) oxygen at 2 atm and \(30^{\circ} \mathrm{C}\) e) nitrogen at \(2 \mathrm{~atm}\) and \(15^{\circ} \mathrm{C}\)

Two identical containers hold equal masses of gas, oxygen in one and nitrogen in the other. The gases are held at the same temperature. How does the pressure of the oxygen compare to that of the nitrogen? a) \(p_{\mathrm{O}}>p_{\mathrm{N}}\) b) \(p_{\mathrm{O}}=p_{\mathrm{N}}\) c) \(p_{\mathrm{O}}

One mole of an ideal gas, at a temperature of \(0^{\circ} \mathrm{C}\), is confined to a volume of \(1.0 \mathrm{~L}\). The pressure of this gas is a) \(1.0 \mathrm{~atm}\). b) 22.4 atm. c) \(1 / 22.4 \mathrm{~atm}\) d) \(11.2 \mathrm{~atm}\).

One hundred milliliters of liquid nitrogen with a mass of \(80.7 \mathrm{~g}\) is sealed inside a 2 - \(\mathrm{L}\) container. After the liquid nitrogen heats up and turns into a gas, what is the pressure inside the container? a) 0.05 atm b) 0.08 atm c) \(0.09 \mathrm{~atm}\) d) 9.1 atm e) 18 atm

Consider a box filled with an ideal gas. The box undergoes a sudden free expansion from \(V_{1}\) to \(V_{2}\). Which of the following correctly describes this process? a) Work done by the gas during the expansion is equal to \(n R T \ln \left(V_{2} / V_{1}\right)\) b) Heat is added to the box. c) Final temperature equals initial temperature times \(\left(V_{2} / V_{1}\right)\). d) The internal energy of the gas remains constant.

Compare the average kinetic energy at room temperature of a nitrogen molecule to that of a nitrogen atom. Which has the larger kinetic energy? a) nitrogen atom b) nitrogen molecule c) They have the same energy. d) It depends upon the pressure.

Hot air is less dense than cold air and therefore experiences a net buoyant force and rises. Since hot air rises, the higher the elevation, the warmer the air should be. Therefore, the top of Mount Everest should be very warm. Explain why Mount Everest is colder than Death Valley.

The Maxwell speed distribution assumes that the gas is in equilibrium. Thus, if a gas, all of whose molecules were moving at the same speed, were given enough time, they would eventually come to satisfy the speed distribution. But the kinetic theory derivations in the text assumed that when a gas molecule hits the wall of a container, it bounces back with the same energy it had before the collision and that gas molecules exert no forces on each other. If gas molecules exchange energy neither with the walls of their container nor with each other, how can they ever come to equilibrium? Is it not true that if they all had the same speed initially, some would have to slow down and others speed up, according to the Maxwell speed distribution?

When you blow hard on your hand, it feels cool, but when you breathe softly, it feels warm. Why?

Explain why the average velocity of air molecules in a closed auditorium is zero but their root-mean-square speed or average speed is not zero.

In a diesel engine, the fuel-air mixture is compressed rapidly. As a result, the temperature rises to the spontaneous combustion temperature for the fuel, causing the fuel to ignite. How is the temperature rise possible, given the fact that the compression occurs so rapidly that there is not enough time for a significant amount of heat to flow into or out of the fuel-air mixture?

Show that the adiabatic bulk modulus, defined as \(B=-V(d P / d V),\) for an ideal gas is equal to \(\gamma P\).

A monatomic ideal gas expands isothermally from \(\left\\{p_{1}, V_{1}, T_{1}\right\\}\) to \(\left\\{p_{2}, V_{2}, T_{1}\right\\} .\) Then it undergoes an isochoric process, which takes it from \(\left\\{p_{2}, V_{2}, T_{1}\right\\}\) to \(\left\\{p_{1}, V_{2}, T_{2}\right\\}\) Finally the gas undergoes an isobaric compression, which takes it back to \(\left\\{p_{1}, V_{1}, T_{1}\right\\}\) a) Use the First Law of Thermodynamics to find \(Q\) for each of these processes. b) Write an expression for total \(Q\) in terms of \(p_{1}, p_{2}, V_{1},\) and \(V_{2}\).

A relationship that gives the pressure, \(p\), of a substance as a function of its density, \(\rho\), and temperature, \(T\), is called an equation of state. For a gas with molar mass \(M\), write the Ideal Gas Law as an equation of state.

The compression and rarefaction associated with a sound wave propogating in a gas are so much faster than the flow of heat in the gas that they can be treated as adiabatic processes. a) Find the speed of sound, \(v_{s}\), in an ideal gas of molar mass \(M\). b) In accord with Einstein's refinement of Newtonian mechanics, \(v_{\mathrm{s}}\) cannot exceed the speed of light in vacuum, \(c\). This fact implies a maximum temperature for an ideal gas. Find this temperature. c) Evaluate the maximum temperature of part (b) for monatomic hydrogen gas (H). d) What happens to the hydrogen at this maximum temperature?

The kinetic theory of an ideal gas takes into account not only translational motion of atoms or molecules but also, for diatomic and polyatomic gases, vibration and rotation. Will the temperature increase from a given amount of energy being supplied to a monatomic gas differ from the temperature increase due to the same amount of energy being supplied to a diatomic gas? Explain.

A glass of water at room temperature is left on the kitchen counter overnight. In the morning, the amount of water in the glass is smaller due to evaporation. The water in the glass is below the boiling point, so how is it possible for some of the liquid water to have turned into a gas?

A tire has a gauge pressure of \(300 . \mathrm{kPa}\) at \(15.0^{\circ} \mathrm{C}\). What is the gauge pressure at \(45.0^{\circ} \mathrm{C}\) ? Assume that the change in volume of the tire is negligible.

A tank of compressed helium for inflating balloons is advertised as containing helium at a pressure of 2400 psi, which, when allowed to expand at atmospheric pressure, will occupy a volume of \(244 \mathrm{ft}^{3}\). Assuming no temperature change takes place during the expansion, what is the volume of the tank in cubic feet?

A \(1.00-\mathrm{L}\) volume of a gas undergoes first an isochoric process in which its pressure doubles, followed by an isothermal process until the original pressure is reached. Determine the final volume of the gas.

A quantity of liquid water comes into equilibrium with the air in a closed container, without completely evaporating, at a temperature of \(25.0^{\circ} \mathrm{C} .\) How many grams of water vapor does a liter of the air contain in this situation? The vapor pressure of water at \(25.0^{\circ} \mathrm{C}\) is \(3.1690 \mathrm{kPa}\).

One \(1.00 \mathrm{~mol}\) of an ideal gas is held at a constant volume of \(2.00 \mathrm{~L}\). Find the change in pressure if the temperature increases by \(100 .{ }^{\circ} \mathrm{C}\).

Liquid nitrogen, which is used in many physics research labs, can present a safety hazard if a large quantity evaporates in a confined space. The resulting nitrogen gas reduces the oxygen concentration, creating the risk of asphyxiation. Suppose \(1.00 \mathrm{~L}\) of liquid nitrogen \(\left(\rho=808 \mathrm{~kg} / \mathrm{m}^{3}\right)\) evaporates and comes into equilibrium with the air at \(21.0^{\circ} \mathrm{C}\) and \(101 \mathrm{kPa}\). How much volume will it occupy?

Liquid bromine \(\left(\mathrm{Br}_{2}\right)\) is spilled in a laboratory accident and evaporates. Assuming the vapor behaves as an ideal gas, with a temperature \(20.0^{\circ} \mathrm{C}\) and a pressure of \(101.0 \mathrm{kPa}\), find its density.

A sample of gas at \(p=1000 . \mathrm{Pa}, V=1.00 \mathrm{~L},\) and \(T=300 . \mathrm{K}\) is confined in a cylinder. a) Find the new pressure if the volume is reduced to half of the original volume at the same temperature. b) If the temperature is raised to \(400 . \mathrm{K}\) in the process of part (a), what is the new pressure? c) If the gas is then heated to \(600 . \mathrm{K}\) from the initial value and the pressure of the gas becomes \(3000 . \mathrm{Pa},\) what is the new volume?

Air at 1.00 atm is inside a cylinder \(20.0 \mathrm{~cm}\) in radius and \(20.0 \mathrm{~cm}\) in length that sits on a table. The top of the cylinder is sealed with a movable piston. A \(20.0-\mathrm{kg}\) block is dropped onto the piston. From what height above the piston must the block be dropped to compress the piston by \(1.00 \mathrm{~mm} ? 2.00 \mathrm{~mm} ? 1.00 \mathrm{~cm} ?\)

Interstellar space far from any stars is usually filled with atomic hydrogen (H) at a density of 1 atom/cm \(^{3}\) and a very low temperature of \(2.73 \mathrm{~K}\). a) Determine the pressure in interstellar space. b) What is the root-mean-square speed of the atoms? c) What would be the edge length of a cube that would contain atoms with a total of \(1.00 \mathrm{~J}\) of energy?

a) What is the root-mean-square speed for a collection of helium- 4 atoms at \(300 . \mathrm{K} ?\) b) What is the root-mean-square speed for a collection of helium- 3 atoms at 300 . K?

Two isotopes of uranium, \({ }^{235} \mathrm{U}\) and \({ }^{238} \mathrm{U},\) are separated by a gas diffusion process that involves combining them with flourine to make the compound \(\mathrm{UF}_{6} .\) Determine the ratio of the root-mean-square speeds of UF \(_{6}\) molecules for the two isotopes. The masses of \({ }^{235} \mathrm{UF}_{6}\) and \({ }^{238} \mathrm{UF}_{6}\) are \(249 \mathrm{amu}\) and \(252 \mathrm{amu}\).

The electrons in a metal that produce electric currents behave approximately as molecules of an ideal gas. The mass of an electron is \(m_{\mathrm{e}} \doteq 9.109 \cdot 10^{-31} \mathrm{~kg} .\) If the temperature of the metal is \(300.0 \mathrm{~K},\) what is the root-mean-square speed of the electrons?

In a period of \(6.00 \mathrm{~s}, 9.00 \cdot 10^{23}\) nitrogen molecules strike a section of a wall with an area of \(2.00 \mathrm{~cm}^{2}\). If the molecules move with a speed of \(400.0 \mathrm{~m} / \mathrm{s}\) and strike the wall head on in elastic collisions, what is the pressure exerted on the wall? (The mass of one \(\mathrm{N}_{2}\) molecule is \(4.68 \cdot 10^{-26} \mathrm{~kg}\).)

Assuming the pressure remains constant, at what temperature is the root-mean- square speed of a helium atom equal to the root-mean-square speed of an air molecule at STP?

At room temperature, identical gas cylinders contain 10 moles of nitrogen gas and argon gas, respectively. Determine the ratio of energies stored in the two systems. Assume ideal gas behavior.

Calculate the change in internal energy of 1.00 mole of a diatomic ideal gas that starts at room temperature \((293 \mathrm{~K})\) when its temperature is increased by \(2.00 \mathrm{~K}\).

Treating air as an ideal gas of diatomic molecules, calculate how much heat is required to raise the temperature of the air in an \(8.00 \mathrm{~m}\) by \(10.0 \mathrm{~m}\) by \(3.00 \mathrm{~m}\) room from \(20.0^{\circ} \mathrm{C}\) to \(22.0^{\circ} \mathrm{C}\) at \(101 \mathrm{kPa}\). Neglect the change in the number of moles of air in the room.

What is the approximate energy required to raise the temperature of \(1.00 \mathrm{~L}\) of air by \(100 .{ }^{\circ} \mathrm{C} ?\) The volume is held constant.

You are designing an experiment that requires a gas with \(\gamma=1.60 .\) However, from your physics lectures, you remember that no gas has such a \(\gamma\) value. However, you also remember that mixing monatomic and diatomic gases can yield a gas with such a \(\gamma\) value. Determine the fraction of diatomic molecules a mixture has to have to obtain this value.

Suppose \(15.0 \mathrm{~L}\) of an ideal monatomic gas at a pressure of \(1.50 \cdot 10^{5} \mathrm{kPa}\) is expanded adiabatically (no heat transfer) until the volume is doubled. a) What is the pressure of the gas at the new volume? b) If the initial temperature of the gas was \(300 . \mathrm{K},\) what is its final temperature after the expansion?

A diesel engine works at a high compression ratio to compress air until it reaches a temperature high enough to ignite the diesel fuel. Suppose the compression ratio (ratio of volumes) of a specific diesel engine is 20 to \(1 .\) If air enters a cylinder at 1.00 atm and is compressed adiabatically, the compressed air reaches a pressure of 66.0 atm. Assuming that the air enters the engine at room temperature \(\left(25.0^{\circ} \mathrm{C}\right)\) and that the air can be treated as an ideal gas, find the temperature of the compressed air.

Air in a diesel engine cylinder is quickly compressed from an initial temperature of \(20.0^{\circ} \mathrm{C}\), an initial pressure of \(1.00 \mathrm{~atm}\), and an initial volume of \(600 . \mathrm{cm}^{3}\) to a final volume of \(45.0 \mathrm{~cm}^{3}\). Assuming the air to be an ideal diatomic gas, find the final temperature and pressure.

6.00 liters of a monatomic ideal gas, originally at \(400 . \mathrm{K}\) and a pressure of \(3.00 \mathrm{~atm}\) (called state 1 ), undergo the following processes: \(1 \rightarrow 2\) isothermal expansion to \(V_{2}=4 V_{1}\) \(2 \rightarrow 3\) isobaric compression \(3 \rightarrow 1\) adiabatic compression to its original state Find the pressure, volume, and temperature of the gas in states 2 and \(3 .\) How many moles of the gas are there?

Chapter 13 examined the variation of pressure with altitude in the Earth's atmosphere, assuming constant temperature-a model known as the isothermal atmosphere. A better approximation is to treat the pressure variations with altitude as adiabatic. Assume that air can be treated as a diatomic ideal gas with effective molar mass \(M_{\text {air }}=28.97 \mathrm{~g} / \mathrm{mol}\) a) Find the air pressure and temperature of the atmosphere as functions of altitude. Let the pressure at sea level be \(p_{0}=101.0 \mathrm{kPa}\) and the temperature at sea level be \(20.0^{\circ} \mathrm{C}\) b) Determine the altitude at which the air pressure and density are half their sea-level values. What is the temperature at this altitude, in this model? c) Compare these results with the isothermal model of Chapter \(13 .\)

Consider nitrogen gas, \(\mathrm{N}_{2}\), at \(20.0^{\circ} \mathrm{C}\). What is the root-mean-square speed of the nitrogen molecules? What is the most probable speed? What percentage of nitrogen molecules have a speed within \(1.00 \mathrm{~m} / \mathrm{s}\) of the most probable speed? (Hint: Assume the probability of neon atoms having speeds between \(200.00 \mathrm{~m} / \mathrm{s}\) and \(202.00 \mathrm{~m} / \mathrm{s}\) is constant. \()\)

As noted in the text, the speed distribution of molecules in the Earth's atmosphere has a significant impact on its composition. a) What is the average speed of a nitrogen molecule in the atmosphere, at a temperature of \(18.0^{\circ} \mathrm{C}\) and a (partial) pressure of \(78.8 \mathrm{kPa} ?\) b) What is the average speed of a hydrogen molecule at the same temperature and pressure?

A sealed container contains 1.00 mole of neon gas at STP. Estimate the number of neon atoms having speeds in the range from \(200.00 \mathrm{~m} / \mathrm{s}\) to \(202.00 \mathrm{~m} / \mathrm{s}\). (Hint: Assume the probability of neon atoms having speeds between \(200.00 \mathrm{~m} / \mathrm{s}\) and \(202.00 \mathrm{~m} / \mathrm{s}\) is constant.

1 .00 mol of molecular nitrogen gas expands in volume very quickly, so no heat is exchanged with the environment during the process. If the volume increases from \(1.00 \mathrm{~L}\) to \(1.50 \mathrm{~L},\) determine the work done on the environment if the gas's temperature dropped from \(22.0^{\circ} \mathrm{C}\) to \(18.0^{\circ} \mathrm{C}\). Assume ideal gas behavior.

Calculate the root-mean-square speed of air molecules at room temperature \(\left(22.0^{\circ} \mathrm{C}\right)\) from the kinetic theory of an ideal gas.