Estimate the rms speed and the average kinetic energy of a

True or false: (a) The zeroth law of thermodynamics states that two objects in thermal equilibrium with each other must be in thermal equilibrium with a third object. (b) The Fahrenheit and Celsius temperature scales differ only in the choice of the ice-point temperature. (c) The Celsius degree and the kelvin are the same size.

How can you determine if two objects are in thermal equilibrium with each other when putting them into physical contact with each other would have undesirable effects? (For example, if you put a piece of sodium in contact with water there would be a violent chemical reaction.)

Yesterday I woke up and it was in my bedroom, said Mert to his old friend Mort. Thats nothing, replied Mort. My room was Who had the colder room, Mert or Mort?

Two identical vessels contain different ideal gases at the same pressure and temperature. It follows that (a) the number of gas molecules is the same in both vessels, (b) the total mass of gas is the same in both vessels, (c) the average speed of the gas molecules is the same in both vessels, (d) None of the above.

Figure 17-18 shows a plot of volume versus absolute temperature for a process that takes a fixed amount of an ideal gas from point A to point B. What happens to the pressure of the gas during this process? SSM

Figure 17-19 shows a plot of pressure versus absolute temperature for a process that takes a sample of an ideal gas from point A to point B. What happens to the volume of the gas during this process?

If a vessel contains equal amounts, by mass, of helium and argon, which of the following are true? (a) The partial pressure exerted by each of the two gases on the walls of the container is the same. (b) The average speed of a helium atom is the same as that of an argon atom. (c) The number of helium atoms and argon atoms in the vessel are equal. (d) None of the above.

By what factor must the absolute temperature of a gas be increased to double the rms speed of its molecules?

Two different gases are at the same temperature. What can be said about the average translational kinetic energies of the molecules? What can be said about the rms speeds of the gas molecules?

A vessel holds a mixture of helium (He) and methane The ratio of the rms speed of the He atoms to that of the molecules is (a) 1, (b) 2, (c) 4, (d) 16.

True or false: If the pressure of a fixed amount of gas increases, the temperature of the gas must increase.

Why might the Celsius and Fahrenheit scales be more convenient than the absolute scale for ordinary, nonscientific purposes?

An astronomer claims that the temperature at the center of the Sun is about degrees. Do you think that this temperature is in kelvins, degrees Celsius, or does it not matter?

Imagine that you have a fixed amount of ideal gas in a container that expands to maintain constant pressure. If you double the absolute temperature of the gas, the average speed of the molecules (a) remains constant, (b) doubles, (c) quadruples, (d) increases by a factor of

Suppose that you compress an ideal gas to half its original volume, while also halving its absolute temperature. During this process, the pressure of the gas (a) halves, (b) remains constant, (c) doubles, (d) quadruples.

The average translational kinetic energy of the molecules of a gas depends on (a) the number of moles and the temperature, (b) the pressure and the temperature, (c) the pressure only, (d) the temperature only.

Which speed is greater, the speed of sound in a gas or the rms speed of the molecules of the gas? Justify your answer, using the appropriate formulas, and explain why your answer is intuitively plausible.

Imagine that you increase the temperature of a gas while holding its volume fixed. Explain in terms of molecular motion why the pressure of the gas on the walls of its container increases.

Imagine that you compress a gas while holding it at a fixed temperature (perhaps by immersing the container in cool water). Explain in terms of molecular motion why the pressure of the gas on the walls of its container increases.

Oxygen has a molar mass of and nitrogen has a molar mass of The oxygen and nitrogen molecules in a room have: (a) equal average translational kinetic energies, but the oxygen molecules have a larger average speed than the nitrogen molecules have. (b) equal average translational kinetic energies, but the oxygen molecules have a smaller average speed than the nitrogen molecules have. (c) equal average translational kinetic energies and equal average speeds. (d) equal average speeds, but the oxygen molecules have a larger average translational kinetic energy than the nitrogen molecules have. (e) equal average speeds, but the oxygen molecules have a smaller average translational kinetic energy than the nitrogen molecules have. (f) None of the above.

Liquid nitrogen is relatively cheap, while liquid helium is relatively expensive. One reason for the difference in price is that while nitrogen is the most common constituent of the atmosphere, only small traces of helium can be found in the atmosphere. Use ideas from this chapter to explain why it is that only small traces of helium can be found in the atmosphere.

Estimate the total number of air molecules in your classroom.

Estimate the density of dry air at sea level on a warm summer day.

A stoppered test tube that has a volume of 10.0 mL has 1.00 mL of water at its bottom. The water has a temperature of and is initially at a pressure of 1.00 atm. The test tube is held over a flame until the water has completely boiled away. Estimate the final pressure inside the test tube.

In Chapter 11, we found that the escape speed at the surface of a planet of radius is where is the acceleration due to gravity at the surface of the planet. If the rms speed of a gas is greater than about 15 to 20 percent of the escape speed of a planet, virtually all of the molecules of that gas will escape the atmosphere of the planet. (a) At what temperature is for equal to 15 percent of the escape speed for Earth? (b) At what temperature is for equal to 15 percent of the escape speed for Earth? (c) Temperatures in the upper atmosphere reach 1000 K. How does this help account for the low abundance of hydrogen in Earths atmosphere? (d) Compute the temperatures for which the rms speeds of and are equal to 15 percent of the escape speed at the surface of the moon, where is about one-sixth of its value on Earth and How does this account for the absence of an atmosphere on the moon?

The escape speed for gas molecules in the atmosphere of Mars is and the surface temperature of Mars is typically Calculate the rms speeds for (a) (b) and (c) at this temperature. (d) Are and likely to be found in the atmosphere of Mars?

The escape speed for gas molecules in the atmosphere of Jupiter is and the surface temperature of Jupiter is typically Calculate the rms speeds for (a) (b) and (c) at this temperature. (d) Are and likely to be found in the atmosphere of Jupiter? SSM

Estimate the average pressure on the front wall of a racquetball court, due to the collisions of the ball with the wall during a game. Use any reasonable numbers for the mass of the ball, its typical speed, and the dimensions of the court. Is the average pressure from the ball significant compared to that from the air?

To a first approximation, the Sun consists of a gas of equal numbers of protons and electrons. (The masses of these particles can be found in Appendix B.) The temperature at the center of the Sun is about and the density of the Sun is about Because the temperature is so high, the protons and electrons are separate particles (rather than being joined together to form hydrogen atoms). (a) Estimate the pressure at the center of the Sun. (b) Estimate the rms speeds of the protons and the electrons at the center of the Sun.

CONTEXT-RICH, ENGINEERING APPLICATION You are designing a vacuum chamber for fabricating reflective coatings. Inside this chamber, a small sample of metal will be vaporized so that its atoms travel in straight lines (the effects of gravity are negligible during the brief time of flight) to a surface where they land to form a very thin film. The sample of metal is 30 cm from the surface to which the metal atoms will adhere. How low must the pressure in the chamber be so that the metal atoms only rarely collide with air molecules before they land on the surface?

BIOLOGICAL APPLICATION In normal breathing conditions, approximately 5 percent of each exhaled breath is carbon dioxide. Given this information, and neglecting any difference in water-vapor content, estimate the typical difference in mass between an inhaled breath and an exhaled breath.

A certain ski wax is rated for use between and What is this temperature range on the Fahrenheit scale?

The melting point of gold is Express this temperature in degrees Celsius.

A weather report indicates that the temperature is expected to drop by over the next four hours. By how many degrees on the Fahrenheit scale will the temperature drop?

The length of the column of mercury in a thermometer is 4.00 cm when the thermometer is immersed in ice water at 1 atm of pressure, and 24.0 cm when the thermometer is immersed in boiling water at 1 atm of pressure. Assume that the length of the mercury column varies linearly with temperature. (a) Sketch a graph of the length of the mercury column versus temperature (in degrees Celsius). (b) What is the length of the column at room temperature (c) If the mercury column is 25.4 cm long when the thermometer is immersed in a chemical solution, what is the temperature of the solution?

The temperature of the interior of the Sun is about What is this temperature in (a) Celsius degrees, (b) Fahrenheit degrees?

The boiling point of nitrogen, is 77.35 K. Express this temperature in degrees Fahrenheit.

The pressure of a constant-volume gas thermometer is 0.400 atm at the ice point and 0.546 atm at the steam point. (a) Sketch a graph of pressure versus Celsius temperature for this thermometer. (b) When the pressure is 0.100 atm, what is the temperature? (c) What is the pressure at (the boiling point of sulfur)?

A constant-volume gas thermometer reads 50.0 torr at the triple point of water. (a) Sketch a graph of pressure versus absolute temperature for this thermometer. (b) What will be the pressure when the thermometer measures a temperature of 300 K? (c) What ideal-gas temperature corresponds to a pressure of 678 torr?

Aconstant-volume gas thermometer has a pressure of 30.0 torr when it reads a temperature of 373 K. (a) Sketch a graph of pressure versus absolute temperature for this thermometer. (b) What is its triple-point pressure (c) What temperature corresponds to a pressure of 0.175 torr?

At what temperature do the Fahrenheit and Celsius temperature scales give the same reading?

Sodium melts at 371 K. What is the melting point of sodium on the Celsius and Fahrenheit temperature scales?

The boiling point of oxygen at 1.00 atm is 90.2 K. What is the boiling point of oxygen at 1.00 atm on the Celsius and Fahrenheit scales?

On the Raumur temperature scale, the melting point of ice is and the boiling point of water is Derive expressions for converting temperatures on the Raumur scale to the Celsius and Fahrenheit scales.

ENGINEERING APPLICATION A thermistor is a solid-state device widely used in a variety of engineering applications. Its primary characteristic is that its electrical resistance varies greatly with temperature. Its temperature dependence is given approximately by where is in ohms is in kelvins, and and B are constants that can be determined by measuring at calibration points such as the ice point and the steam point. (a) If at the ice point and at the steam point, find and B. (b) What is the resistance of the thermistor at (c) What is the rate of change of the resistance with temperature at the ice point and the steam point? (d) At which temperature is the thermistor most sensitive?

An ideal gas in a cylinder fitted with a piston (Figure 17-20) is held at fixed pressure. If the temperature of the gas increases from 50 to 100C, by what factor does the volume change? SSM F I G U R E 1 7 - 2 0 Problems 46, 71

A 10.0-L vessel contains gas at a temperature of and a pressure of 4.00 atm. How many moles of gas are in the vessel? How many molecules?

A pressure as low as can be achieved using an oil diffusion pump. How many molecules are there in of a gas at this pressure if its temperature is 300 K?

You copy the following paragraph from a Martian physics textbook: 1 snorf of an ideal gas occupies a volume of 1.35 zaks. At a temperature of 22 glips, the gas has a pressure of 12.5 klads. At a temperature of glips, the same gas now has a pressure of 8.7 klads. Determine the temperature of absolute zero in glips.

Amotorist inflates the tires of her car to a gauge pressure of 180 kPa on a day when the temperature is 8. When she arrives at her destination, the tire pressure has increased to 245 kPa. What is the temperature of the tires if we assume that (a) the tires do not expand, or (b) that the tires expand so the volume of the enclosed air increases by 7 percent?

A room is 6.0 m by 5.0 m by 3.0 m. (a) If the air pressure in the room is 1.0 atm and the temperature is 300 K, find the number of moles of air in the room. (b) If the temperature increases by 5.0 K and the pressure remains constant, how many moles of air leave the room?

Imagine that 10.0 g of liquid helium, initially at 4.20 K, evaporate into an empty balloon that is kept at 1.00-atm pressure. What is the volume of the balloon at (a) 25.0 K, and (b) 293 K?

A closed container with a volume of 6.00 L holds 10.0 g of liquid helium at 25.0 K and enough air to fill the rest of its volume at a pressure of 1.00 atm. The helium then evaporates and the container warms to room temperature (293 K). What is the final pressure inside the container?

An automobile tire is filled to a gauge pressure of 200 kPa when its temperature is (Gauge pressure is the difference between the actual pressure and atmospheric pressure.) After the car has been driven at high speeds, the tire temperature increases to (a) Assuming that the volume of the tire does not change and that air behaves as an ideal gas, find the gauge pressure of the air in the tire. (b) Calculate the gauge pressure if the tire expands so the volume of the enclosed air increases by 10 percent.

After nitrogen and oxygen the most abundant molecule in Earths atmosphere is water, However, the fraction of molecules in a given volume of air varies dramatically, from practically zero percent under the driest conditions to a high of 4 percent where it is very humid. (a) At a given temperature and pressure, would air be denser when its water vapor content is large or small? (b) What is the difference in mass, at room temperature and atmospheric pressure, between a cubic meter of air with no water vapor molecules and a cubic meter of air in which 4 percent of the molecules are water vapor molecules?

A scuba diver is 40 m below the surface of a lake, where the temperature is He releases an air bubble that has a volume of The bubble rises to the surface, where the temperature is Assume that the air in the bubble is always in thermal equilibrium with the surrounding water, and assume that there is no exchange of molecules between the bubble and the surrounding water. What is the volume of the bubble right before it breaks the surface? Hint: Remember that the pressure also changes.

ENGINEERING APPLICATION A hot-air balloon is open at the bottom. The balloon, which has a volume of is filled with air that has an average temperature of The air outside the balloon has a temperature of and a pressure of 1.00 atm. How large a payload (including the envelope of the balloon itself) can the balloon lift? Use for the molar mass of air. (Neglect the volume of both the payload and the envelope of the balloon.)

A helium balloon is used to lift a load of 110 N. The weight of the envelope of the balloon is 50.0 N and the volume of the helium when the balloon is fully inflated is The temperature of the air is and the atmospheric pressure is 1.00 atm. The balloon is inflated with a sufficient amount of helium gas that the net upward force on the balloon and its load is 30.0 N. Neglect any effects due to the changes of temperature as the altitude changes. (a) How many moles of helium gas are contained in the balloon? (b) At what altitude will the balloon be fully inflated? (c) Does the balloon ever reach the altitude at which it is fully inflated? (d) If the answer to Part (c) is Yes, what is the maximum altitude attained by the balloon?

(a) One mole of argon gas is confined to a 1.0-liter container at a pressure of 10 atm. What is the rms speed of the argon atoms? (b) Compare your answer to the rms speed for helium atoms under the same conditions.

Find the total translational kinetic energy of the molecules of 1.0 L of oxygen gas at a temperature of and a pressure of 1.0 atm.

Estimate the rms speed and the average kinetic energy of a hydrogen atom in a gas at a temperature of (At this temperature, which is approximately the temperature in the interior of a star, hydrogen atoms are ionized and become protons.)

Liquid helium has a temperature of only 4.20 K and is in equilibrium with its vapor at atmospheric pressure. Calculate the rms speed of a helium atom in the vapor at this temperature, and comment on the result.

Show that the mean free path for a molecule in an ideal gas at temperature and pressure is given by

ENGINEERING APPLICATION State-of-the-art vacuum equipment can attain pressures as low as \(7.0 \times 10^{-11}\) Pa. Suppose that a chamber contains helium at this pressure and at room temperature (300 K). Estimate the mean free path and the collision time for helium in the chamber. Assume the diameter of a helium atom is \(1.0 \times 10^{-10} \ \mathrm m\)

Oxygen is confined to a cube-shaped container 15 cm on an edge at a temperature of 300 K. Compare the average kinetic energy of a molecule of the gas to the change in its gravitational potential energy if it falls 15 cm (the height of the container).

Use calculus to show that given by Equation 17-36, has its maximum value at a speed

The fractional distribution function is defined in Equation 17-36. Because gives the fraction of molecules that have speeds in the range between and the integral of over all the possible ranges of speeds must equal 1. Given that the integral show that where is given by Equation 17-36.

Given that the integral calculate the average speed of molecules in a gas using the MaxwellBoltzmann distribution function.

MULTISTEP The translational kinetic energies of the molecules of a gas are distributed according to the MaxwellBoltzmann energy distribution, Equation 17-38. (a) Determine the most probable value of the translational kinetic energy (in terms of the temperature ) and compare this value to the average value. (b) Sketch a graph of the translational kinetic energy distribution and label the most probable energy and the average energy. (Do not worry about calibrating the vertical scale of the graph.) (c) Your teacher says, Just looking at the graph allows you to see that the average translational kinetic energy is considerably greater than the most probable translational kinetic energy. What feature(s) of the graph support her claim?

Find the temperature at which the rms speed of a molecule of hydrogen gas equals \(343 \mathrm{~m} / \mathrm{s}\).

(a) If 1.0 mol of a gas in a cylindrical container occupies a volume of 10 L at a pressure of 1.0 atm, what is the temperature of the gas in kelvins? (b) The cylinder is fitted with a piston so that the volume of the gas (Figure 17-20) can vary. When the gas is heated at constant pressure, it expands to a volume of 20 L. What is the temperature of the gas in kelvins? (c) Next, the volume is fixed at 20 L, and the gass temperature is increased to 350 K. What is the pressure of the gas now?

MULTISTEP (a) The volume per molecule of a gas is the reciprocal of the number density (the number of molecules per unit volume). Find the average volume per molecule for dry air at room temperature and atmospheric pressure. (b) Take the cube root of your answer to Part (a) to obtain a rough estimate of the average distance d between air molecules. (c) Find or estimate the average diameter D of an air molecule, and compare it to your answer to Part (b). (d) Sketch the molecules in a cube-shaped volume of air, with the edge length of the cube equal to 3d. Make your figure to scale and place the molecules in what you think is a typical configuration. (e) Use your picture to explain why the mean free path of an air molecule is much greater than the average distance between molecules.

CONCEPTUAL The MaxwellBoltzmann distribution applies not just to gases, but also to the molecular motions within liquids. The fact that not all molecules have the same speed helps us understand the process of evaporation. (a) Explain in terms of molecular motion why a drop of water becomes cooler as molecules evaporate from the drops surface. (Evaporative cooling is an important mechanism for regulating our body temperatures, and is also used to cool buildings in hot, dry locations.) (b) Use the MaxwellBoltzmann distribution to explain why even a slight increase in temperature can greatly increase the rate at which a drop of water evaporates.

A cubic metal box that has 20-cm-long edges contains air at a pressure of 1.0 atm and a temperature of 300 K. The box is sealed so that the enclosed volume remains constant, and it is heated to a temperature of 400 K. Find the force due to the internal air pressure on each wall of the box.

ENGINEERING APPLICATION In attempting to create liquid hydrogen for fuel, one of the proposals is to convert plain old water into and gases by electrolysis. How many moles of each of these gases result from the electrolysis of 2.0 L of water?

A 40-cm-long hollow cylinder of negligible mass rests on its side on a horizontal frictionless table. The cylinder is divided into two equal sections by a vertical nonporous membrane. One section contains nitrogen and the other contains oxygen. The pressure of the nitrogen is twice that of the oxygen. How far will the cylinder move if the membrane breaks?

A cylinder of fixed volume contains a mixture of helium gas (He) and hydrogen gas at a temperature and pressure If the temperature is doubled to the pressure would also double, except for the fact that at this temperature the is essentially 100 percent dissociated into In reality, at pressure the temperature is If the mass of the hydrogen in the cylinder is what is the mass of the helium in the cylinder?

The mean free path for molecules at a temperature of 300 K and at 1.00-atm pressure is Use these data to estimate the size of an molecule.

ENGINEERING APPLICATION Current experiments in atomic trapping and cooling can create low-density gases of rubidium and other atoms with temperatures in the nanokelvin range. These atoms are trapped and cooled using magnetic fields and lasers in ultrahigh vacuum chambers. One method that is used to measure the temperature of a trapped gas is to turn the trap off and measure the time it takes for molecules of the gas to fall a given distance. Consider a gas of rubidium atoms at a temperature of 120 nK. Calculate how long it would take an atom traveling at the rms speed of the gas to fall a distance of 10.0 cm if (a) it were initially moving directly downward, and (b) if it were initially moving directly upward. Assume that the atom does not collide with any other atoms along its trajectory.

A cylinder is filled with 0.10 mol of an ideal gas at standard temperature and pressure, and a 1.4-kg piston seals the gas in the cylinder (Figure 17-21) with a frictionless seal. The trapped column of gas is 2.4 m high. The piston and cylinder are surrounded by air, also at standard temperature and pressure. The piston is released from rest and starts to fall. The motion of the piston ceases after the oscillations stop with the piston and the trapped air in thermal equilibrium with the surrounding air. (a) Find the height of the gas column. (b) Suppose that the piston is pushed down below its equilibrium position by a small amount and then released. Assuming that the temperature of the gas remains constant, find the frequency of vibration of the piston.

SPREADSHEET, MULTISTEP To solve this problem, you will use a spreadsheet to study the distribution of molecular speeds in a gas. Figure 17-22 should help you get started. (a) Enter the values for constants and as shown. Then in column A, enter values of speeds ranging from in increments of (This spreadsheet will be long.) In cell B7, enter the formula for the MaxwellBoltzmann fractional speed distribution. This formula contains parameters v, R, M and T. Substitute A7 for v, B$1 for R, B$2 for M and B$3 for T. Then use the FILL DOWN command to enter the formula in the cells below B7. Create a graph of versus using the data in columns A and B. (b) Explore how the graph changes as you increase and decrease the temperature, and describe the results. (c) Add a third column in which each cell contains the cumulative sum of all values, multiplied by the interval size (which equals 1), in the rows above and including the row in question. What is the physical interpretation of the numbers in this column? (d) For nitrogen gas at 300 K, what percentage of the molecules has speeds less than (e) For nitrogen gas at 300 K, what percentage of the molecules has speeds greater than 700 m>s