The escape speed from the Earth is so that a gas molecule

How does the number of atoms in a 27.5-gram gold ring compare to the number in a silver ring of the same mass?

(a) Room temperature is often taken to be 68F. What is this on the Celsius scale? (b) The temperature of the filament in a lightbulb is about 1900C. What is this on the Fahrenheit scale?

(a) Room temperature is often taken to be 68F. What is this on the Celsius scale? (b) The temperature of the filament in a lightbulb is about 1900C. What is this on the Fahrenheit scale?

Among the highest and lowest natural air temperatures claimed are 136F in the Libyan desert and in Antarctica. What are these temperatures on the Celsius scale?

A thermometer tells you that you have a fever of 38.9C. What is this in Fahrenheit?

(a) 18 below zero on the Celsius scale is what Fahrenheit temperature? (b) 18 below zero on the Fahrenheit scale is what Celsius temperature?

Determine the temperature at which the Celsius and Fahrenheit scales give the same numerical reading ATC = TF

In an alcohol-in-glass thermometer, the alcohol column has length 12.61 cm at 0.0C and length 22.79 cm at 100.0C. What is the temperature if the column has length (a) 18.70 cm, and (b) 14.60 cm?

The Eiffel Tower (Fig. 1331) is built of wrought iron approximately 300 m tall. Estimate how much its height changes between January (average temperature of 2C) and July (average temperature of 25C). Ignore the angles of the iron beams and treat the tower as a vertical beam.

A concrete highway is built of slabs 12 m long (15C). How wide should the expansion cracks between the slabs be (at 15C) to prevent buckling if the range of temperature is

(I) Super Invar™, an alloy of iron and nickel, is a strong material with a very low coefficient of thermal expansion \((0.20 \times 10^{-6}/C^{\circ})\). A 1.8-m-long tabletop made of this alloy is used for sensitive laser measurements where extremely high tolerances are required. How much will this alloy table expand along its length if the temperature increases 6.0 \(C^{\circ}\)? Compare to tabletops made of steel.

(II) To what temperature would you have to heat a brass rod for it to be 1.5% longer than it is at \(25^{\circ}C\)?

To make a secure fit, rivets that are larger than the rivet hole are often used and the rivet is cooled (usually in dry ice) before it is placed in the hole. A steel rivet 1.872 cm in diameter is to be placed in a hole 1.870 cm in diameter in a metal at 22C. To what temperature must the rivet be cooled if it is to fit in the hole?

An ordinary glass is filled to the brim with 450.0 mL of water at 100.0C. If the temperature of glass and water is decreased to 20.0C, how much water could be added to the glass?

An aluminum sphere is 8.75 cm in diameter. What will be its % change in volume if it is heated from 30C to 160C?

It is observed that 55.50 mL of water at 20C completely fills a container to the brim. When the container and the water are heated to 60C, 0.35 g of water is lost. (a) What is the coefficient of volume expansion of the container? (b) What is the most likely material of the container? Density of water at 60C is 0.98324 gmL.

A brass plug is to be placed in a ring made of iron. At 15C, the diameter of the plug is 8.755 cm and that of the inside of the ring is 8.741 cm. They must both be brought to what common temperature in order to fit?

A certain car has 14.0 L of liquid coolant circulating at a temperature of 93C through the engines cooling system. Assume that, in this normal condition, the coolant completely fills the 3.5-L volume of the aluminum radiator and the 10.5-L internal cavities within the aluminum engine. When a car overheats, the radiator, engine, and coolant expand and a small reservoir connected to the radiator catches any resultant coolant overflow. Estimate how much coolant overflows to the reservoir if the system goes from 93C to 105C. Model the radiator and engine as hollow shells of aluminum. The coefficient of volume expansion for coolant is b = 410 * 106C.

An aluminum bar has the desired length when at 12C. How much stress is required to keep it at this length if the temperature increases to 35C? [See Table 91.]

The pendulum in a grandfather clock is made of brass and keeps perfect time at 17C. How much time is gained or lost in a year if the clock is kept at 29C? (Assume the frequency dependence on length for a simple pendulum applies.) [Hint: See Chapter 8.

The pendulum in a grandfather clock is made of brass and keeps perfect time at 17C. How much time is gained or lost in a year if the clock is kept at 29C? (Assume the frequency dependence on length for a simple pendulum applies.) [Hint: See Chapter 8.

(II) Typical temperatures in the interior of the Earth and Sun are about 4000°C and \(15 \times 10^6\ ^{\circ}C\), respectively. (a) What are these temperatures in kelvins? (b) What percent error is made in each case if a person forgets to change \(^{\circ}C\) to K?

(II) Typical temperatures in the interior of the Earth and Sun are about \(4000^\circ \mathrm C\) and \(15 \times 10^6 \ ^\circ \mathrm C\) respectively. (a) What are these temperatures in kelvins? (b) What percent error is made in each case if a person forgets to change \(^\circ \mathrm C\) to K?

In an internal combustion engine, air at atmospheric pressure and a temperature of about 20C is compressed in the cylinder by a piston to of its original volume (compression ). Estimate the temperature of the compressed air, assuming the pressure reaches 40 atm.

If 16.00 mol of helium gas is at 10.0C and a gauge pressure of 0.350 atm, calculate (a) the volume of the helium gas under these conditions, and (b) the temperature if the gas is compressed to precisely half the volume at a gauge pressure of 1.00 atm.

A storage tank contains 21.6 kg of nitrogen at an absolute pressure of 3.45 atm. What will the pressure be if the nitrogen is replaced by an equal mass of at the same temperature?

A storage tank at STP contains 28.5 kg of nitrogen (a) What is the volume of the tank? (b) What is the pressure if an additional 32.2 kg of nitrogen is added without changing the temperature?

A scuba tank is filled with air to a gauge pressure of 204 atm when the air temperature is 29C. A diver then jumps into the ocean and, after a short time on the ocean surface, checks the tanks gauge pressure and finds that it is only 191 atm. Assuming the diver has inhaled a negligible amount of air from the tank, what is the temperature of the ocean water?

What is the pressure inside a 38.0-L container holding 105.0 kg of argon gas at 21.6C?

A sealed metal container contains a gas at 20.0C and 1.00 atm. To what temperature must the gas be heated for the pressure to double to 2.00 atm? (Ignore expansion of the container.)

A tire is filled with air at 15C to a gauge pressure of 230 kPa. If the tire reaches a temperature of 38C, what fraction of the original air must be removed if the original pressure of 230 kPa is to be maintained?

If 61.5 L of oxygen at 18.0C and an absolute pressure of 2.45 atm are compressed to 38.8 L and at the same time the temperature is raised to 56.0C, what will the new pressure be?

You buy an airtight bag of potato chips packaged at sea level, and take the chips on an airplane flight. When you take the potato chips out of your carry-onbag, you notice it has noticeably puffed up. Airplane cabins are typically pressurized at 0.75 atm, and assuming the temperature inside an airplane is about the same as inside a potato chip processing plant, by what percentage has the bag puffed up in comparison to when it was packaged?

A helium-filled balloon escapes a childs hand at sea level and 20.0C. When it reaches an altitude of 3600 m, where the temperature is 5.0C and the pressure only 0.68 atm, how will its volume compare to that at sea level?

(II) Compare the value for the density of water vapor at exactly \(100^{\circ}C\) and 1 atm (Table 10–1) with the value predicted from the ideal gas law.Why would you expect a difference?

A sealed test tube traps of air at a pressure of 1.00 atm and temperature of 18C. The test tubes stopper has a diameter of 1.50 cm and will pop off the test tube if a net upward force of 10.0 N is applied to it. To what temperature would you have to heat the trapped air in order to pop off the stopper? Assume the air surrounding the test tube is always at a pressure of 1.00 atm

(III) An air bubble at the bottom of a lake 41.0 m deep has a volume of 1.00 \(cm^3\) If the temperature at the bottom is \(5.5^{\circ} \mathrm{C}\) and at the top \(18.5^{\circ} \mathrm{C}\), what is the radius of the bubble just before it reaches the surface?

Calculate the number of in an ideal gas at STP

How many moles of water are there in 1.000 L at STP? How many molecules?

(II) Estimate the number of (a) moles and (b) molecules of water in all the Earth’s oceans. Assume water covers 75% of the Earth to an average depth of 3 km.

The lowest pressure attainable using the best available vacuum techniques is about At such a pressure, how many molecules are there per at 0C?

Is a gas mostly empty space? Check by assuming that the spatial extent of the gas molecules in air is about so one gas molecule occupies an approximate volume equal to Assume STP.

(a) What is the average translational kinetic energy of a nitrogen molecule at STP? (b) What is the total translational kinetic energy of 1.0 mol of molecules at 25C?

(I) Calculate the rms speed of helium atoms near the surface of the Sun at a temperature of about 6000 K.

By what factor will the rms speed of gas molecules increase if the temperature is increased from 20C to 160C?

A gas is at 20C. To what temperature must it be raised to triple the rms speed of its molecules?

What speed would a 1.0-g paper clip have if it had the same kinetic energy as a molecule at 22C?

The rms speed of molecules in a gas at 20.0C is to be increased by 4.0%. To what temperature must it be raised?

(II) If the pressure in a gas is tripled while its volume is held constant, by what factor does \(v_{rms}\) change?

(II) Show that the rms speed of molecules in a gas is given by \(v_{rms} = \sqrt{3P/\rho}\), where P is the pressure in the gas and \(\rho\) is the gas density.

Show that for a mixture of two gases at the same temperature, the ratio of their rms speeds is equal to the inverse ratio of the square roots of their molecular masses,

What is the rms speed of nitrogen molecules contained in an volume at 2.9 atm if the total amount of nitrogen is 2100 mol?

(II) Two isotopes of uranium, \(^{235}U\) and \(^{238}U\) (the superscripts refer to their atomic masses), can be separated by a gas diffusion process by combining them with fluorine to make the gaseous compound \(UF_6\). Calculate the ratio of the rms speeds of these molecules for the two isotopes, at constant T. Use Appendix B for masses.

Calculate (a) the rms speed of an oxygen molecule at 0C and (b) determine how many times per second it would move back and forth across a 5.0-m-long room on average, assuming it made no collisions with other molecules.

CO2 exists in what phase when the pressure is 35 atm and the temperature is 35C (Fig. 1323)?

(a) At atmospheric pressure, in what phases can exist? (b) For what range of pressures and temperatures can be a liquid? Refer to Fig. 1323.

Water is in which phase when the pressure is 0.01 atm and the temperature is (a) 90C, (b)

You have a sample of water and are able to control temperature and pressure arbitrarily. (a) Using Fig. 1322, describe the phase changes you would see if you started at a temperature of 85C, a pressure of 180 atm, and decreased the pressure down to 0.004 atm while keeping the temperature fixed. (b) Repeat part (a) with the temperature at 0.0C. Assume that you held the system at the starting conditions long enough for the system to stabilize before making further changes.

(I) What is the partial pressure of water vapor at \(30^\circ \mathrm C\) if the humidity is 75%?

What is the air pressure at a place where water boils at 80C

What is the dew point if the humidity is 65% on a day when the temperature is 25C?

If the air pressure at a particular place in the mountains is 0.80 atm, estimate the temperature at which water boils.

What is the mass of water in a closed room when the temperature is 25C and the relative humidity is 55%?

A pressure cooker is a sealed pot designed to cook food with the steam produced by boiling water somewhat above 100C. The pressure cooker in Fig. 1332 uses a weight of mass m to allow steam to escape at a certain pressure through a small hole (diameter d) in the cookers lid. If what should m be in order to cook food at 120C? Assume that atmospheric pressure outside the cooker is

(II) If the humidity in a sealed room of volume \(420\ m^3\) at \(20^{\circ}C\) is 65%, what mass of water can still evaporate from an open pan?

Air that is at its dew point of 5C is drawn into a building where it is heated to 22C. What will be the relative humidity at this temperature? Assume constant pressure of 1.0 atm. Take into account the expansion of the air.

When using a mercury barometer (Section 106), the vapor pressure of mercury is usually assumed to be zero. At room temperature mercurys vapor pressure is about 0.0015 mm-Hg. At sea level, the height h of mercury in a barometer is about 760 mm. (a) If the vapor pressure of mercury is neglected, is the true atmospheric pressure greater or less than the value read from the barometer? (b) What is the percent error? (c) What is the percent error if you use a water barometer and ignore waters saturated vapor pressure at STP?

Estimate the time needed for a glycine molecule (see Table 134) to diffuse a distance of in water at 20C if its concentration varies over that distance from to Compare this speed to its rms (thermal) speed. The molecular mass of glycine is about 75 u

Oxygen diffuses from the surface of insects to the interior through tiny tubes called tracheae. An average trachea is about 2 mm long and has cross-sectional area of Assuming the concentration of oxygen inside is half what it is outside in the atmosphere, (a) show that the concentration of oxygen in the air (assume 21% is oxygen) at 20C is about then (b) calculate the diffusion rate J, and (c) estimate the average time for a molecule to diffuse in. Assume the diffusion constant is 1 * 105 m2 s.

A Pyrex measuring cup was calibrated at normal room temperature. How much error will be made in a recipe calling for 375 mL of cool water, if the water and the cup are hot, at \(95^\circ \mathrm C\), instead of at room temperature? Neglect the glass expansion.

A precise steel tape measure has been calibrated at \(14^\circ \mathrm C\). At \(37^\circ \mathrm C\), (a) will it read high or low, and (b) what will be the percentage error?

A cubic box of volume \(6.15 \times 10^{-2}\ m^3\) is filled with air at atmospheric pressure at \(15^{\circ}\). The box is closed and heated to \(165^{\circ}\). What is the net force on each side of the box?

The gauge pressure in a helium gas cylinder is initially 32 atm. After many balloons have been blown up, the gauge pressure has decreased to 5 atm. What fraction of the original gas remains in the cylinder?

. If a scuba diver fills his lungs to full capacity of 5.5 L when 9.0 m below the surface, to what volume would his lungs expand if he quickly rose to the surface? Is this advisable?

A house has a volume of (a) What is the total mass of air inside the house at 15C? (b) If the temperature drops to what mass of air enters or leaves the house?

Estimate the number of air molecules in a room of length 6.0 m, width 3.0 m, and height 2.5 m. Assume the temperature is \(22^{\circ}C\). How many moles does that correspond to?

An iron cube floats in a bowl of liquid mercury at \(0^{\circ}C\). (a) If the temperature is raised to \(25^{\circ}C\), will the cube float higher or lower in the mercury? (b) By what percent will the fraction of volume submerged change? [Hint: See Chapter 10.]

A helium balloon, assumed to be a perfect sphere, has a radius of 24.0 cm. At room temperature \((20^\circ \mathrm C)\), its internal pressure is 1.08 atm. Determine the number of moles of helium in the balloon, and the mass of helium needed to inflate the balloon to these values.

A standard cylinder of oxygen used in a hospital has gauge pressure = 2000 psi (13,800 kPa) and \(\text {volume} = 14 \ \mathrm L (0.014 \ \mathrm m^3)\) at T = 295 K. How long will the cylinder last if the flow rate, measured at atmospheric pressure, is constant at 2.1 L/min?

A brass lid screws tightly onto a glass jar at \(15^\circ \mathrm C\). To help open the jar, it can be placed into a bath of hot water. After this treatment, the temperatures of the lid and the jar are both \(55^\circ \mathrm C\). The inside diameter of the lid is 8.0 cm. Find the size of the gap (difference in radius) that develops by this procedure.

The density of gasoline at 0C is (a) What is the density on a hot day, when the temperature is 33C? (b) What is the percent change in density?

The first real length standard, adopted more than 200 years ago, was a platinum bar with two very fine marks separated by what was defined to be exactly one meter. If this standard bar was to be accurate to within how carefully would the trustees have needed to control the temperature? The coefficient of linear expansion is 9 * 106 C.

If a steel band were to fit snugly around the Earth’s equator at \(25^\circ \mathrm C\), but then was heated to \(55^\circ \mathrm C\), how high above the Earth would the band be (assume equal everywhere)?

In outer space the density of matter is about one atom per mainly hydrogen atoms, and the temperature is about 2.7 K. Calculate the rms speed of these hydrogen atoms, and the pressure (in atmospheres).

(a) Estimate the rms speed of an amino acid, whose molecular mass is 89 u, in a living cell at 37C. (b) What would be the rms speed of a protein of molecular mass 85,000 u at 37C?

The escape speed from the Earth is so that a gas molecule traveling away from Earth near the outer boundary of the Earths atmosphere would, at this speed, be able to escape from the Earths gravitational field and be lost to the atmosphere. At what temperature is the rms speed of (a) oxygen molecules, and (b) helium atoms equal to (c) Can you explain why our atmosphere contains oxygen but not helium?

Consider a container of oxygen gas at a temperature of 23C that is 1.00 m tall. Compare the gravitational potential energy of a molecule at the top of the container (assuming the potential energy is zero at the bottom) with the average kinetic energy of the molecules. Is it reasonable to neglect the potential energy?

A space vehicle returning from the Moon enters the Earth’s atmosphere at a speed of about 42,000 km/h. Molecules (assume nitrogen) striking the nose of the vehicle with this speed correspond to what temperature? (Because of this high temperature, the nose of a space vehicle must be made of special materials; indeed, part of it does vaporize, and this is seen as a bright blaze upon reentry.)

A sauna has of air volume, and the temperature is 85C. The air is perfectly dry. How much water (in kg) should be evaporated if we want to increase the relative humidity from 0% to 10%? (See Table 133.)

A 0.50-kg trash-can lid is suspended against gravity by tennis balls thrown vertically upward at it. How many tennis balls per second must rebound from the lid elastically, assuming they have a mass of 0.060 kg and are thrown at 15 m/s?

In humid climates, people constantly dehumidify their cellars to prevent rot and mildew. If the cellar in a house (kept at 20C) has of floor space and a ceiling height of 2.4 m, what is the mass of water that must be removed from it in order to drop the humidity from 95% to a more reasonable 40%?

Problem 1COQ A hot-air balloon, open at one end (see photos above), rises when the air inside is heated by a flame. For the following properties, is the air inside the balloon higher, lower, or the same as for the air outside the balloon? (i) Temperature. (ii) Pressure. (iii) Density.

Problem 1MCQ Rod A has twice the diameter of rod B, but both are made of iron and have the same initial length. Both rods are now subjected to the same change in temperature (but remain solid). How would the change in the rods’ lengths compare? (a)Rod A > rod B. B. (b)Rod B > rod A. (c)Rod A = rod B. (d) Need to know whether the rods were cooled or heated.

Problem 1P (I) How does the number of atoms in a 27.5-gram gold ring compare to the number in a silver ring of the same mass?

Problem 1Q Which has more atoms: 1 kg of lead or 1 kg of copper? (See the Periodic Table or Appendix B.) Explain why.

Problem 1SL Rod A has twice the diameter of rod B, but both are made of iron and have the same initial length. Both rods are now subjected to the same change in temperature (but remain solid). How would the change in the rods’ lengths compare? (a)Rod A > rod B. (b)Rod B > rod A. (c)Rod A = rod B. (d) Need to know whether the rods were cooled or heated.

The linear expansion of a material depends on which of the following? (a) The length of the material. (b) The change in temperature of the material. (c) The type of material. (d) All of the above. (e) Both (b) and (c).

Problem 2P (I) How many atoms are there in a 3.4-g copper coin?

Name several properties of materials that could be used to make a thermometer.

A scuba tank when fully charged has a pressure of 180 atm at \(18^\circ \mathrm C\). The volume of the tank is 11.3 L. (a) What would the volume of the air be at 1.00 atm and at the same temperature? (b) Before entering the water, a person consumes 2.0 L of air in each breath, and breathes 12 times a minute. At this rate, how long would the tank last? (c) At a depth of 23.0 m in sea water at a temperature of \(10^\circ \mathrm C\), how long would the same tank last assuming the breathing rate does not change?

Problem 3P (I) (a) “Room temperature” is often taken to be 68°F. What is this on the Celsius scale? (b) The temperature of the filament in a lightbulb is about 1900°C. What is this on the Fahrenheit scale?

Problem 3Q Which is larger, 1 C° or 1 F°? Explain why.

A \(28.4-\mathrm{kg}\) solid aluminum cylindrical wheel of radius \(0.41 \mathrm{~m}\) is rotating about its axle in frictionless bearings with angular velocity \(\omega=32.8 \mathrm{rad} / \mathrm{s}\). If its temperature is then raised from \(15.0^{\circ} \mathrm{C}\) to \(95.0^{\circ} \mathrm{C}\), what is the fractional change in \(\omega\)?

Problem 4P (I) Among the highest and lowest natural air temperatures claimed are 136°F in the Libyan desert and-129°F in Antarctica. What are these temperatures on the Celsius scale?

In the relation \(\Delta \ell=\alpha \ell_{0} \Delta T\), should be \(\ell_{0}\) the initial length, the final length, or does it matter? Equation Transcription: Text Transcription: \Delta \ell=\alpha \ell_{0} \Delta T \ell_{0}

Problem 4SL One mole of an ideal gas in a sealed rigid container is initially at a temperature of 100°C. The temperature is then increased to 200°C. The pressure in the gas (a) remains constant. (b) increases by about 25%. (c) doubles. (d) triples.

(I) A thermometer tells you that you have a fever of \(38.9^{\circ}C\). What is this in Fahrenheit?

A flat bimetallic strip consists of a strip of aluminum riveted to a strip of iron. When heated, the strip will bend. Which metal will be on the outside of the curve? Why? [Hint: See Table 13–1.]

Problem 5SL Estimate how many molecules of air are in each 2.0-L breath you inhale that were also in the last breath Galileo took. Assume the atmosphere is about 10 km high and of constant density. What other assumptions did you make?

Problem 6P (I) (a) 18° below zero on the Celsius scale is what Fahrenheit temperature? (b) 18° below zero on the Fahrenheit scale is what Celsius temperature?

Long steam pipes that are fixed at the ends often have a section in the shape of a \(\cup\). Why? Equation Transcription: Text Transcription: \cup

Problem 6SL (a) The second postulate of kinetic theory is that the molecules are, on average, far apart from one another. That is, their average separation is much greater than the diameter of each molecule. Is this assumption reasonable? To check, calculate the average distance between molecules of a gas at STP, and compare it to the diameter of a typical gas molecule, about 0.3 nm. (b) If the molecules were the diameter of ping-pong balls, say 4 cm, how far away would the next ping-pong ball be on average? (c) Repeat part a, but now assume the gas has been compressed so that the pressure is now 3 atm but still at 273 K. (d) Estimate what % of the total volume of gas is taken up by molecules themselves in parts a and c. [Note that the volume of the molecules themselves can become a significant part of the total volume at lower temperatures and higher pressures. Hence the actual volume the molecules have to bounce around in is less than the total volume. This contributes to the effect shown in Fig. 13–21 at high pressures where real gases (solid red lines) deviate from ideal gas behavior (dashed lines A’ and B’).]

Problem 7P (II) Determine the temperature at which the Celsius and Fahrenheit scales give the same numerical reading (TC =TF).

Figure 13–29 shows a diagram of a simple bimetallic thermostat used to control a furnace (or other heating or cooling system). The electric switch (attached to the bimetallic strip) is a glass vessel containing liquid mercury that conducts electricity when it touches both contact wires. Explain how this device controls the furnace and how it can be set at different temperatures.

Problem 8MCQ An ideal gas is in a sealed rigid container. The average kinetic energy of the gas molecules depends most on (a) the size of the container. (b) the number of molecules in the container. (c) the temperature of the gas. (d) the mass of the molecules.

Problem 8P (II) In an alcohol-in-glass thermometer, the alcohol column has length 12.61 cm at 0.0°C and length 22.79 cm at 100.0°C. What is the temperature if the column has length (a) 18.70 cm, and (b) 14.60 cm?

A glass container may break if one part of it is heated or cooled more rapidly than adjacent parts. Explain.

Two ideal gases, A and B, are at the same temperature. If the molecular mass of the molecules in gas A is twice that of the molecules in gas B, the molecules’ root-mean-square speed is (a) the same in both gases. (b) twice as great in A. (c) 1.4 times greater in A. (d) twice as great in B. (e) 1.4 times greater in B.

(I) The Eiffel Tower (Fig. 13–31) is built of wrought iron approximately 300 m tall. Estimate how much its height changes between January (average temperature of \(2^{0} \mathrm{C}\)) and July (average temperature of \(25^{0} \mathrm{C}\)). Ignore the angles of the iron beams and treat the tower as a vertical beam. Equation transcription: Text transcription: 2^{0}{C} 25^{0}{C}

Problem 9Q Explain why it is advisable to add water to an overheated automobile engine only slowly, and only with the engine running.

In a mixture of the gases oxygen and helium, which statement is valid? (a) The helium atoms will be moving faster than the oxygen molecules, on average. (b) Both will be moving at the same speed. (c) The oxygen molecules will, on average, be moving more rapidly than the helium atoms. (d) The kinetic energy of helium atoms will exceed that of oxygen molecules. (e) None of the above.

The units for the coefficient of linear expansion \(\alpha\) are \(\left(C^{\circ}\right)^{-1}\), and there is no mention of a length unit such as meters. Would the expansion coefficient change if we used feet or millimeters instead of meters? Explain. Equation Transcription: Text Transcription: \alpha \left(C^{\circ}\right)^{-1}

Problem 10P (I) A concrete highway is built of slabs 12 m long (15°C). How wide should the expansion cracks between the slabs be (at 15°C) to prevent buckling if the range of temperature Is -30°C to +50°C?

Problem 11MCQ Which of the following is not true about an ideal gas? (a) The average kinetic energy of the gas molecules increases as the temperature increases. (b) The volume of an ideal gas increases with temperature if the pressure is held constant. (c) The pressure of an ideal gas increases with temperature if the volume is held constant. (d) All gas molecules have the same speed at a particular temperature. (e) The molecules are assumed to be far apart compared to their size.

Problem 11P (I) Super Invar™, an alloy of iron and nickel, is a strong material with a very low coefficient of thermal expansion (0.20 X 10-6/C°). A 1.8-m-long tabletop made of this alloy is used for sensitive laser measurements where extremely high tolerances are required. How much will this alloy table expand along its length if the temperature increases 6.0 C°? Compare to tabletops made of steel.

Problem 11Q When a cold alcohol-in-glass thermometer is first placed in a hot tub of water, the alcohol initially descends a bit and then rises. Explain.

When using the ideal gas law, which of the following rules must be obeyed? (a) Always use temperature in kelvins and absolute pressure. (b) Always use volume in \(\mathrm m^3\) and temperature in kelvins. (c) Always use gauge pressure and temperature in degrees Celsius. (d) Always use gauge pressure and temperature in kelvins. (e) Always use volume in \(\mathrm m^3\) and gauge pressure.

Problem 12P (II) To what temperature would you have to heat a brass rod for it to be 1.5% longer than it is at 25°C?

The principal virtue of Pyrex glass is that its coefficient of linear expansion is much smaller than that for ordinary glass (Table 13–1). Explain why this gives rise to the higher heat resistance of Pyrex.

Return to the Chapter-Opening Question, page 359, and answer it again now. Try to explain why you may have answered differently the first time.

Problem 13EB An ideal gas is contained in a steel sphere at 27.0°C and 1.00 atm absolute pressure. If no gas is allowed to escape and the temperature is raised to 127°C, what will be the new pressure? (a) 0.21 atm; (b) 0.75 atm; (c) 1.00 atm; (d) 1.33

What is the volume of 1.00 mol of ideal gas at \(546\ K\ (=2 \times 273\ K)\) and 2.0 atm absolute pressure? (a) 11.2 L; (b) 22.4 L; (c) 44.8 L; (d) 67.2 L; (e) 89.6 L.

Problem 13ED At 20°C, would there be (a) more, (b) less, or (c) the same mass of air in a room than at 0°C?

If molecules of hydrogen gas and oxygen gas were placed in the same balloon at room temperature, how would the average kinetic energies of the molecules compare? (a) They would be the same. (b) The hydrogen molecules would have greater kinetic energy. (c) The oxygen molecules would have greater kinetic energy. (d) Need more information.

By what factor must the absolute temperature change to double \(v_{r m s}\)?(a) \(\sqrt{2}\); b) 2; (c) \(2 \sqrt{2}\); d) 4; (e) 16. Equation transcription: Text transcription: v{r m s} sqrt{2} 2 sqrt{2}

The rms speed of the molecules of an ideal gas (a) is the same as the most probable speed of the molecules. (b) is always equal to \(\sqrt{2}\) times the maximum molecular speed. (c) will increase as the temperature of a gas increases. (d) All of the above. Equation transcription: Text transcription: sqrt{2}

(II) To make a secure fit, rivets that are larger than the rivet hole are often used and the rivet is cooled (usually in dry ice) before it is placed in the hole. A steel rivet 1.872 cm in diameter is to be placed in a hole 1.870 cm in diameter in a metal at \(22^\circ \mathrm C\). To what temperature must the rivet be cooled if it is to fit in the hole?

Problem 13Q Will a clock using a pendulum supported on a long thin brass rod that is accurate at 20°C run fast or slow on a hot day (30°C)? Explain.

(II) An ordinary glass is filled to the brim with 450.0 mL of water at \(100.0^{\circ}C\). If the temperature of glass and water is decreased to \(20.0^{\circ}C\), how much water could be added to the glass?

Problem 14Q Freezing a can of soda will cause its bottom and top to bulge so badly the can will not stand up. What has happened?

Problem 15P (II) An aluminum sphere is 8.75 cm in diameter. What will be its %change in volume if it is heated from 30°C to 160°C?

Will the buoyant force on an aluminum sphere submerged in water increase, decrease, or remain the same, if the temperature is increased from \(20^\circ \mathrm C\) to \(40^\circ \mathrm C\)? Explain.

Problem 16P (II) It is observed that 55.50 mL of water at 20°C completely fills a container to the brim. When the container and the water are heated to 60°C, 0.35 g of water is lost. (a) What is the coefficient of volume expansion of the container? (b) What is the most likely material of the container? Density of water at 60°C is 0.98324 g/mL.

Problem 16Q Can you determine the temperature of a vacuum? Explain.

(II) A brass plug is to be placed in a ring made of iron. At \(15^\circ \mathrm C\), the diameter of the plug is 8.755 cm and that of the inside of the ring is 8.741 cm. They must both be brought to what common temperature in order to fit?

Escape velocity from the Earth refers to the minimum speed an object must have to leave the Earth and never return. (a) The escape velocity from the Moon is about one-fifth what it is for the Earth, due to the Moon’s smaller mass. Explain why the Moon has practically no atmosphere. (b) If hydrogen was once in the Earth’s atmosphere, why would it have probably escaped?

Problem 18P (II) A certain car has 14.0 L of liquid coolant circulating at a temperature of 93°C through the engine’s cooling system. Assume that, in this normal condition, the coolant completely fills the 3.5-L volume of the aluminum radiator and the 10.5-L internal cavities within the aluminum engine. When a car overheats, the radiator, engine, and coolant expand and a small reservoir connected to the radiator catches any resultant coolant overflow. Estimate how much coolant overflows to the reservoir if the system goes from 93°C to 105°C. Model the radiator and engine as hollow shells of aluminum. The coefficient of volume expansion for coolant is ? = 410 X 10-6/C°.

Problem 18Q What exactly does it mean when we say that oxygen boils at -183oC?

Problem 19P (II) An aluminum bar has the desired length when at 12°C. How much stress is required to keep it at this length if the temperature increases to 35°C?

Problem 19Q A length of thin wire is placed over a block of ice (or an ice cube) at 0°C. The wire hangs down both sides of the ice, and weights are hung from the ends of the wire. It is found that the wire cuts its way through the ice cube, but leaves a solid block of ice behind it. This process is called regelation. Explain how this happens by inferring how the freezing point of water depends on pressure.

(III) The pendulum in a grandfather clock is made of brass and keeps perfect time at \(17^\circ \mathrm C\). How much time is gained or lost in a year if the clock is kept at \(29^\circ \mathrm C\)? (Assume the frequency dependence on length for a simple pendulum applies.) [Hint: See Chapter 8.]

Problem 20Q (a) Why does food cook faster in a pressure cooker? (b) Why does pasta or rice need to boil longer at high altitudes? (c) Is it harder to boil water at high altitudes?

Problem 21P (I) Absolute zero is what temperature on the Fahrenheit scale?

Is it possible to boil water at room temperature \((20^{\circ}C)\) without heating it? Explain.

(II) Typical temperatures in the interior of the Earth and Sun are about \(4000^\circ \mathrm C\) and \(15 \times 10^6 \ ^{\circ}\mathrm C\) respectively. (a) What are these temperatures in kelvins? (b) What percent error is made in each case if a person forgets to change \(^{\circ}\mathrm C\) to K?

Why does exhaled air appear as a little white cloud in the winter (Fig. 13–30)?

Problem 23P (I) If 3.50 m3 of a gas initially at STP is placed under a pressure of 3.20 atm, the temperature of the gas rises to 38.0°C. What is the volume?

Problem 23Q Explain why it is dangerous to open the radiator cap of an overheated automobile engine.

(I) In an internal combustion engine, air at atmospheric pressure and a temperature of about \(20^{\circ} \mathrm{C}\) is compressed in the cylinder by a piston to \(\frac{1}{9}\) of its original volume (compression ratio = 9.0). Estimate the temperature of the compressed air, assuming the pressure reaches 40 atm.

Problem 25P (II) If 16.00 mol of helium gas is at 10.0°C and a gauge pressure of 0.350 atm, calculate (a) the volume of the helium gas under these conditions, and (b) the temperature if the gas is compressed to precisely half the volume at a gauge pressure of 1.00 atm.

(II) A storage tank contains 21.6 kg of nitrogen \((N_2)\) at an absolute pressure of 3.45 atm. What will the pressure be if the nitrogen is replaced by an equal mass of \(CO_2\) at the same temperature?

Problem 27P (II) A storage tank at STP contains 28.5 kg of nitrogen (N2) (a) What is the volume of the tank? (b) What is the pressure if an additional 32.2 kg of nitrogen is added without changing the temperature?

(II) A scuba tank is filled with air to a gauge pressure of 204 atm when the air temperature is \(29^{\circ}C\).A diver then jumps into the ocean and, after a short time on the ocean surface, checks the tank’s gauge pressure and finds that it is only 191 atm. Assuming the diver has inhaled a negligible amount of air from the tank, what is the temperature of the ocean water?

(II) What is the pressure inside a 38.0-L container holding 105.0 kg of argon gas at \(21.6^{\circ}C\)?

Problem 30P (II) A sealed metal container contains a gas at 20.0°C and 1.00 atm. To what temperature must the gas be heated for the pressure to double to 2.00 atm? (Ignore expansion of the container.)

Problem 31P (II) A tire is filled with air at 15°C to a gauge pressure of 230 kPa. If the tire reaches a temperature of 38°C, what fraction of the original air must be removed if the original pressure of 230 kPa is to be maintained?

Problem 32P (II) If 61.5 L of oxygen at 18.0°C and an absolute pressure of 2.45 atm are compressed to 38.8 L and at the same time the temperature is raised to 56.0°C, what will the new pressure be?

Problem 33P (II) You buy an “airtight” bag of potato chips packaged at sea level, and take the chips on an airplane flight. When you take the potato chips out of your “carry-on”bag, you notice it has noticeably “puffed up.” Airplane cabins are typically pressurized at 0.75 atm, and assuming the temperature inside an airplane is about the same as inside a potato chip processing plant, by what percentage has the bag “puffed up” in comparison to when it was packaged?

Problem 34P (II) A helium-filled balloon escapes a child’s hand at sea level and 20.0°C. When it reaches an altitude of 3600 m, where the temperature is 5.0°C and the pressure only 0.68 atm, how will its volume compare to that at sea level?

Problem 35P (II) Compare the value for the density of water vapor at exactly 100°C and 1 atm (Table 10–1) with the value predicted from the ideal gas law. Why would you expect a difference?

Problem 36P (III) A sealed test tube traps 25.0 cm3 of air at a pressure of 1.00 atm and temperature of 18°C. The test tube’s stopper has a diameter of 1.50 cm and will “pop off” the test tube if a net upward force of 10.0 N is applied to it. To what temperature would you have to heat the trapped air in order to “pop off” the stopper? Assume the air surrounding the test tube is always at a pressure of 1.00 atm.

Problem 37P (III) An air bubble at the bottom of a lake 41.0 m deep has a volume of 1.00 cm3. If the temperature at the bottom is 5.5°C and at the top 18.5°C, what is the radius of the bubble just before it reaches the surface?

Problem 38P (I) Calculate the number of molecules/m3 in an ideal gas at STP.

Problem 39P (I) How many moles of water are there in 1.000 L at STP? How many molecules?

Problem 40P (II) Estimate the number of (a) moles and (b) molecules of water in all the Earth’s oceans. Assume water covers 75% of the Earth to an average depth of 3 km.

Problem 41P (II) The lowest pressure attainable using the best available vacuum techniques is about 10-12 N/m2, At such a pressure, how many molecules are there per cm3 at 0°C?

(II) Is a gas mostly empty space? Check by assuming that the spatial extent of the gas molecules in air is about \(\ell_{0}=0.3 n m\) so one gas molecule occupies an approximate volume equal to \(\ell_{0}^{3}\). Assume STP. Equation transcription: Text transcription: ell{0}=0.3 n m ell{0}^{3}

Problem 43P (I) (a) What is the average translational kinetic energy of a nitrogen molecule at STP? (b) What is the total translational kinetic energy of 1.0 mol of N2 molecules at 25°C?

Problem 44P (I) Calculate the rms speed of helium atoms near the surface of the Sun at a temperature of about 6000 K.

Problem 45P (I) By what factor will the rms speed of gas molecules increase if the temperature is increased from 20°C to 160°C?

(I) A gas is at \(20^\circ \mathrm C\). To what temperature must it be raised to triple the rms speed of its molecules?

Problem 47P (I) What speed would a 1.0-g paper clip have if it had the same kinetic energy as a molecule at 22°C?

Problem 48P (II) The rms speed of molecules in a gas at 20.0°C is to be increased by 4.0%. To what temperature must it be raised?

Problem 49P (II) If the pressure in a gas is tripled while its volume is held constant, by what factor does v rms change?

(II) Show that the rms speed of molecules in a gas is given by \(v_{rms} = \sqrt{3P/\rho}\), where P is the pressure in the gas and \(\rho\) is the gas density.

(II) Show that for a mixture of two gases at the same temperature, the ratio of their rms speeds is equal to the inverse ratio of the square roots of their molecular masses, \(v_{1} / v_{2}=\sqrt{M_{2} / M_{1}}\). Equation transcription: Text transcription: v{1} / v{2}=sqrt{M{2} / M{1}}

(II) What is the rms speed of nitrogen molecules contained in an \(8.5-m^3\) volume at 2.9 atm if the total amount of nitrogen is 2100 mol?

(II) Two isotopes of uranium, \(^{235}\mathrm U\) and \(^{238}\mathrm U\) (the superscripts refer to their atomic masses), can be separated by a gas diffusion process by combining them with fluorine to make the gaseous compound \(\mathrm {UF}_6\). Calculate the ratio of the rms speeds of these molecules for the two isotopes, at constant T. Use Appendix B for masses.

Problem 54P (III) Calculate (a) the rms speed of an oxygen molecule at 0°C and (b) determine how many times per second it would move back and forth across a 5.0-m-long room on average, assuming it made no collisions with other molecules.

(I) \(\mathrm{CO}_{2}\) exists in what phase when the pressure is 35 atm and the temperature is \(35^\circ \mathrm{C}\) (Fig. 13–23)?

(I) (a) At atmospheric pressure, in what phases can \(\mathrm{CO}_{2}\) exist? (b) For what range of pressures and temperatures can \(\mathrm{CO}_{2}\) be a liquid? Refer to Fig. 13–23. Equation transcription: Text transcription: {CO}{2}

(I) Water is in which phase when the pressure is 0.01 atm and the temperature is (a) \(90^\circ \mathrm C\), (b) \(-20^\circ \mathrm C\)?

Problem 58P (II) You have a sample of water and are able to control temperature and pressure arbitrarily. (a) Using Fig. 13–22, describe the phase changes you would see if you started at a temperature of 85°C, a pressure of 180 atm, and decreased the pressure down to 0.004 atm while keeping the temperature fixed. (b) Repeat part (a) with the temperature at 0.0°C. Assume that you held the system at the starting conditions long enough for the system to stabilize before making further changes.

(I) What is the partial pressure of water vapor at \(30^{\circ}C\) if the humidity is 75%?

Problem 60 (I) What is the air pressure at a place where water boils at 80°C?

(II) If the air pressure at a particular place in the mountains is 0.80 atm, estimate the temperature at which water boils.

What is the mass of water in a closed room \(\mathrm{5.0~m \times 6.0~m \times 2.4~ m}\) when the temperature is \(25^\circ \mathrm C\) and the relative humidity is 55%?

(II) A pressure cooker is a sealed pot designed to cook food with the steam produced by boiling water somewhat above \(100^{\circ} \mathrm{C}\). The pressure cooker in Fig. 13–32 uses a weight of mass to allow steam to escape at a certain pressure through a small hole (diameter ) in the cooker’s lid. If = 3.0mm, what should be in order to cook food at \(120^{0} \mathrm{C}\)? Assume that atmospheric pressure outside the cooker is \(1.01 x 10^{6} \mathrm{~Pa}\). Equation transcription: Text transcription: 100^{circ}{C} 120^{0}{C} 1.01 x 10^{6}{~Pa}

(III) When using a mercury barometer (Section 10–6), the vapor pressure of mercury is usually assumed to be zero. At room temperature mercury’s vapor pressure is about 0.0015 mm-Hg. At sea level, the height h of mercury in a barometer is about 760 mm. (a) If the vapor pressure of mercury is neglected, is the true atmospheric pressure greater or less than the value read from the barometer? (b) What is the percent error? (c) What is the percent error if you use a water barometer and ignore water’s saturated vapor pressure at STP?

(II) Estimate the time needed for a glycine molecule (see Table 13–4) to diffuse a distance of \(25\ \mu m\) in water at \(20^{\circ}C\) if its concentration varies over that distance from \(1.00 mol/m^3\) to \(0.50 mol/m^3\)? Compare this “speed” to its rms (thermal) speed. The molecular mass of glycine is about 75 u.

Problem 69P (II) Oxygen diffuses from the surface of insects to the interior through tiny tubes called tracheae. An average trachea is about 2 mm long and has cross-sectional area of 2X 10-9 m2 Assuming the concentration of oxygen inside is half what it is outside in the atmosphere, (a) show that the concentration of oxygen in the air (assume 21% is oxygen) at 20°C is about 8.7 mol/m3 then (b) calculate the diffusion rate J, and (c) estimate the average time for a molecule to diffuse in. Assume the diffusion constant is 1X 10-5 m2/s.

A Pyrex measuring cup was calibrated at normal room temperature. How much error will be made in a recipe calling for 375 mL of cool water, if the water and the cup are hot, at \(95^\circ \mathrm C\), instead of at room temperature? Neglect the glass expansion.

A cubic box of volume \(6.15 \times 10^{-2}\ m^{-3}\) is filled with air at atmospheric pressure at \(15^{\circ}C\). The box is closed and heated to \(165^{\circ}C\). What is the net force on each side of the box?

Problem 73GP The gauge pressure in a helium gas cylinder is initially 32 atm. After many balloons have been blown up, the gauge pressure has decreased to 5 atm. What fraction of the original gas remains in the cylinder?

Problem 74GP If a scuba diver fills his lungs to full capacity of 5.5 L when 9.0 m below the surface, to what volume would his lungs expand if he quickly rose to the surface? Is this advisable?

Problem 75GP A house has a volume of 1200 m3 (a) What is the total mass of air inside the house at 15°C? (b) If the temperature drops to -15o C, what mass of air enters or leaves the house?

Problem 76GP Estimate the number of air molecules in a room of length 6.0 m, width 3.0 m, and height 2.5 m. Assume the temperature is 22°C. How many moles does that correspond to?

An iron cube floats in a bowl of liquid mercury at \(0^{\circ}C\). (a) If the temperature is raised to \(25^{\circ}C\), will the cube float higher or lower in the mercury? (b) By what percent will the fraction of volume submerged change? [Hint: See Chapter 10.]

Problem 78GP A helium balloon, assumed to be a perfect sphere, has a radius of 24.0 cm. At room temperature (20°C), its internal pressure is 1.08 atm. Determine the number of moles of helium in the balloon, and the mass of helium needed to inflate the balloon to these values.

A standard cylinder of oxygen used in a hospital has gauge pressure = 2000 psi (13,800 kPa) and volume = \(14\ L\ (0.014\ m^3)\) at T = 295 K. How long will the cylinder last if the flow rate, measured at atmospheric pressure, is constant at 2.1 L/min?

A brass lid screws tightly onto a glass jar at \(15^{\circ}C\). To help open the jar, it can be placed into a bath of hot water. After this treatment, the temperatures of the lid and the jar are both \(55^{\circ}C\). The inside diameter of the lid is 8.0 cm. Find the size of the gap (difference in radius) that develops by this procedure.

Problem 81GP The density of gasoline at 0°C is 0.68*103 kg/3.(a)What is the density on a hot day, when the temperature is 33°C? (b) What is the percent change in density?

The first real length standard, adopted more than 200 years ago, was a platinum bar with two very fine marks separated by what was defined to be exactly one meter. If this standard bar was to be accurate to within \(\pm\ 1.0\ \mu m\), how carefully would the trustees have needed to control the temperature? The coefficient of linear expansion is \(9 \times 10^{-6}/C^{\circ}\).

Problem 83GP If a steel band were to fit snugly around the Earth’s equator at 25°C, but then was heated to 55°C, how high above the Earth would the band be (assume equal everywhere)?

In outer space the density of matter is about one atom per \(cm^3\), mainly hydrogen atoms, and the temperature is about 2.7 K. Calculate the rms speed of these hydrogen atoms, and the pressure (in atmospheres).

(a) Estimate the rms speed of an amino acid, whose molecular mass is 89 u, in a living cell at \(37^{\circ}C\). (b) What would be the rms speed of a protein of molecular mass 85,000 u at \(37^{\circ}C\)?

Problem 86GP The escape speed from the Earth is 1.12*104 m/s, so that a gas molecule traveling away from Earth near the outer boundary of the Earth’s atmosphere would, at this speed, be able to escape from the Earth’s gravitational field and be lost to the atmosphere. At what temperature is the rms speed of (a) oxygen molecules, and (b) helium atoms equal to1.12*104 m/s? (c) Can you explain why our atmosphere contains oxygen but not helium?

Problem 87GP Consider a container of oxygen gas at a temperature of 23°C that is 1.00 m tall. Compare the gravitational potential energy of a molecule at the top of the container (assuming the potential energy is zero at the bottom) with the average kinetic energy of the molecules. Is it reasonable to neglect the potential energy?

Problem 88GP A space vehicle returning from the Moon enters the Earth’s atmosphere at a speed of about 42000 km/h. Molecules (assume nitrogen) striking the nose of the vehicle with this speed correspond to what temperature? (Because of this high temperature, the nose of a space vehicle must be made of special materials; indeed, part of it does vaporize, and this is seen as a bright blaze upon reentry.)

A sauna has \(8.5\ m^3\) of air volume, and the temperature is \(85^{\circ}C\). The air is perfectly dry. How much water (in kg) should be evaporated if we want to increase the relative humidity from 0% to 10%? (See Table 13–3.)

A 0.50-kg trash-can lid is suspended against gravity by tennis balls thrown vertically upward at it. How many tennis balls per second must rebound from the lid elastically, assuming they have a mass of 0.060 kg and are thrown at 15 m/s?

Problem 91GP In humid climates, people constantly dehumidify their cellars to prevent rot and mildew. If the cellar in a house (kept at 20°C) has 105 m2 of floor space and a ceiling height of 2.4 m, what is the mass of water that must be removed from it in order to drop the humidity from 95% to a more reasonable 40%?

How does the number of atoms in a 27.5-gram gold ring compare to the number in a silver ring of the same mass?

(a) Room temperature is often taken to be 68F. What is this on the Celsius scale? (b) The temperature of the filament in a lightbulb is about 1900C. What is this on the Fahrenheit scale?

(a) Room temperature is often taken to be 68F. What is this on the Celsius scale? (b) The temperature of the filament in a lightbulb is about 1900C. What is this on the Fahrenheit scale?

Among the highest and lowest natural air temperatures claimed are 136F in the Libyan desert and in Antarctica. What are these temperatures on the Celsius scale?

A thermometer tells you that you have a fever of 38.9C. What is this in Fahrenheit?

(a) 18 below zero on the Celsius scale is what Fahrenheit temperature? (b) 18 below zero on the Fahrenheit scale is what Celsius temperature?

Determine the temperature at which the Celsius and Fahrenheit scales give the same numerical reading ATC = TF

In an alcohol-in-glass thermometer, the alcohol column has length 12.61 cm at 0.0C and length 22.79 cm at 100.0C. What is the temperature if the column has length (a) 18.70 cm, and (b) 14.60 cm?

The Eiffel Tower (Fig. 1331) is built of wrought iron approximately 300 m tall. Estimate how much its height changes between January (average temperature of 2C) and July (average temperature of 25C). Ignore the angles of the iron beams and treat the tower as a vertical beam.

(I) A concrete highway is built of slabs 12 m long (\(15^{\circ}C\)). How wide should the expansion cracks between the slabs be (at \(15^{circ}C\)) to prevent buckling if the range of temperature is \(-30^{\circ}C\) to \(+50^{\circ}C\)?

(I) Super Invar\(^\mathrm{TM}\), an alloy of iron and nickel, is a strong material with a very low coefficient of thermal expansion \((0.20 \times 10^{-6}/\mathrm C^\circ)\). A 1.8-m-long tabletop made of this alloy is used for sensitive laser measurements where extremely high tolerances are required. How much will this alloy table expand along its length if the temperature increases \(6.0 \ \mathrm C^\circ\)? Compare to tabletops made of steel.

To what temperature would you have to heat a brass rod for it to be 1.5% longer than it is at 25C?

(II) To make a secure fit, rivets that are larger than the rivet hole are often used and the rivet is cooled (usually in dry ice) before it is placed in the hole. A steel rivet 1.872 cm in diameter is to be placed in a hole 1.870 cm in diameter in a metal at \(22^\circ \mathrm C\). To what temperature must the rivet be cooled if it is to fit in the hole?

An ordinary glass is filled to the brim with 450.0 mL of water at 100.0C. If the temperature of glass and water is decreased to 20.0C, how much water could be added to the glass?

An aluminum sphere is 8.75 cm in diameter. What will be its % change in volume if it is heated from 30C to 160C?

It is observed that 55.50 mL of water at 20C completely fills a container to the brim. When the container and the water are heated to 60C, 0.35 g of water is lost. (a) What is the coefficient of volume expansion of the container? (b) What is the most likely material of the container? Density of water at 60C is 0.98324 gmL.

A brass plug is to be placed in a ring made of iron. At 15C, the diameter of the plug is 8.755 cm and that of the inside of the ring is 8.741 cm. They must both be brought to what common temperature in order to fit?

A certain car has 14.0 L of liquid coolant circulating at a temperature of 93C through the engines cooling system. Assume that, in this normal condition, the coolant completely fills the 3.5-L volume of the aluminum radiator and the 10.5-L internal cavities within the aluminum engine. When a car overheats, the radiator, engine, and coolant expand and a small reservoir connected to the radiator catches any resultant coolant overflow. Estimate how much coolant overflows to the reservoir if the system goes from 93C to 105C. Model the radiator and engine as hollow shells of aluminum. The coefficient of volume expansion for coolant is b = 410 * 106C.

(II) An aluminum bar has the desired length when at \(12^{\circ}C\). How much stress is required to keep it at this length if the temperature increases to \(35^{\circ}C\)? [See Table 9–1.]

The pendulum in a grandfather clock is made of brass and keeps perfect time at 17C. How much time is gained or lost in a year if the clock is kept at 29C? (Assume the frequency dependence on length for a simple pendulum applies.) [Hint: See Chapter 8.

The pendulum in a grandfather clock is made of brass and keeps perfect time at 17C. How much time is gained or lost in a year if the clock is kept at 29C? (Assume the frequency dependence on length for a simple pendulum applies.) [Hint: See Chapter 8.

Typical temperatures in the interior of the Earth and Sun are about 4000C and respectively. (a) What are these temperatures in kelvins? (b) What percent error is made in each case if a person forgets to change C to K?

Typical temperatures in the interior of the Earth and Sun are about 4000C and respectively. (a) What are these temperatures in kelvins? (b) What percent error is made in each case if a person forgets to change C to K?

In an internal combustion engine, air at atmospheric pressure and a temperature of about 20C is compressed in the cylinder by a piston to of its original volume (compression ). Estimate the temperature of the compressed air, assuming the pressure reaches 40 atm.

If 16.00 mol of helium gas is at 10.0C and a gauge pressure of 0.350 atm, calculate (a) the volume of the helium gas under these conditions, and (b) the temperature if the gas is compressed to precisely half the volume at a gauge pressure of 1.00 atm.

(II) A storage tank contains 21.6 kg of nitrogen \((N_2)\) at an absolute pressure of 3.45 atm. What will the pressure be if the nitrogen is replaced by an equal mass of \(CO_2\) at the same temperature?

A storage tank at STP contains 28.5 kg of nitrogen (a) What is the volume of the tank? (b) What is the pressure if an additional 32.2 kg of nitrogen is added without changing the temperature?

A scuba tank is filled with air to a gauge pressure of 204 atm when the air temperature is 29C. A diver then jumps into the ocean and, after a short time on the ocean surface, checks the tanks gauge pressure and finds that it is only 191 atm. Assuming the diver has inhaled a negligible amount of air from the tank, what is the temperature of the ocean water?

What is the pressure inside a 38.0-L container holding 105.0 kg of argon gas at 21.6C?

A sealed metal container contains a gas at 20.0C and 1.00 atm. To what temperature must the gas be heated for the pressure to double to 2.00 atm? (Ignore expansion of the container.)

A tire is filled with air at 15C to a gauge pressure of 230 kPa. If the tire reaches a temperature of 38C, what fraction of the original air must be removed if the original pressure of 230 kPa is to be maintained?

If 61.5 L of oxygen at 18.0C and an absolute pressure of 2.45 atm are compressed to 38.8 L and at the same time the temperature is raised to 56.0C, what will the new pressure be?

You buy an airtight bag of potato chips packaged at sea level, and take the chips on an airplane flight. When you take the potato chips out of your carry-onbag, you notice it has noticeably puffed up. Airplane cabins are typically pressurized at 0.75 atm, and assuming the temperature inside an airplane is about the same as inside a potato chip processing plant, by what percentage has the bag puffed up in comparison to when it was packaged?

(II) A helium-filled balloon escapes a child’s hand at sea level and \(20.0^\circ \mathrm C\). When it reaches an altitude of 3600 m, where the temperature is \(5.0^\circ \mathrm C\) and the pressure only 0.68 atm, how will its volume compare to that at sea level?

Compare the value for the density of water vapor at exactly 100C and 1 atm (Table 101) with the value predicted from the ideal gas law. Why would you expect a difference?

A sealed test tube traps of air at a pressure of 1.00 atm and temperature of 18C. The test tubes stopper has a diameter of 1.50 cm and will pop off the test tube if a net upward force of 10.0 N is applied to it. To what temperature would you have to heat the trapped air in order to pop off the stopper? Assume the air surrounding the test tube is always at a pressure of 1.00 atm

(III) An air bubble at the bottom of a lake 41.0 m deep has a volume of \(1.00\ cm^3\). If the temperature at the bottom is \(5.5^{\circ}C\) and at the top \(18.5^{\circ}C\), what is the radius of the bubble just before it reaches the surface?

Calculate the number of in an ideal gas at STP

How many moles of water are there in 1.000 L at STP? How many molecules?

Estimate the number of (a) moles and (b) molecules of water in all the Earth’s oceans. Assume water covers 75% of the Earth to an average depth of 3 km.

The lowest pressure attainable using the best available vacuum techniques is about At such a pressure, how many molecules are there per at 0C?

Is a gas mostly empty space? Check by assuming that the spatial extent of the gas molecules in air is about so one gas molecule occupies an approximate volume equal to Assume STP.

(I) (a) What is the average translational kinetic energy of a nitrogen molecule at STP? (b) What is the total translational kinetic energy of 1.0 mol of \(\mathrm N_2\) molecules at \(25^\circ \mathrm C\)?

(I) Calculate the rms speed of helium atoms near the surface of the Sun at a temperature of about 6000 K.

By what factor will the rms speed of gas molecules increase if the temperature is increased from 20C to 160C?

A gas is at \(20^{\circ} \mathrm{C}\). To what temperature must it be raised to triple the rms speed of its molecules?

What speed would a 1.0-g paper clip have if it had the same kinetic energy as a molecule at 22C?

The rms speed of molecules in a gas at 20.0C is to be increased by 4.0%. To what temperature must it be raised?

If the pressure in a gas is tripled while its volume is held constant, by what factor does change?

(II) Show that the rms speed of molecules in a gas is given by \(v_{rms} = \sqrt{3P/\rho}\), where P is the pressure in the gas and \(\rho\) is the gas density.

Show that for a mixture of two gases at the same temperature, the ratio of their rms speeds is equal to the inverse ratio of the square roots of their molecular masses, \(v_1 v_2=\sqrt{M_2/M_1}\).

What is the rms speed of nitrogen molecules contained in an volume at 2.9 atm if the total amount of nitrogen is 2100 mol?

Two isotopes of uranium, and (the superscripts refer to their atomic masses), can be separated by a gas diffusion process by combining them with fluorine to make the gaseous compound Calculate the ratio of the rms speeds of these molecules for the two isotopes, at constant T. Use Appendix B for masses

Calculate (a) the rms speed of an oxygen molecule at 0C and (b) determine how many times per second it would move back and forth across a 5.0-m-long room on average, assuming it made no collisions with other molecules.

CO2 exists in what phase when the pressure is 35 atm and the temperature is 35C (Fig. 1323)?

(a) At atmospheric pressure, in what phases can exist? (b) For what range of pressures and temperatures can be a liquid? Refer to Fig. 1323.

Water is in which phase when the pressure is 0.01 atm and the temperature is (a) 90C, (b)

You have a sample of water and are able to control temperature and pressure arbitrarily. (a) Using Fig. 1322, describe the phase changes you would see if you started at a temperature of 85C, a pressure of 180 atm, and decreased the pressure down to 0.004 atm while keeping the temperature fixed. (b) Repeat part (a) with the temperature at 0.0C. Assume that you held the system at the starting conditions long enough for the system to stabilize before making further changes.

What is the partial pressure of water vapor at 30C if the humidity is 75%?

What is the air pressure at a place where water boils at 80C

What is the dew point if the humidity is 65% on a day when the temperature is 25C?

(II) If the air pressure at a particular place in the mountains is 0.80 atm, estimate the temperature at which water boils.

What is the mass of water in a closed room when the temperature is 25C and the relative humidity is 55%?

(II) A pressure cooker is a sealed pot designed to cook food with the steam produced by boiling water somewhat above \(100^{\circ}C\). The pressure cooker in Fig. 13–32 uses a weight of mass m to allow steam to escape at a certain pressure through a small hole (diameter d) in the cooker’s lid. If d = 3.0 mm, what should m be in order to cook food at \(120^{\circ}C\)? Assume that atmospheric pressure outside the cooker is \(1.01 \times 10^5\ Pa\).

If the humidity in a sealed room of volume at 20C is 65%, what mass of water can still evaporate from an open pan?

Air that is at its dew point of 5C is drawn into a building where it is heated to 22C. What will be the relative humidity at this temperature? Assume constant pressure of 1.0 atm. Take into account the expansion of the air.

When using a mercury barometer (Section 106), the vapor pressure of mercury is usually assumed to be zero. At room temperature mercurys vapor pressure is about 0.0015 mm-Hg. At sea level, the height h of mercury in a barometer is about 760 mm. (a) If the vapor pressure of mercury is neglected, is the true atmospheric pressure greater or less than the value read from the barometer? (b) What is the percent error? (c) What is the percent error if you use a water barometer and ignore waters saturated vapor pressure at STP?

Estimate the time needed for a glycine molecule (see Table 134) to diffuse a distance of in water at 20C if its concentration varies over that distance from to Compare this speed to its rms (thermal) speed. The molecular mass of glycine is about 75 u

(II) Oxygen diffuses from the surface of insects to the interior through tiny tubes called tracheae. An average trachea is about 2 mm long and has cross-sectional area of \(2 \times 10^{-9}\ m^2\). Assuming the concentration of oxygen inside is half what it is outside in the atmosphere, (a) show that the concentration of oxygen in the air (assume 21% is oxygen) at 20°C is about \(8.7\ mol/m^3\), then (b) calculate the diffusion rate J, and (c) estimate the average time for a molecule to diffuse in. Assume the diffusion constant is \(1 \times 10^{-5}\ m^2/s\).

A Pyrex measuring cup was calibrated at normal room temperature. How much error will be made in a recipe calling for 375 mL of cool water, if the water and the cup are hot, at 95C, instead of at room temperature? Neglect the glass expansion.

A precise steel tape measure has been calibrated at \(14^\circ \mathrm C\). At \(37^\circ \mathrm C\), (a) will it read high or low, and (b) what will be the percentage error?

A cubic box of volume is filled with air at atmospheric pressure at 15C. The box is closed and heated to 165C. What is the net force on each side of the box?

The gauge pressure in a helium gas cylinder is initially 32 atm. After many balloons have been blown up, the gauge pressure has decreased to 5 atm. What fraction of the original gas remains in the cylinder?

. If a scuba diver fills his lungs to full capacity of 5.5 L when 9.0 m below the surface, to what volume would his lungs expand if he quickly rose to the surface? Is this advisable?

A house has a volume of (a) What is the total mass of air inside the house at 15C? (b) If the temperature drops to what mass of air enters or leaves the house?

Estimate the number of air molecules in a room of length 6.0 m, width 3.0 m, and height 2.5 m. Assume the temperature is 22C. How many moles does that correspond to?

An iron cube floats in a bowl of liquid mercury at 0C. (a) If the temperature is raised to 25C, will the cube float higher or lower in the mercury? (b) By what percent will the fraction of volume submerged change? [Hint: See Chapter 10.]

A helium balloon, assumed to be a perfect sphere, has a radius of 24.0 cm. At room temperature (20C), its internal pressure is 1.08 atm. Determine the number of moles of helium in the balloon, and the mass of helium needed to inflate the balloon to these values.

A standard cylinder of oxygen used in a hospital has gauge (13,800 kPa) and 14 L at How long will the cylinder last if the flow rate, measured at atmospheric pressure, is constant at 2.1 L/min?

A brass lid screws tightly onto a glass jar at \(15^{\circ}C\). To help open the jar, it can be placed into a bath of hot water. After this treatment, the temperatures of the lid and the jar are both \(55^{\circ}C\). The inside diameter of the lid is 8.0 cm. Find the size of the gap (difference in radius) that develops by this procedure.

The density of gasoline at 0C is (a) What is the density on a hot day, when the temperature is 33C? (b) What is the percent change in density?

The first real length standard, adopted more than 200 years ago, was a platinum bar with two very fine marks separated by what was defined to be exactly one meter. If this standard bar was to be accurate to within how carefully would the trustees have needed to control the temperature? The coefficient of linear expansion is 9 * 106 C.

If a steel band were to fit snugly around the Earths equator at 25C, but then was heated to 55C, how high above the Earth would the band be (assume equal everywhere)?

In outer space the density of matter is about one atom per \(cm^3\), mainly hydrogen atoms, and the temperature is about 2.7 K. Calculate the rms speed of these hydrogen atoms, and the pressure (in atmospheres).

(a) Estimate the rms speed of an amino acid, whose molecular mass is 89 u, in a living cell at \(37^{\circ}C\). (b) What would be the rms speed of a protein of molecular mass 85,000 u at \(37^{\circ}C\)?

The escape speed from the Earth is so that a gas molecule traveling away from Earth near the outer boundary of the Earths atmosphere would, at this speed, be able to escape from the Earths gravitational field and be lost to the atmosphere. At what temperature is the rms speed of (a) oxygen molecules, and (b) helium atoms equal to (c) Can you explain why our atmosphere contains oxygen but not helium?

Consider a container of oxygen gas at a temperature of 23C that is 1.00 m tall. Compare the gravitational potential energy of a molecule at the top of the container (assuming the potential energy is zero at the bottom) with the average kinetic energy of the molecules. Is it reasonable to neglect the potential energy?

A space vehicle returning from the Moon enters the Earths atmosphere at a speed of about Molecules (assume nitrogen) striking the nose of the vehicle with this speed correspond to what temperature? (Because of this high temperature, the nose of a space vehicle must be made of special materials; indeed, part of it does vaporize, and this is seen as a bright blaze upon reentry.)

A sauna has of air volume, and the temperature is 85C. The air is perfectly dry. How much water (in kg) should be evaporated if we want to increase the relative humidity from 0% to 10%? (See Table 133.)

A 0.50-kg trash-can lid is suspended against gravity by tennis balls thrown vertically upward at it. How many tennis balls per second must rebound from the lid elastically, assuming they have a mass of 0.060 kg and are thrown at 15 m/s?

In humid climates, people constantly dehumidify their cellars to prevent rot and mildew. If the cellar in a house (kept at 20C) has of floor space and a ceiling height of 2.4 m, what is the mass of water that must be removed from it in order to drop the humidity from 95% to a more reasonable 40%?