Nuclear fusion, the energy source of the Sun, hydrogen

Problem 72PE Unreasonable Results Suppose the relative humidity is 80% on a day when the temperature is 30. 0ºC . (a) What will the relative humidity be if the air cools to 25. 0ºC and the vapor density remains constant? (b) What is unreasonable about this result? (c) Which premise is responsible?

Problem 1PE What is the Fahrenheit temperature of a person with a 39.0ºC fever?

Problem 1CQ What does it mean to say that two systems are in thermal equilibrium?

When a cold alcohol thermometer is placed in a hot liquid, the column of alcohol goes down slightly before going up. Explain why.

Problem 2PE Frost damage to most plants occurs at temperatures of 28.0ºF or lower. What is this temperature on the Kelvin scale?

Problem 4CQ If you add boiling water to a cup at room temperature, what would you expect the final equilibrium temperature of the unit to be? You will need to include the surroundings as part of the system. Consider the zeroth law of thermodynamics.

Problem 5CQ Thermal stresses caused by uneven cooling can easily break glass cookware. Explain why Pyrex®, a glass with a small coefficient of linear expansion, is less susceptible.

Problem 4PE A tungsten light bulb filament may operate at 2900 K. What is its Fahrenheit temperature? What is this on the Celsius scale?

Problem 5PE The surface temperature of the Sun is about 5750 K. What is this temperature on the Fahrenheit scale?

Problem 6CQ Water expands significantly when it freezes: a volume increase of about 9% occurs. As a result of this expansion and because of the formation and growth of crystals as water freezes, anywhere from 10% to 30% of biological cells are burst when animal or plant material is frozen. Discuss the implications of this cell damage for the prospect of preserving human bodies by freezing so that they can be thawed at some future date when it is hoped that all diseases are curable.

Problem 6PE One of the hottest temperatures ever recorded on the surface of Earth was 134ºF in Death Valley, CA. What is this temperature in Celsius degrees? What is this temperature in Kelvin?

Problem 7CQ One method of getting a tight fit, say of a metal peg in a hole in a metal block, is to manufacture the peg slightly larger than the hole. The peg is then inserted when at a different temperature than the block. Should the block be hotter or colder than the peg during insertion? Explain your answer.

Problem 7PE (a) Suppose a cold front blows into your locale and drops the temperature by 40.0 Fahrenheit degrees. How many degrees Celsius does the temperature decrease when there is a 40.0ºF decrease in temperature? (b) Show that any change in temperature in Fahrenheit degrees is nine-fifths the change in Celsius degrees.

Problem 8CQ Does it really help to run hot water over a tight metal lid on a glass jar before trying to open it? Explain your answer.

Problem 9CQ Liquids and solids expand with increasing temperature, because the kinetic energy of a body’s atoms and molecules increases. Explain why some materials shrink with increasing temperature.

Problem 8PE (a) At what temperature do the Fahrenheit and Celsius scales have the same numerical value? (b) At what temperature do the Fahrenheit and Kelvin scales have the same numerical value?

Problem 9PE The height of the Washington Monument is measured to be 170 m on a day when the temperature is 35.0ºC . What will its height be on a day when the temperature falls to –10.0ºC ? Although the monument is made of limestone, assume that its thermal coefficient of expansion is the same as marble’s.

Problem 10PE How much taller does the Eiffel Tower become at the end of a day when the temperature has increased by 15ºC ? Its original height is 321 m and you can assume it is made of steel.

Problem 11CQ Under what circumstances would you expect a gas to behave significantly differently than predicted by the ideal gas law?

Problem 10CQ Find out the human population of Earth. Is there a mole of people inhabiting Earth? If the average mass of a person is 60 kg, calculate the mass of a mole of people. How does the mass of a mole of people compare with the mass of Earth?

What is the change in length of a 3.00-cm-long column of mercury if its temperature changes from \(37.0^{\circ} \mathrm{C}\) to \(40.0^{\circ} \mathrm{C}\) , assuming the mercury is unconstrained?

Problem 12CQ A constant-volume gas thermometer contains a fixed amount of gas. What property of the gas is measured to indicate its temperature?

Problem 12PE How large an expansion gap should be left between steel railroad rails if they may reach a maximum temperature 35.0ºC greater than when they were laid? Their original length is 10.0 m.

Problem 13CQ How is momentum related to the pressure exerted by a gas? Explain on the atomic and molecular level, considering the behavior of atoms and molecules.

Problem 14CQ A pressure cooker contains water and steam in equilibrium at a pressure greater than atmospheric pressure. How does this greater pressure increase cooking speed?

Problem 14PE Global warming will produce rising sea levels partly due to melting ice caps but also due to the expansion of water as average ocean temperatures rise. To get some idea of the size of this effect, calculate the change in length of a column of water 1.00 km high for a temperature increase of 1.00ºC. Note that this calculation is only approximate because ocean warming is not uniform with depth.

Problem 15CQ Why does condensation form most rapidly on the coldest object in a room—for example, on a glass of ice water?

Problem 13PE You are looking to purchase a small piece of land in Hong Kong. The price is “only” $60,000 per square meter! The land title says the dimensions are 20 m × 30 m. By how much would the total price change if you measured the parcel with a steel tape measure on a day when the temperature was 20ºC above normal?

Problem 15PE Show that 60.0 L of gasoline originally at 15.0ºC will expand to 61.1 L when it warms to 35.0ºC, as claimed in Example 13.4. Example 13.4: Calculating Thermal Expansion: Gas vs. Gas Tank Suppose your 60.0-L (15.9-gal) steel gasoline tank is full of gas, so both the tank and the gasoline have a temperature of 15.0oC . How much gasoline has spilled by the time they warm to 35.0oC ?

Problem 16CQ What is the vapor pressure of solid carbon dioxide (dry ice) at – 78.5ºC ?

Can carbon dioxide be liquefied at room temperature \(\left(20^{\circ} \mathrm{C}\right)\)? If so, how? If not, why not? (See Figure \(13.36\).) Equation Transcription: Text Transcription: (20 degrees C) 13.36

Problem 17PE (a) If a 500-mL glass beaker is filled to the brim with ethyl alcohol at a temperature of 5.00ºC, how much will overflow when its temperature reaches 22.0ºC ? (b) How much less water would overflow under the same conditions?

Problem 18CQ Oxygen cannot be liquefied at room temperature by placing it under a large enough pressure to force its molecules together. Explain why this is.

Problem 18PE Most automobiles have a coolant reservoir to catch radiator fluid that may overflow when the engine is hot. A radiator is made of copper and is filled to its 16.0-L capacity when at 10.0ºC. What volume of radiator fluid will overflow when the radiator and fluid reach their 95.0ºC operating temperature, given that the fluid’s volume coefficient of expansion is ? = 400×10 – 6 / ºC ? Note that this coefficient is approximate, because most car radiators have operating temperatures of greater than 95.0ºC.

Problem 19PE A physicist makes a cup of instant coffee and notices that, as the coffee cools, its level drops 3.00 mm in the glass cup. Show that this decrease cannot be due to thermal contraction by calculating the decrease in level if the 350 cm3 of coffee is in a 7.00-cm-diameter cup and decreases in temperature from 95.0ºC to 45.0ºC. (Most of the drop in level is actually due to escaping bubbles of air.)

Problem 19CQ What is the distinction between gas and vapor?

Because humidity depends only on water’s vapor pressure and temperature, are the saturation vapor densities listed in Table \(13.5\) valid in an atmosphere of helium at a pressure of \(1.01 \times 10^{5} \mathrm{~N} / \mathrm{m}^{2}\) , rather than air? Are those values affected by altitude on Earth? Equation Transcription: Text Transcription: 13.5 1.01 times 10^5 N/m^2

Problem 20PE (a) The density of water at 0ºC is very nearly 1000 kg/m3 (it is actually 999.84 kg/m3 ), whereas the density of ice at 0ºC is 917 kg/m3 . Calculate the pressure necessary to keep ice from expanding when it freezes, neglecting the effect such a large pressure would have on the freezing temperature. (This problem gives you only an indication of how large the forces associated with freezing water might be.) (b) What are the implications of this result for biological cells that are frozen?

Problem 21CQ Why does a beaker of 40.0ºC water placed in a vacuum chamber start to boil as the chamber is evacuated (air is pumped out of the chamber)? At what pressure does the boiling begin? Would food cook any faster in such a beaker?

Problem 21PE Show that ? ? 3?, by calculating the change in volume ?V of a cube with sides of length L. 13.3 The Ideal Gas Law

Problem 22CQ Why does rubbing alcohol evaporate much more rapidly than water at STP (standard temperature and pressure)?

Problem 22PE The gauge pressure in your car tires is 2.50×105 N/m2 at a temperature of 35.0ºC when you drive it onto a ferry boat to Alaska. What is their gauge pressure later, when their temperature has dropped to – 40.0ºC ?

Problem 23PE Convert an absolute pressure of 7.00×105 N/m2 to gauge pressure in lb/in2 . (This value was stated to be just less than 90.0 lb/in2 in Example 13.9. Is it?) Example 13.9: Calculating Number of Moles: Gas in a Bike Tire How many moles of gas are in a bike tire with a volume of 2.00×10 – 3 m3(2.00 L), a pressure of 7.00×105 Pa (a gauge pressure of just under 90.0 lb/in2 ), and at a temperature of 18.0oC ?

Problem 24PE Suppose a gas-filled incandescent light bulb is manufactured so that the gas inside the bulb is at atmospheric pressure when the bulb has a temperature of 20.0ºC . (a) Find the gauge pressure inside such a bulb when it is hot, assuming its average temperature is 60.0ºC (an approximation) and neglecting any change in volume due to thermal expansion or gas leaks. (b) The actual final pressure for the light bulb will be less than calculated in part (a) because the glass bulb will expand. What will the actual final pressure be, taking this into account? Is this a negligible difference?

Problem 25PE Large helium-filled balloons are used to lift scientific equipment to high altitudes. (a) What is the pressure inside such a balloon if it starts out at sea level with a temperature of 10.0ºC and rises to an altitude where its volume is twenty times the original volume and its temperature is – 50.0ºC ? (b) What is the gauge pressure? (Assume atmospheric pressure is constant.)

Problem 26PE Confirm that the units of nRT are those of energy for each value of R : (a) 8.31 J/mol ? K , (b) 1.99 cal/mol ? K , and (c) 0.0821 L ? atm/mol ? K .

Problem 27PE In the text, it was shown that N / V = 2.68×1025 m ?3 for gas at STP. (a) Show that this quantity is equivalent to N / V = 2.68×1019 cm?3 , as stated. (b) About how many atoms are there in one ?m3 (a cubic micrometer) at STP? (c) What does your answer to part (b) imply about the separation of atoms and molecules?

Problem 28PE Calculate the number of moles in the 2.00-L volume of air in the lungs of the average person. Note that the air is at 37.0ºC (body temperature).

Problem 29PE An airplane passenger has 100 cm3 of air in his stomach just before the plane takes off from a sea-level airport. What volume will the air have at cruising altitude if cabin pressure drops to 7.50×104 N/m2 ?

Problem 30PE (a) What is the volume (in km3 ) of Avogadro’s number of sand grains if each grain is a cube and has sides that are 1.0 mm long? (b) How many kilometers of beaches in length would this cover if the beach averages 100 m in width and 10.0 m in depth? Neglect air spaces between grains.

Problem 31PE An expensive vacuum system can achieve a pressure as low as 1.00×10 – 7 N/m2 at 20ºC . How many atoms are there in a cubic centimeter at this pressure and temperature?

Problem 32PE The number density of gas atoms at a certain location in the space above our planet is about 1.00×1011 m ?3 , and the pressure is 2.75×10 – 10 N/m2 in this space. What is the temperature there?

Problem 33PE A bicycle tire has a pressure of 7.00×105 N/m2 at a temperature of 18.0ºC and contains 2.00 L of gas. What will its pressure be if you let out an amount of air that has a volume of 100 cm3 at atmospheric pressure? Assume tire temperature and volume remain constant.

Problem 34PE A high-pressure gas cylinder contains 50.0 L of toxic gas at a pressure of 1.40×107 N/m2 and a temperature of 25.0ºC . Its valve leaks after the cylinder is dropped. The cylinder is cooled to dry ice temperature (–78.5ºC) to reduce the leak rate and pressure so that it can be safely repaired. (a) What is the final pressure in the tank, assuming a negligible amount of gas leaks while being cooled and that there is no phase change? (b) What is the final pressure if one-tenth of the gas escapes? (c) To what temperature must the tank be cooled to reduce the pressure to 1.00 atm (assuming the gas does not change phase and that there is no leakage during cooling)? (d) Does cooling the tank appear to be a practical solution?

Problem 36PE Calculate the depth to which Avogadro’s number of table tennis balls would cover Earth. Each ball has a diameter of 3.75 cm. Assume the space between balls adds an extra 25.0% to their volume and assume they are not crushed by their own weight.

Problem 37PE (a) What is the gauge pressure in a 25.0ºC car tire containing 3.60 mol of gas in a 30.0 L volume? (b) What will its gauge pressure be if you add 1.00 L of gas originally at atmospheric pressure and 25.0ºC ? Assume the temperature returns to 25.0ºC and the volume remains constant.

Problem 38PE (a) In the deep space between galaxies, the density of atoms is as low as 106 atoms/m3 , and the temperature is a frigid 2.7 K. What is the pressure? (b) What volume (in m 3 ) is occupied by 1 mol of gas? (c) If this volume is a cube, what is the length of its sides in kilometers?

Problem 35PE Find the number of moles in 2.00 L of gas at 35.0ºC and under 7.41×107 N/m2 of pressure.

Problem 40PE Average atomic and molecular speeds (vrms) are large, even at low temperatures. What is vrms for helium atoms at 5.00 K, just one degree above helium’s liquefaction temperature?

Problem 41PE (a) What is the average kinetic energy in joules of hydrogen atoms on the 5500ºC surface of the Sun? (b) What is the average kinetic energy of helium atoms in a region of the solar corona where the temperature is 6.00×105 K ?

Problem 43PE The escape velocity from the Moon is much smaller than from Earth and is only 2.38 km/s. At what temperature would hydrogen molecules (molecular mass is equal to 2.016 g/mol) have an average velocity vrms equal to the Moon’s escape velocity?

Problem 44PE Nuclear fusion, the energy source of the Sun, hydrogen bombs, and fusion reactors, occurs much more readily when the average kinetic energy of the atoms is high—that is, at high temperatures. Suppose you want the atoms in your fusion experiment to have average kinetic energies of 6.40×10– 14 J . What temperature is needed?

Problem 42PE The escape velocity of any object from Earth is 11.2 km/s. (a) Express this speed in m/s and km/h. (b) At what temperature would oxygen molecules (molecular mass is equal to 32.0 g/mol) have an average velocity vrms equal to Earth’s escape velocity of 11.1 km/s?

Problem 45PE Suppose that the average velocity (vrms) of carbon dioxide molecules (molecular mass is equal to 44.0 g/mol) in a flame is found to be 1.05×105 m/s . What temperature does this represent?

Problem 48PE There are two important isotopes of uranium— 235U and 238U ; these isotopes are nearly identical chemically but have different atomic masses. Only 235U is very useful in nuclear reactors. One of the techniques for separating them (gas diffusion) is based on the different average velocities vrms of uranium hexafluoride gas, UF6 . (a) The molecular masses for 235U UF6 and 238U UF6 are 349.0 g/mol and 352.0 g/mol, respectively. What is the ratio of their average velocities? (b) At what temperature would their average velocities differ by 1.00 m/s? (c) Do your answers in this problem imply that this technique may be difficult?

Problem 49PE Dry air is 78.1% nitrogen. What is the partial pressure of nitrogen when the atmospheric pressure is 1.01×105 N/m2 ?

Problem 47PE Much of the gas near the Sun is atomic hydrogen. Its temperature would have to be 1.5×107 K for the average velocity vrms to equal the escape velocity from the Sun. What is that velocity?

Problem 51PE Pressure cookers increase cooking speed by raising the boiling temperature of water above its value at atmospheric pressure. (a) What pressure is necessary to raise the boiling point to 120.0ºC ? (b) What gauge pressure does this correspond to?

Problem 50PE (a) What is the vapor pressure of water at 20.0ºC ? (b) What percentage of atmospheric pressure does this correspond to? (c) What percent of 20.0ºC air is water vapor if it has 100% relative humidity? (The density of dry air at 20.0ºC is 1.20 kg/m3 .)

Problem 46PE Hydrogen molecules (molecular mass is equal to 2.016 g/ mol) have an average velocity vrms equal to 193 m/s. What is the temperature?

Problem 52PE (a) At what temperature does water boil at an altitude of 1500 m (about 5000 ft) on a day when atmospheric pressure is 8.59×104 N/m2 ? (b) What about at an altitude of 3000 m (about 10,000 ft) when atmospheric pressure is 7.00×104 N/m2 ?

Problem 54PE At a spot in the high Andes, water boils at 80.0ºC , greatly reducing the cooking speed of potatoes, for example. What is atmospheric pressure at this location?

Problem 55PE What is the relative humidity on a 25.0ºC day when the air contains 18.0 g/m3 of water vapor?

Problem 56PE What is the density of water vapor in g/m3 on a hot dry day in the desert when the temperature is 40.0ºC and the relative humidity is 6.00%?

Problem 53PE What is the atmospheric pressure on top of Mt. Everest on a day when water boils there at a temperature of 70.0ºC?

Problem 57PE A deep-sea diver should breathe a gas mixture that has the same oxygen partial pressure as at sea level, where dry air contains 20.9% oxygen and has a total pressure of 1.01×105 N/m2 . (a) What is the partial pressure of oxygen at sea level? (b) If the diver breathes a gas mixture at a pressure of 2.00×106 N/m2 , what percent oxygen should it be to have the same oxygen partial pressure as at sea level?

Problem 58PE The vapor pressure of water at 40.0ºC is 7.34×103 N/m2 . Using the ideal gas law, calculate the density of water vapor in g/m3 that creates a partial pressure equal to this vapor pressure. The result should be the same as the saturation vapor density at that temperature (51.1 g/m3).

Problem 59PE Air in human lungs has a temperature of 37.0ºC and a saturation vapor density of 44.0 g/m3 . (a) If 2.00 L of air is exhaled and very dry air inhaled, what is the maximum loss of water vapor by the person? (b) Calculate the partial pressure of water vapor having this density, and compare it with the vapor pressure of 6.31×103 N/m2 .

Problem 61PE Late on an autumn day, the relative humidity is 45.0% and the temperature is 20.0ºC . What will the relative humidity be that evening when the temperature has dropped to 10.0ºC , assuming constant water vapor density?

Problem 60PE If the relative humidity is 90.0% on a muggy summer morning when the temperature is 20.0ºC , what will it be later in the day when the temperature is 30.0ºC , assuming the water vapor density remains constant?

Problem 63PE What is the dew point (the temperature at which 100% relative humidity would occur) on a day when relative humidity is 39.0% at a temperature of 20.0ºC ?

Problem 62PE Atmospheric pressure atop Mt. Everest is 3.30×104 N/m2. (a) What is the partial pressure of oxygen there if it is 20.9% of the air? (b) What percent oxygen should a mountain climber breathe so that its partial pressure is the same as at sea level, where atmospheric pressure is 1.01×105 N/m2 ? (c) One of the most severe problems for those climbing very high mountains is the extreme drying of breathing passages. Why does this drying occur?

Problem 64PE On a certain day, the temperature is 25.0ºC and the relative humidity is 90.0%. How many grams of water must condense out of each cubic meter of air if the temperature falls to 15.0ºC ? Such a drop in temperature can, thus, produce heavy dew or fog.

Problem 65PE Integrated Concepts The boiling point of water increases with depth because pressure increases with depth. At what depth will fresh water have a boiling point of 150ºC , if the surface of the water is at sea level?

Problem 66PE Integrated Concepts (a) At what depth in fresh water is the critical pressure of water reached, given that the surface is at sea level? (b) At what temperature will this water boil? (c) Is a significantly higher temperature needed to boil water at a greater depth?

Problem 67PE Integrated Concepts To get an idea of the small effect that temperature has on Archimedes’ principle, calculate the fraction of a copper block’s weight that is supported by the buoyant force in 0ºC water and compare this fraction with the fraction supported in 95. 0ºC water.

Problem 69PE Unreasonable Results (a) How many moles per cubic meter of an ideal gas are there at a pressure of 1.00×1014 N/m 2 and at 0ºC ? (b) What is unreasonable about this result? (c) Which premise or assumption is responsible?

Problem 70PE Unreasonable Results (a) An automobile mechanic claims that an aluminum rod fits loosely into its hole on an aluminum engine block because the engine is hot and the rod is cold. If the hole is 10.0% bigger in diameter than the 22. 0ºC rod, at what temperature will the rod be the same size as the hole? (b) What is unreasonable about this temperature? (c) Which premise is responsible?

Problem 68PE Integrated Concepts If you want to cook in water at 150ºC , you need a pressure cooker that can withstand the necessary pressure. (a) What pressure is required for the boiling point of water to be this high? (b) If the lid of the pressure cooker is a disk 25.0 cm in diameter, what force must it be able to withstand at this pressure?

Problem 71PE Unreasonable Results The temperature inside a supernova explosion is said to be 2.00×1013 K . (a) What would the average velocity vrms of hydrogen atoms be? (b) What is unreasonable about this velocity? (c) Which premise or assumption is responsible?

Problem 39PE Some incandescent light bulbs are filled with argon gas. What is vrms for argon atoms near the filament, assuming their temperature is 2500 K?

Problem 3PE To conserve energy, room temperatures are kept at 68.0ºF in the winter and 78.0ºF in the summer. What are these temperatures on the Celsius scale?