Chapter Questions
(I) How many atoms are there in a 3.4 gram copper penny?
Name several properties of materials that could be exploited to make a thermometer.
(I) $(a)$ "Room temperature " is often taken to be $68^{\circ} \mathrm{F}$ . What is this on the Celsius scale? (b) The temperature of the filament in a lightbulb is about $1800^{\circ} \mathrm{C}$ . What is this on the Fahrenheit scale?
(I) Among the highest and lowest temperatures recorded are $136^{\circ} \mathrm{F}$ in the Libyan desert and $-129^{\circ} \mathrm{F}$ in Antarctica. What are these temperatures on the Celsius scale?
(I) $(a) 15^{\circ}$ below zero on the Celsius scale is what Fahrenheit temperature? (b) $15^{\circ}$ below zero on the Fahrenheit scale is what Celsius temperature?
(II) In an alcohol-in-glass thermometer, the alcohol column has length 11.82 $\mathrm{cm}$ at $0.0^{\circ} \mathrm{C}$ and length 22.85 $\mathrm{cm}$ at $100.0^{\circ} \mathrm{C} .$ What is the temperature if the column has length (a) $16.70 \mathrm{cm},$ and $(b) 20.50 \mathrm{cm} ?$
(I) A concrete highway is built of slabs 12 $\mathrm{m}$ long $\left(20^{\circ} \mathrm{C}\right) .$How wide should the expansion cracks between the slabs be (at $20^{\circ} \mathrm{C} )$ to prevent buckling if the range of temperature is $-30^{\circ} \mathrm{C}$ to $+50^{\circ} \mathrm{C} ?$
(I) Super Invar'M, an alloy of iron and nickel, is a strong material with a very low coefficient of linear expansion $\left[0.2 \times 10^{-6}\left(\mathrm{C}^{\circ}\right)^{-1}\right] .$ A $2.0-\mathrm{m}$ -long tabletop made of this alloy is used for sensitive laser measurements where extremely high tolerances are required. How much will this table expand along its length if the temperature increases 5.0 $\mathrm{C}^{\circ} ?$ Compare to tabletops made of steel.
(I) The Eiffel Tower (Fig. $13-29$ ) is built of wrought iron approximately 300 $\mathrm{m}$ tall. Esti-mate how much its height changes between July (average temperature of $25^{\circ} \mathrm{C} )$ and January (average temperature of $2^{\circ} \mathrm{C} )$ . Ignore the angles of the iron beams, and treat the tower as a vertical beam.figure can't copy
(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 1.871 $\mathrm{cm}$ in diameter is to be placed in a hole 1.869 $\mathrm{cm}$ in diameter at $20^{\circ} \mathrm{C}$ . To what temperature must the rivet be cooled if it is to fit in the hole?
(II) The density of water at $4^{\circ} \mathrm{C}$ is $1.00 \times 10^{3} \mathrm{kg} / \mathrm{m}^{3}$ . What is water's density at $94^{\circ} \mathrm{C}$ ?
(II) A quartz sphere is 8.75 $\mathrm{cm}$ in diameter. What will be its change in volume if it is heated from $30^{\circ} \mathrm{C}$ to $200^{\circ} \mathrm{C} ?$
(II) An ordinary glass is filled to the brim with 350.0 $\mathrm{mL}$ of water at $100.0^{\circ} \mathrm{C}$ . If the temperature decreased to $20.0^{\circ} \mathrm{C}$ , how much water could be added to the glass?
(II) It is observed that 55.50 $\mathrm{mL}$ of water at $20^{\circ} \mathrm{C}$ completely fills a container to the brim. When the container and the water are heated to $60^{\circ} \mathrm{C}, 0.35 \mathrm{g}$ 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? The density of water at $60^{\circ} \mathrm{C}$ is 0.98324 $\mathrm{g} / \mathrm{mL}$ .
(II) $(a)$ A brass plug is to be placed in a ring made of iron. At $20^{\circ} \mathrm{C}$ , the diameter of the plug is 8.753 $\mathrm{cm}$ and that of the inside of the ring is 8.743 $\mathrm{cm}$ . They must both be brought to what common temperature in order to fit? (b) What if the plug were iron and the ring brass?
(II) If a fluid is contained in a long, narrow vessel so it can expand in essentially one direction only, show that the effective coefficient of linear expansion $\alpha$ is approximately equal to the coefficient of volume expansion $\beta$ .
(II) $(a)$ Show that the change in the density $\rho$ of a substance, when the temperature changes by $\Delta T,$ is given by $\Delta \rho=-\beta \rho \Delta T .(b)$ What is the fractional change in density of a lead sphere whose temperature decreases from $25^{\circ} \mathrm{C}$ to $-40^{\circ} \mathrm{C} ?$
(II) A uniform rectangular plate of length $l$ and width $w$ has coefficient of linear expansion $\alpha$ . Show that, if we neglect very small quantities, the change in area of the plate due to a temperature change $\Delta T$ is $\Delta A=2 \alpha l w \Delta T .$ figure can't copy
(III) Show that for an isotropic solid, $\beta=3 \alpha,$ if the amount of expansion is small. $\beta$ and $\alpha$ are the coefficients of volume and linear expansion, respectively. $[\text { Hint: }$ Consider a cubical solid, and neglect very small quantities. See also Problem 18 and Fig. $13-30.1$
(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 $25^{\circ} \mathrm{C}$ ? (Assume the frequency dependence on length for a simple pendulum applies.)
(III) $(a)$ The tube of a mercury thermometer has an inside diameter of 0.140 $\mathrm{mm}$ . The bulb has a volume of 0.255 $\mathrm{cm}^{3}$ . How far will the thread of mercury move when the temperature changes from $11.5^{\circ} \mathrm{C}$ to $33.0^{\circ} \mathrm{C}$ ? Take into account expansion of the Pyrex glass. (b) Determine a formula for the change in length of the mercury column in terms of relevant variables. Ignore tube volume compared to bulb volume.
(III) A 23.4 -kg solid aluminum cylindrical wheel of radius 0.41 $\mathrm{m}$ is rotating about its axle on frictionless bearings with angular velocity $\omega=32.8 \mathrm{rad} / \mathrm{s}$ . If its temperature is now raised from $20.0^{\circ} \mathrm{C}$ to $75.0^{\circ} \mathrm{C}$ , what is the fractional change in $\omega ?$
(II) An aluminum bar has the desired length when at $15^{\circ} \mathrm{C}$ . How much stress is required to keep it at this length if the temperature increases to $35^{\circ} \mathrm{C} ?$
(II) $(a)$ A horizontal steel I-beam of cross-sectional area 0.041 $\mathrm{m}^{2}$ is rigidly connected to two vertical steel girders. If the beam was installed when the temperature was $30^{\circ} \mathrm{C}$ , what stress is developed in the beam when the temperature drops to $-30^{\circ} \mathrm{C} ?(b)$ Is the ultimate strength of the steel exceeded? (c) What stress is developed if the beam is concrete and has a cross-sectional area of 0.13 $\mathrm{m}^{2} ?$ Will it fracture?
(III) A barrel of diameter 134.122 $\mathrm{cm}$ at $20^{\circ} \mathrm{C}$ is to be enclosed by an iron band. The circular band has an inside diameter of 134.110 $\mathrm{cm}$ at $20^{\circ} \mathrm{C}$ . It is 7.4 $\mathrm{cm}$ wide and 0.65 $\mathrm{cm}$ thick. ( $a$ ) To what temperature must the band beheated so that it will fit over the barrel? (b) What will be the tension in the band when it cools to $20^{\circ} \mathrm{C} ?$
(I) What are the following temperatures on the Kelvin scale: $(a) \quad 86^{\circ} \mathrm{C}, \quad(b) \quad 78^{\circ} \mathrm{F}, \quad(c) \quad-100^{\circ} \mathrm{C}, \quad(d) \quad 5500^{\circ} \mathrm{C}$ $(e)-459^{\circ} \mathrm{F} ?$
(I) Absolute zero is what temperature on the Fahrenheit scale?
(II) Typical temperatures in the interior of the Earth and Sun are about $4000^{\circ} \mathrm{C}$ and $15 \times 10^{6}$ "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 $\mathrm{K}$ ?
(I) If 3.00 $\mathrm{m}^{3}$ of a gas initially at STP is placed under a pressure of 3.20 $\mathrm{atm}$ , the temperature of the gas rises to $38.0^{\circ} \mathrm{C}$ . What is the volume?
(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 $\mathrm{atm}$ .
(II) Calculate the density of oxygen at STP using the ideal gas law.
(II) A storage tank contains 21.6 $\mathrm{kg}$ of nitrogen $\left(\mathrm{N}_{2}\right)$ at anabsolute pressure of $3.65 \mathrm{atm} .$ What will the pressure be if the nitrogen is replaced by an equal mass of $\mathrm{CO}_{2} ?$
(II) A storage tank at STP contains 18.5 $\mathrm{kg}$ of nitrogen $\left(\mathrm{N}_{2}\right) .(a)$ What is the volume of the tank? $(b)$ What is the pressure if an additional 15.0 $\mathrm{kg}$ of nitrogen is added without changing the temperature?
(II) If 18.75 mol of helium gas is at $10.0^{\circ} \mathrm{C}$ and a gauge pressure of 0.350 atm, (a) calculate the volume of the helium gas under these conditions. (b) Calculate the temperature if the gas is compressed to precisely half the volume at a gauge pressure of 1.00 $\mathrm{atm}$ .
(II) What is the pressure inside a 35.0 - L container holding 105.0 $\mathrm{kg}$ of argon gas at 385 $\mathrm{K}$ ?
(II) A tank contains 26.0 $\mathrm{kg}$ of $\mathrm{O}_{2}$ gas at a gauge pressure of $8.70 \mathrm{atm} .$ If the oxygen is replaced by helium, how many kilograms of the latter will be needed to produce a gauge pressure of 7.00 $\mathrm{atm}$ ?
(II) A hot-air balloon achieves its buoyant lift by heating the air inside the balloon, which makes it less dense than the air outside. Suppose the volume of a balloon is 1800 $\mathrm{m}^{3}$ and the required lift is 2700 $\mathrm{N}$ (rough estimate of the weight of the equipment and passenger). Calculate the temperature of the air inside the balloon which will produce the required lift. Assume that the outside air temperature is $0^{\circ} \mathrm{C}$ and that air is an ideal gas under these conditions. What factors limit the maximum altitude attainable by this method for a given load? (Neglect variables like wind.)
(II) A tire is filled with air at $15^{\circ} \mathrm{C}$ to a gauge pressure of 220 $\mathrm{kPa}$ . If the tire reaches a temperature of $38^{\circ} \mathrm{C},$ what fraction of the original air must be removed if the original pressure of 220 $\mathrm{kPa}$ is to be maintained?
(II) If 61.5 $\mathrm{L}$ of oxygen at $18.0^{\circ} \mathrm{C}$ and an absolute pressure of 2.45 atm are compressed to 48.8 $\mathrm{L}$ and at the same time the temperature is raised to $50.0^{\circ} \mathrm{C}$ , what will the new pressure be?
(III) A helium-filled balloon escapes a child's hand at sea level and $20.0^{\circ} \mathrm{C}$ . When it reaches an altitude of 3000 $\mathrm{m}$ , where the temperature is $5.0^{\circ} \mathrm{C}$ and the pressure is only 0.70 atm, how will its volume compare to that at sea level?
(I) Calculate the number of molecules/m' in an ideal gas at STP.
(I) How many moles of water are there in 1.000 $\mathrm{L}$ ? 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 $\mathrm{km}$ .
(II) A cubic box of volume $5.1 \times 10^{-2} \mathrm{m}^{3}$ is filled with air at atmospheric pressure at $20^{\circ} \mathrm{C}$ . The box is closed and heated to $180^{\circ} \mathrm{C}$ . What is the net force on each side of the box?
(III) Estimate how many molecules of air are in each 2.0 -L breath you inhale that were also in the last breath Galileo took. [Hint: Assume the atmosphere is about 10 $\mathrm{km}$ high and of constant density.]
(I) $(a)$ What is the average translational kinetic energy of an oxygen molecule at STP? (b) What is the total translational kinetic energy of 2.0 $\mathrm{mol}$ of $\mathrm{O}_{2}$ molecules at $20^{\circ} \mathrm{C}$ ?
(I) Calculate the rms speed of helium atoms near the surface of the Sun at a temperature of about 6000 $\mathrm{K}$ .
(I) By what factor will the rms speed of gas molecules increase if the temperature is increased from $0^{\circ} \mathrm{C}$ to $100^{\circ} \mathrm{C}$ ?
(I) A gas is at $20^{\circ} \mathrm{C}$ . To what temperature must it be raised to double the rms speed of its molecules?
(I) Twelve molecules have the following speeds, given in units of $\mathrm{km} / \mathrm{s} : 6,2,4,6,0,4,1,8,5,3,7,$ and $8 .$ Calculate the rms speed.
(II) The rms speed of molecules in a gas at $20.0^{\circ} \mathrm{C}$ is to be increased by 1.0$\%$ . To what temperature must it be raised?
(II) If the pressure of a gas is doubled while its volume is held constant, by what factor does $v_{\mathrm{rms}}$ change?
(II) Show that the rms speed of molecules in a gas is given by $v_{\mathrm{rms}}=\sqrt{3 P / \rho}$ , 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.
(II) What is the rms speed of nitrogen molecules contained in an $8.5-\mathrm{m}^{3}$ volume at 2.1 $\mathrm{atm}$ if the total amount of nitrogen is 1300 $\mathrm{mol} ?$
(II) Calculate $(a)$ the rms speed of an oxygen molecule at $0^{\circ} \mathrm{C}$ and $(b)$ determine how many times per second it would move back and forth across a 7.0 -m-long room on the average, assuming it made very few collisions with other molecules.
(II) What is the average distance between nitrogen molecules at STP?
(II) $(a)$ Estimate the rms speed of an amino acid whose molecular mass is 89 $\mathrm{u}$ in a living cell at $37^{\circ} \mathrm{C}$ . (b) What would be the rms speed of a protein of molecular mass $50,000 \mathrm{u}$ at $37^{\circ} \mathrm{C}$ ?
(II) Show that the pressure $P$ of a gas can be written $P=\frac{1}{3} \rho v^{2},$ where $\rho$ is the density of the gas and $v$ is the rms speed of the molecules.
(III) The two isotopes of uranium, $^{235} \mathrm{U}$ and $^{238} \mathrm{U}$ (the superscripts refer to their atomic mass), can be separated by a gas-diffusion process by combining them with fluorine to make the gaseous compound UF, Calculate the ratio of the rms speeds of these molecules for the two isotopes, at constant $T$ .
(I) $(a)$ At atmospheric pressure, in what phases can CO $_{2}$ exist? $(b)$ For what range of pressures and temperatures can $\mathrm{CO}_{2}$ be a liquid? Refer to Fig. $13-21$ .
(I) Water is in which phase when the pressure is 0.01 atm and the temperature is $(a) 90^{\circ} \mathrm{C},(b)-20^{\circ} \mathrm{C} ?$
(I) What is the dew point (approximately) if the humidity is 50$\%$ on a day when the temperature is $25^{\circ} \mathrm{C}$ ?
(1) What is the air pressure at a place where water boils at $90^{\circ} \mathrm{C} ?$
(I) If the air pressure at a particular place in the mountains is 0.72 atm, estimate the temperature at which water boils.
(I) What is the temperature on a day when the partial pressure of water is 530 $\mathrm{Pa}$ and the relative humidity is 40$\% ?$
(I) What is the partial pressure of water on a day when the temperature is $25^{\circ} \mathrm{C}$ and the relative humidity is 35$\% ?$
(I) What is the approximate pressure inside a pressure cooker if the water is boiling at a temperature of $120^{\circ} \mathrm{C}$ ? Assume no air escaped during the heating process, which started at $20^{\circ} \mathrm{C}$ .
(II) If the humidity in a room of volume 680 $\mathrm{m}^{3}$ at $25^{\circ} \mathrm{C}$ is $80 \%,$ what mass of water can still evaporate from an open pan?
(III) Air that is at its dew point of $5^{\circ} \mathrm{C}$ is drawn into a building where it is heated to $25^{\circ} \mathrm{C}$ . What will be the relative humidity at this temperature? Assume constant pressure of $1.0 \mathrm{atm} .$ Take into account the expansion of the air.
(II) Estimate the time needed for a glycine molecule (see Table $13-4$ ) to diffuse a distance of 15$\mu m$ in water at $20^{\circ} \mathrm{C}$ if its concentration varies over that distance from 1.00 $\mathrm{mol} / \mathrm{m}^{3}$ to $0.40 \mathrm{mol} / \mathrm{m}^{3} .$ Compare this "speed" to its $\mathrm{rms}$ (thermal) speed. The molecular mass of glycine is about 75 $\mathrm{u}$ .
(II) Oxygen diffuses from the surface of insects to the interior through tiny tubes called tracheae. An average trachea is about 2 $\mathrm{mm}$ long and has cross-sectional area of $2 \times 10^{-9} \mathrm{m}^{2} .$ 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 $20^{\circ} \mathrm{C}$ is about 8.7 $\mathrm{mol} / \mathrm{m}^{3}$ , then(b) calculate the diffusion rate $J,$ and $(c)$ estimate the average time for a molecule to diffuse in. Assume the diffusion constant is $1 \times 10^{-5} \mathrm{m}^{2} / \mathrm{s} .$
A precise steel tape measure has been calibrated at $20^{\circ} \mathrm{C}$ . At $34^{\circ} \mathrm{C},(a)$ will it read high or low, and $(b)$ what will be the percentage error?
A Pyrex measuring cup was calibrated at normal room temperature. How much error will be made in a recipe calling for 300 $\mathrm{mL}$ of cool water, if the water and the cup are hot, at $80^{\circ} \mathrm{C},$ instead of at $20^{\circ} \mathrm{C} ?$ Neglect the glass expansion.
The gauge pressure in a helium gas cylinder is initially 28 atm. After many balloons have been blown up, the gauge pressure has decreased to 5 $\mathrm{atm}$ . What fraction of the original gas remains in the cylinder?
Estimate the number of air molecules in a room of length $6.5 \mathrm{m},$ width 3.1 $\mathrm{m}$ , and height 2.5 $\mathrm{m}$ . Assume the temperature is $22^{\circ} \mathrm{C}$ . How many moles does that correspond to?
In outer space the density of matter is about one atom per $\mathrm{cm}^{3},$ mainly hydrogen atoms, and the temperature is about 2.7 $\mathrm{K}$ . Calculate the rms speed of these hydrogen atoms, and the pressure (in atmospheres).
The lowest pressure attainable using the best available vacuum techniques is about $10^{-12} \mathrm{N} / \mathrm{m}^{2} .$ At such a pressure, how many molecules are there per $\mathrm{cm}^{3}$ at $0^{\circ} \mathrm{C} ?$
If a scuba diver fills his lungs to full capacity of 5.5 $\mathrm{L}$ when 10 $\mathrm{m}$ below the surface, to what volume would his lungs expand if he quickly rose to the surface? Is this advisable?
A space vehicle returning from the Moon enters Earth's atmosphere at a speed of about $40,000 \mathrm{km} / \mathrm{h}$ . 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.)
The temperature of an ideal gas is increased from $110^{\circ} \mathrm{C}$ to $360^{\circ} \mathrm{C}$ while the volume and the number of moles stay constant. By what factor does the pressure change? By what factor does $v_{\mathrm{rms}}$ change?
A house has a volume of 770 $\mathrm{m}^{3}$ . (a) What is the total mass of air inside the house at $20^{\circ} \mathrm{C}$ ? $(b)$ If the temperature drops to $-10^{\circ} \mathrm{C},$ what mass of air enters or leaves the house?
From the known value of atmospheric pressure at the surface of the Earth, estimate the total number of air molecules in the Earth's atmosphere.
What is the rms speed of nitrogen molecules contained in a $7.6-m^{3}$ volume at 4.2 atm if the total amount of nitrogen is 1800 mol?
A standard cylinder of oxygen used in a hospital has gauge pressure $=2000$ psi $(13,800 \mathrm{kPa})$ and volume $=$ 16 $\mathrm{L}\left(0.016 \mathrm{m}^{3}\right)$ at $T=295 \mathrm{K}$ . How long will the cylinder last if the flow rate, measured at atmospheric pressure, is constant at 2.4 $\mathrm{L} / \mathrm{min} ?$
An iron cube floats in a bowl of liquid mercury at $0^{\circ} \mathrm{C}$ $(a)$ If the temperature is raised to $25^{\circ} \mathrm{C},$ will the cube float higher or lower in the mercury? (b) By what percent will the fraction of volume submerged change?
The density of gasoline at $0^{\circ} \mathrm{C}$ is $0.68 \times 10^{3} \mathrm{kg} / \mathrm{m}^{3} .$ What is the density on a hot day, when the temperature is $38^{\circ} \mathrm{C} ?$ What is the percentage change?
If a steel band were to fit snugly around the Earth's equator at $25^{\circ} \mathrm{C},$ but then was heated to $45^{\circ} \mathrm{C}$ , how high above the Earth would the band be (assume equal everywhere)?
A brass lid screws tightly onto a glass jar at $20^{\circ} \mathrm{C}$ . 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 $60^{\circ} \mathrm{C}$ The inside diameter of the lid is 8.0 $\mathrm{cm}$ at $20^{\circ} \mathrm{C}$ . Find the size of the gap (difference in radius) that develops by this procedure.
The first length standard, adopted in the $18^{\text { th }}$ century, was a platinum bar with two very fine marks separated by what was defined to be exactly 1 $\mathrm{m}$ . If this standard bar was to be accurate to within $\pm 1.0 \mu \mathrm{m},$ how carefully would the trustees have needed to control the temperature? The coefficient of linear expansion for platinum is $9 \times 10^{-6} \mathrm{C}^{-1} .$
A scuba tank, when fully charged, has a pressure of 195 atm at $20^{\circ} \mathrm{C}$ . The volume of the tank is 11.3 $\mathrm{L}$ . (a) What would the volume of the air be at 1.00 atm and at the same temperature? (b) Before entering the water, and a person consumes 2.0 $\mathrm{L}$ 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 20.0 $\mathrm{m}$ of sea water and temperature of $10^{\circ} \mathrm{C}$ , how long would the same tank last assuming the breathing rate does not change?
The escape speed from the Earth is $1.12 \times 10^{4} \mathrm{m} / \mathrm{s},$ so a gas molecule travelling 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. At what temperature is the average speed of $(a)$ oxygen molecules, and $(b)$ helium atoms equal to $1.12 \times 10^{4} \mathrm{m} / \mathrm{s} ?(c)$ Can you see why our atmosphere contains oxygen but not helium?
A 1.0 -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 $\mathrm{kg}$ and are thrown at 12 $\mathrm{m} / \mathrm{s} ?$
A scuba diver releases a 3.00 -cm-diameter (spherical) bubble of air from a depth of 14.0 $\mathrm{m}$ in a lake. Assume the temperature is constant at 298 $\mathrm{K}$ , and the air behaves as a perfect gas. How large is the bubble when it reaches the surface?
Calculate the total water vapor pressure in the air on the following two days: $(a)$ a hot summer day, with the temperature $30^{\circ} \mathrm{C}$ and the relative humidity at $40 \% ;$ (b) a cold winter day, with the temperature $5^{\circ} \mathrm{C}$ and the relative humidity at $80 \% .$
A sauna has 7.0 $\mathrm{m}^{3}$ of air volume, and the temperature is $90^{\circ} \mathrm{C}$ . The air is perfectly dry. How much water (in $\mathrm{kg} )$ should be evaporated if we want to increase the relative humidity from 0$\%$ to 10$\% ?$ (See Table $13-3 . )$
Estimate the percent difference in the density of iron at $\mathrm{STP},$ and when it is a solid deep in the Earth where the temperature is $2000^{\circ} \mathrm{C}$ and under 5000 $\mathrm{atm}$ of pressure. Assume the bulk modulus $\left(90 \times 10^{9} \mathrm{N} / \mathrm{m}^{2}\right)$ and the coefficient of volume expansion do not vary with temperature and are the same as at STP.
(a) Use the ideal gas law to show that, for an ideal gas at constant pressure, the coefficient of volume expansion is equal to $\beta=1 / T,$ where $T$ is the temperature in kelvins. Compare to Table $13-1$ for gases at $T=293 \mathrm{K}$ . $(b)$ Show that the bulk modulus (Section $9-5$ ) for an ideal gas held atconstant temperature is $B=P,$ where $P$ is the pressure.
In humid climates, people constantly dehumidify their cellars to prevent rot and mildew. If the cellar in a house (kept at $20^{\circ} \mathrm{C} )$ has 95 $\mathrm{m}^{2}$ of floor space and a ceiling height of $2.8 \mathrm{m},$ what is the mass of water that must be removed from it to drop the humidity from 95$\%$ to a more reasonable 30$\% ?$