Chapter 3: Thermodynamics

.. DATA The Dew Point and Clouds. The vapour pressure of water (see Exercise 18.44) decreases as the temperature decreases. The table lists the vapour pressure of water at various temperatures:

If the amount of water vapour in the air is kept constant as the air is cooled, the dew pointtemperature is reached, at which the

partial pressure and vapour pressure coincide and the vapour is saturated. If the air is cooled further, the vapour condenses to liquid until the partial pressure again equals the vapour pressure at that temperature. The temperature in a room is . (a) A meteorologist cools a metal can by gradually adding cold water. When the can’s temperature reacheswater droplets form on its outside surface. What is the relative humidity of the air in the room? On a spring day in the midwestern United States, the air temperature at the surface is . Puffy cumulus clouds form at an altitude where the air temperature equals the dew point. If the air temperature decreases with altitude at a rate of , at approximately what height above the ground will clouds form if the relative humidity at the surface is (b) (c) ?

A typical doughnut contains 2.0 g of protein, 17.0 g of carbohydrates, and7.0 g of fat. Average food energy values are 4.0 kcal/g for protein andcarbohydrates and 9.0 kcal/g for fat. (a) During heavy exercise, anaverage person uses energy at a rate of 510 kcal/h. How long would you have to exercise to “work off” one doughnut? (b) If the energy in the doughnut could somehow be converted into the kinetic energy of your body as a whole, how fast could you move after eating the doughnut? Take your mass to be 60 kg, and express your answer in m/s and in km/h.

The statistical quantities “average value” and “root-mean-square value” can be applied to any distribution. Figure P18.82 shows the scores of a class of 150 students on a 100 point quiz. (a) Find the average score for the class. (b) Find the rms score for the class. (c) Which is higher: the average score or the rms score? Why?

... CP Dark Nebulae and the Interstellar Medium.The dark area in Fig. P18.83 that appears devoid of stars is a dark nebula,a cold gas cloud in interstellar space that contains enough material to block out light from the stars behind it. A typical dark nebula is about light-years in diameter and contains about hydrogen atoms per cubic centimetre (monatomic hydrogen, nott) at about . (A light-year is the distance light travels in vacuum in one year and is equal to.) (a) Estimate the mean free path for a hydrogen atom in a dark nebula. The Radius of a hydrogen atom is . (b) Estimate the rms speed of a hydrogen atom and the mean free time (the average time between collisions for a given atom). Based on this result, do you think that atomic collisions, such as those leading to molecule formation, are very important in determining the composition of the nebula? (c) Estimate the pressure inside a dark nebula. (d) Compare the rms speed of a hydrogen atom to the escape speed at the surface of the nebula (assumed spherical). If the space around the nebula were a vacuum, would such a cloud be stable or would it tend to evaporate? (e) The stability of dark nebulae is explained by the presence of the interstellar medium(ISM), an even thinner gas that permeates space and in which the dark nebulae are embedded. Show that for dark nebulae to be in equilibrium with the ISM, the numbers of atoms per volume and the emperatures of dark nebulae and the ISM must be related by nebula

f) In the vicinity of the sun, the ISM contains about 1 hydrogen atom per . Estimate the temperature of the ISM in the vicinity of the sun. Compare to the temperature of the sun’s surface, about . Would a spacecraft coasting through interstellar space burn up? Why or why not?

Shivering is your body’s way of generating heatto restore its internal temperature to the normal $\mathbf{37}\mathbf{°}\mathbf{C}$ and it produces approximately 290 W of heat power per square meter of body area. A 68-kg, 1.78-m-tall woman has approximately1.8 m²of surface area. How long would this woman have to shiverto raise her body temperature by$\mathbf{1}\mathbf{.}\mathbf{0}\mathbf{°}\mathbf{C}$, assuming that the body loses none of this heat? The body’s specific heat capacity is about$\mathbf{3500}\mathbf{}\mathbf{J}\mathbf{/}\mathbf{kg}\mathbf{·}\mathbf{K}$

... CALC Earth’s Atmosphere. In the troposphere,the part of the atmosphere that extends from earth’s surface to an altitude of about The temperature is not uniform but decreases with increasing elevation. (a) Show that if the temperature variation is approximated by the linear relationship where is thetemperature at the earth’s surface and Tis the temperature at height y, the pressure pat height yis ln a

where is the pressure at the earth’s surface and Mis the molar mass for air. The coefficient a is called the lapse rate of temperature. It varies with atmospheric conditions, but an average value is about . (b) Show that the above result reduces to the result of Example 18.4 (Section 18.1) in the limit that (c) With calculate pfor and compare your answer to the result of Example 18.4. Take and .

You cool a 100.0-g slug of red-hot iron (temperature ${\mathbf{745}}^{\mathbf{o}}\mathbf{}\mathit{C}$) by dropping it into an insulated cup of negligible mass containing 85.0 g of water at $\mathbf{20}\mathbf{.}{\mathbf{0}}^{\mathbf{o}}\mathbf{}\mathit{C}$. Assuming no heat exchange with the surroundings, (a) what is the final temperature of the water and (b) what is the final mass of the iron and the remaining water?

What is one reason the noble gases are preferable to air (which is mostly nitrogen and oxygen) as an insulating material?

(a) Noble gases are monatomic, so no rotational modes contribute to their molar heat capacity

b) noble gases are monatomic, so they have lower molecular masses than do nitrogen and oxygen

(c) molecular radii in noble gases are much larger than those of gases that consist of diatomic molecules

(d) because noble gases are monatomic, they have many more degrees of freedom than do diatomic molecules, and their molar heat capacity is reduced by the number of degrees of freedom.

A person of mass 70.0 kg is sitting in the bathtub. The bathtub is 190.0 cm by 80.0 cm; before the person got in, the water was 24.0 cm deep. The water is at$\mathbf{37}\mathbf{.}{\mathbf{0}}^{\mathbf{°}}\mathit{C}$. Suppose that the water were to cool down spontaneously to form ice at$\mathbf{0}\mathbf{.}{\mathbf{0}}^{\mathbf{°}}\mathit{C}\mathbf{,}$, and that all the energy released was used to launch the hapless bather vertically into the air. How high would the bather go? (As you will see in Chapter 20, this event is allowed by energy conservation but is prohibited by the second law of thermodynamics.)

Estimate the ratio of the thermal conductivity of Xe to that of He.

(a) 0.015; (b) 0.061; (c) 0.10; (d) 0.17.