Chapter 1

The Fahrenheit temperature scale is defined so that ice melts at\(32^{\circ} \mathrm{F}\) and water boils at \(212^{\circ} \mathrm{F}\) (a) Derive the formulas for converting from Fahrenheit to Celsius and back. (b) What is absolute zero on the Fahrenheit scale?

The Rankine temperature scale (abbreviated \(^{\circ} \mathrm{R}\) ) uses the same size degrees as Fahrenheit, but measured up from absolute zero like kelvin (so Rankine is to Fahrenheit as kelvin is to Celsius). Find the conversion formula between Rankine and Fahrenheit, and also between Rankine and kelvin. What is room temperature on the Rankine scale?

Does it ever make sense to say that one object is "twice as hot" as another? Does it matter whether one is referring to Celsius or kelvin temperatures? Explain.

Give an example to illustrate why you cannot accurately judge the temperature of an object by how hot or cold it feels to the touch.

Rooms \(A\) and \(B\) are the same size, and are connected by an open door. Room \(A\), however, is warmer (perhaps because its windows face the sun). Which room contains the greater mass of air? Explain carefully.

Calculate the mass of a mole of dry air, which is a mixture of \(\mathrm{N}_{2}\) \((78 \% \text { by volume }), \mathrm{O}_{2}(21 \%),\) and argon \((1 \%)\).

Even at low density, real gases don't quite obey the ideal gas law. A systematic way to account for deviations from ideal behavior is the virial expansion, $$P V=n R T\left(1+\frac{B(T)}{(V / n)}+\frac{C(T)}{(V / n)^{2}}+\cdots\right)$$ where the functions \(B(T), C(T),\) and so on are called the virial coefficients. When the density of the gas is fairly low, so that the volume per mole is large, each term in the series is much smaller than the one before. In many situations it's sufficient to omit the third term and concentrate on the second, whose coefficient \(B(T)\) is called the second virial coefficient (the first coefficient being 1 ). Here are some measured values of the second virial coefficient for nitrogen \(\left(\mathrm{N}_{2}\right):\) $$\begin{array}{cc} T(\mathrm{K}) & B\left(\mathrm{cm}^{3} / \mathrm{mol}\right) \\ \hline 100 & -160 \\ 200 & -35 \\ 300 & -4.2 \\ 400 & 9.0 \\ 500 & 16.9 \\ 600 & 21.3 \end{array}$$ (a) For each temperature in the table, compute the second term in the virial equation, \(B(T) /(V / n),\) for nitrogen at atmospheric pressure. Discuss the validity of the ideal gas law under these conditions. (b) Think about the forces between molecules, and explain why we might expect \(B(T)\) to be negative at low temperatures but positive at high temperatures. (c) Any proposed relation between \(P, V,\) and \(T,\) like the ideal gas law or the virial equation, is called an equation of state. Another famous equation of state, which is qualitatively accurate even for dense fluids, is the van der Waals equation, $$\left(P+\frac{a n^{2}}{V^{2}}\right)(V-n b)=n R T$$ where \(a\) and \(b\) are constants that depend on the type of gas. Calculate the second and third virial coefficients \((B \text { and } C)\) for a gas obeying the van der Waals equation, in terms of \(a\) and \(b\). (Hint: The binomial expansion says that \((1+x)^{p} \approx 1+p x+\frac{1}{2} p(p-1) x^{2},\) provided that \(|p x| \ll 1 .\) Apply this approximation to the quantity \([1-(n b / V)]^{-1}\).) (d) Plot a graph of the van der Waals prediction for \(B(T),\) choosing \(a\) and \(b\) so as to approximately match the data given above for nitrogen. Discuss the accuracy of the van der Waals equation over this range of conditions. (The van der Waals equation is discussed much further in Section 5.3.)

Suppose you have a gas containing hydrogen molecules and oxygen molecules, in thermal equilibrium. Which molecules are moving faster, on average? By what factor?

Uranium has two common isotopes, with atomic masses of 238 and \(235 .\) One way to separate these isotopes is to combine the uranium with fluorine to make uranium hexafluoride gas, UF \(_{6}\), then exploit the difference in the average thermal speeds of molecules containing the different isotopes. Calculate the rms speed of each type of molecule at room temperature, and compare them.

During a hailstorm, hailstones with an average mass of 2 g and a speed of \(15 \mathrm{m} / \mathrm{s}\) strike a window pane at a \(45^{\circ}\) angle. The area of the window is \(0.5 \mathrm{m}^{2}\) and the hailstones hit it at a rate of 30 per second. What average pressure do they exert on the window? How does this compare to the pressure of the atmosphere?

If you poke a hole in a container full of gas, the gas will start leaking out. In this problem you will make a rough estimate of the rate at which gas escapes through a hole. (This process is called effusion, at least when the hole is sufficiently small.) (a) Consider a small portion (area \(=A\) ) of the inside wall of a container full of gas. Show that the number of molecules colliding with this surface in a time interval \(\Delta t\) is \(P A \Delta t /(2 m \overline{v_{x}}),\) where \(P\) is the pressure, \(m\) is the average molecular mass, and \(\overline{v_{x}}\) is the average \(x\) velocity of those molecules that collide with the wall. (b) It's not easy to calculate \(\overline{v_{x}},\) but a good enough approximation is \((\overline{v_{x}^{2}})^{1 / 2}\) where the bar now represents an average over all molecules in the gas. Show that \((\overline{v_{x}^{2}})^{1 / 2}=\sqrt{k T / m}\) (c) If we now take away this small part of the wall of the container, the molecules that would have collided with it will instead escape through the hole. Assuming that nothing enters through the hole, show that the number \(N\) of molecules inside the container as a function of time is governed by the differential equation $$\frac{d N}{d t}=-\frac{A}{2 V} \sqrt{\frac{k T}{m}} N$$ Solve this equation (assuming constant temperature) to obtain a formula of the form \(N(t)=N(0) e^{-t / \tau},\) where \(\tau\) is the "characteristic time" for \(N\) \((\text { and } P)\) to drop by a factor of \(e\) (d) Calculate the characteristic time for a gas to escape from a 1 -liter container punctured by a \(1-\mathrm{mm}^{2}\) hole. (e) Your bicycle tire has a slow leak, so that it goes flat within about an hour after being inflated. Roughly how big is the hole? (Use any reasonable estimate for the volume of the tire.) (f) In Jules Verne's Round the Moon, the space travelers dispose of a dog's corpse by quickly opening a window, tossing it out, and closing the window. Do you think they can do this quickly enough to prevent a significant amount of air from escaping? Justify your answer with some rough estimates and calculations.

Calculate the total thermal energy in a gram of lead at room temperature, assuming that none of the degrees of freedom are "frozen out" (this happens to be a good assumption in this case).

List all the degrees of freedom, or as many as you can, for a molecule of water vapor. (Think carefully about the various ways in which the molecule can vibrate.)

A battery is connected in series to a resistor, which is immersed in water (to prepare a nice hot cup of tea). Would you classify the flow of energy from the battery to the resistor as "heat" or "work"? What about the flow of energy from the resistor to the water?

Give an example of a process in which no heat is added to a system, but its temperature increases. Then give an example of the opposite: a process in which heat is added to a system but its temperature does not change.

Estimate how long it should take to bring a cup of water to boiling temperature in a typical 600 -watt microwave oven, assuming that all the energy ends up in the water. (Assume any reasonable initial temperature for the water.) Explain why no heat is involved in this process.

A cup containing 200 g of water is sitting on your dining room table. After carefully measuring its temperature to be \(20^{\circ} \mathrm{C}\), you leave the room. Returning ten minutes later, you measure its temperature again and find that it is now \(25^{\circ} \mathrm{C}\). What can you conclude about the amount of heat added to the water? (Hint: This is a trick question.)

Imagine some helium in a cylinder with an initial volume of 1 liter and an initial pressure of 1 atm. Somehow the helium is made to expand to a final volume of 3 liters, in such a way that its pressure rises in direct proportion to its volume.

In the course of pumping up a bicycle tire, a liter of air at atmospheric pressure is compressed adiabatically to a pressure of 7 atm. (Air is mostly diatomic nitrogen and oxygen.) (a) What is the final volume of this air after compression? (b) How much work is done in compressing the air? (c) If the temperature of the air is initially \(300 \mathrm{K},\) what is the temperature after compression?

Two identical bubbles of gas form at the bottom of a lake, then rise to the surface. Because the pressure is much lower at the surface than at the bottom, both bubbles expand as they rise. However, bubble \(A\) rises very quickly, so that no heat is exchanged between it and the water. Meanwhile, bubble \(B\) rises slowly (impeded by a tangle of seaweed), so that it always remains in thermal equilibrium with the water (which has the same temperature everywhere). Which of the two bubbles is larger by the time they reach the surface? Explain your reasoning fully.

By applying Newton's laws to the oscillations of a continuous medium, one can show that the speed of a sound wave is given by $$c_{s}=\sqrt{\frac{B}{\rho}}$$ where \(\rho\) is the density of the medium (mass per unit volume) and \(B\) is the bulk modulus, a measure of the medium's stiffness. More precisely, if we imagine applying an increase in pressure \(\Delta P\) to a chunk of the material, and this increase results in a (negative) change in volume \(\Delta V\), then \(B\) is defined as the change in pressure divided by the magnitude of the fractional change in volume: $$B \equiv \frac{\Delta P}{-\Delta V / V}$$ This definition is still ambiguous, however, because I haven't said whether the compression is to take place isothermally or adiabatically (or in some other way). (a) Compute the bulk modulus of an ideal gas, in terms of its pressure \(P,\) for both isothermal and adiabatic compressions. (b) Argue that for purposes of computing the speed of a sound wave, the adiabatic \(B\) is the one we should use. (c) Derive an expression for the speed of sound in an ideal gas, in terms of its temperature and average molecular mass. Compare your result to the formula for the rms speed of the molecules in the gas. Evaluate the speed of sound numerically for air at room temperature. (d) When Scotland's Battlefield Band played in Utah, one musician remarked that the high altitude threw their bagpipes out of tune. Would you expect altitude to affect the speed of sound (and hence the frequencies of the standing waves in the pipes)? If so, in which direction? If not, why not?

To measure the heat capacity of an object, all you usually have to do is put it in thermal contact with another object whose heat capacity you know. As an example, suppose that a chunk of metal is immersed in boiling water \(\left(100^{\circ} \mathrm{C}\right),\) then is quickly transferred into a Styrofoam cup containing \(250 \mathrm{g}\) of water at \(20^{\circ} \mathrm{C}\). After a minute or so, the temperature of the contents of the cup is \(24^{\circ} \mathrm{C} .\) Assume that during this time no significant energy is transferred between the contents of the cup and the surroundings. The heat capacity of the cup itself is negligible. (a) How much heat is lost by the water? (b) How much heat is gained by the metal? (c) What is the heat capacity of this chunk of metal? (d) If the mass of the chunk of metal is \(100 \mathrm{g}\), what is its specific heat capacity?

Problem 1.45. As an illustration of why it matters which variables you hold fixed when taking partial derivatives, consider the following mathematical example. Let \(w=x y\) and \(x=y z\) (a) Write \(w\) purely in terms of \(x\) and \(z,\) and then purely in terms of \(y\) and \(z\) (b) Compute the partial derivatives $$ \left(\frac{\partial w}{\partial x}\right)_{y} \quad \text { and } \quad\left(\frac{\partial w}{\partial x}\right)_{z} $$ and show that they are not equal. (Hint: To compute \((\partial w / \partial x)_{y},\) use a formula for \(w\) in terms of \(x\) and \(y,\) not \(z .\) Similarly, compute \((\partial w / \partial x)_{z}\) from a formula for \(w\) in terms of only \(x\) and \(z .\) ) (c) Compute the other four partial derivatives of \(w\) (two each with respect to \(y \text { and } z),\) and show that it matters which variable is held fixed.

When spring finally arrives in the mountains, the snow pack may be two meters deep, composed of \(50 \%\) ice and \(50 \%\) air. Direct sunlight provides about 1000 watts/m \(^{2}\) to earth's surface, but the snow might reflect \(90 \%\) of this energy. Estimate how many weeks the snow pack should last, if direct solar radiation is the only source of energy.

A 60-kg hiker wishes to climb to the summit of Mt. Ogden, an ascent of 5000 vertical feet \((1500 \mathrm{m})\) (a) Assuming that she is \(25 \%\) efficient at converting chemical energy from food into mechanical work, and that essentially all the mechanical work is used to climb vertically, roughly how many bowls of corn flakes (standard serving size 1 ounce, 100 kilocalories) should the hiker eat before setting out? (b) As the hiker climbs the mountain, three-quarters of the energy from the corn flakes is converted to thermal energy. If there were no way to dissipate this energy, by how many degrees would her body temperature increase? (c) In fact, the extra energy does not warm the hiker's body significantly; instead, it goes (mostly) into evaporating water from her skin. How many liters of water should she drink during the hike to replace the lost fluids? (At \(25^{\circ} \mathrm{C},\) a reasonable temperature to assume, the latent heat of vaporization of water is \(580 \mathrm{cal} / \mathrm{g}, 8 \%\) more than at \(100^{\circ} \mathrm{C} .\) )

Heat capacities are normally positive, but there is an important class of exceptions: systems of particles held together by gravity, such as stars and star clusters. (a) Consider a system of just two particles, with identical masses, orbiting in circles about their center of mass. Show that the gravitational potential energy of this system is -2 times the total kinetic energy. (b) The conclusion of part (a) turns out to be true, at least on average, for any system of particles held together by mutual gravitational attraction: $$ \bar{U}_{\text {potential }}=-2 \bar{U}_{\text {kinetic }} $$ Here each \(\bar{U}\) refers to the total energy (of that type) for the entire system, averaged over some sufficiently long time period. This result is known as the virial theorem. (For a proof, see Carroll and Ostlie (1996), Section 2.4.) Suppose, then, that you add some energy to such a system and then wait for the system to equilibrate. Does the average total kinetic energy increase or decrease? Explain. (c) A star can be modeled as a gas of particles that interact with each other only gravitationally. According to the equipartition theorem, the average kinetic energy of the particles in such a star should be \(\frac{3}{2} k T,\) where \(T\) is the average temperature. Express the total energy of a star in terms of its average temperature, and calculate the heat capacity. Note the sign. (d) Use dimensional analysis to argue that a star of mass \(M\) and radius \(R\) should have a total potential energy of \(-G M^{2} / R,\) times some constant of order 1. (e) Estimate the average temperature of the sun, whose mass is \(2 \times 10^{30} \mathrm{kg}\) and whose radius is \(7 \times 10^{8} \mathrm{m}\). Assume, for simplicity, that the sun is made entirely of protons and electrons.

According to a standard reference table, the \(R\) value of a 3.5 inch-thick vertical air space (within a wall) is 1.0 (in English units), while the \(R\) value of a 3.5 -inch thickness of fiberglass batting is \(10.9 .\) Calculate the \(R\) value of a 3.5 -inch thickness of still air, then discuss whether these two numbers are reasonable. (Hint: These reference values include the effects of convection.)

Problem 1.61. Geologists measure conductive heat flow out of the earth by drilling holes (a few hundred meters deep) and measuring the temperature as a function of depth. Suppose that in a certain location the temperature increases by \(20^{\circ} \mathrm{C}\) per kilometer of depth and the thermal conductivity of the rock is \(2.5 \mathrm{W} / \mathrm{m} \cdot \mathrm{K}\) What is the rate of heat conduction per square meter in this location? Assuming that this value is typical of other locations over all of earth's surface, at approximately what rate is the earth losing heat via conduction? (The radius of the earth is \(6400 \mathrm{km} .\) )

Consider a uniform rod of material whose temperature varies only along its length, in the \(x\) direction. By considering the heat flowing from both directions into a small segment of length \(\Delta x,\) derive the heat equation, $$ \frac{\partial T}{\partial t}=K \frac{\partial^{2} T}{\partial x^{2}} $$ where \(K=k_{t} / c \rho, c\) is the specific heat of the material, and \(\rho\) is its density. (Assume that the only motion of energy is heat conduction within the rod; no energy enters or leaves along the sides.) Assuming that \(K\) is independent of temperature, show that a solution of the heat equation is $$ T(x, t)=T_{0}+\frac{A}{\sqrt{t}} e^{-x^{2} / 4 K t} $$ where \(T_{0}\) is a constant background temperature and \(A\) is any constant. Sketch (or use a computer to plot) this solution as a function of \(x,\) for several values of \(t\) Interpret this solution physically, and discuss in some detail how energy spreads through the rod as time passes.