1. The Fahrenheit temperature scale is defined so that ice melts at
32°F and water boils at 212°F.(a) Derive the formulas for converting
from Fahrenheit to Celsius and back.(b) What is absolute zero on the
Fahrenheit scale? Get solution
2. The Rankine temperature scale (abbreviated °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? Get solution
3. Determine the kelvin temperature for each of the following:(a) human body temperature;(b) the boiling point of water (at the standard pressure of 1 atm);(c) the coldest day you can remember;(d) the boiling point of liquid nitrogen (−196°C);(e) the melting point of lead (327°C). Get solution
4. 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. Get solution
5. When you’re sick with a fever and you take your temperature with a thermometer, approximately what is the relaxation time? Get solution
6. 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. Get solution
7. When the temperature of liquid mercury increases by one degree Celsius (or one kelvin), its volume increases by one part in 5500. The fractional increase in volume per unit change in temperature (when the pressure is held fixed) is called the thermal expansion coefficient, β:...(where V is volume, T is temperature, and Δ signifies a change, which in this case should really be infinitesimal if β is to be well defined). So for mercury, β = 1/5500 K−1 = 1.81 × 10−4 K−1. (The exact value varies with temperature, but between 0°C and 200°C the variation is less than 1%.)(a) Get a mercury thermometer, estimate the size of the bulb at the bottom, and then estimate what the inside diameter of the tube has to be in order for the thermometer to work as required. Assume that the thermal expansion of the glass is negligible.(b) The thermal expansion coefficient of water varies significantly with temperature: It is 7.5 × 10−4 K−1 at 100°C, but decreases as the temperature is lowered until it becomes zero at 4°C. Below 4°C it is slightly negative, reaching a value of − 0.68 ×10−4 K−1 at 0°C. (This behavior is related to the fact that ice is less dense than water.) With this behavior in mind, imagine the process of a lake freezing over, and discuss in some detail how this process would be different if the thermal expansion coefficient of water were always positive. Get solution
8. For a solid, we also define the linear thermal expansion coefficient, α, as the fractional increase in length per degree:...(a) For steel, α is 1.1 × 10−5 K−1. Estimate the total variation in length of a 1-km steel bridge between a cold winter night and a hot summer day.(b) The dial thermometer in Figure uses a coiled metal strip made of two different metals laminated together. Explain how this works.(c) Prove that the volume thermal expansion coefficient of a solid is equal to the sum of its linear expansion coefficients in the three directions: β = αx + αy + αz. (So for an isotropic solid, which expands the same in all directions, β = 3α.)Figure: A selection of thermometers. In the center are two liquid-in-glass thermometers, which measure the expansion of mercury (for higher temperatures) and alcohol (for lower temperatures). The dial thermometer to the right measures the turning of a coil of metal, while the bulb apparatus behind it measures the pressure of a fixed volume of gas. The digital thermometer at left-rear uses a thermocouple—a junction of two metals—which generates a small temperature dependent voltage. At left-front is a set of three potter’s cones, which melt and droop at specified clay-firing temperatures.... Get solution
10. Estimate the number of air molecules in an average-sized room. Get solution
11. 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. Get solution
12. Calculate the average volume per molecule for an ideal gas at room temperature and atmospheric pressure. Then take the cube root to get an estimate of the average distance between molecules. How does this distance compare to the size of a small molecule like N2 or H2O? Get solution
13. A mole is approximately the number of protons in a gram of protons. The mass of a neutron is about the same as the mass of a proton, while the mass of an electron is usually negligible in comparison, so if you know the total number of protons and neutrons in a molecule (i.e., its “atomic mass”), you know the approximate mass (in grams) of a mole of these molecules. Referring to the periodic table at the back of this book, find the mass of a mole of each of the following: water, nitrogen (N2), lead, quartz (SiO2). Get solution
14. Calculate the mass of a mole of dry air, which is a mixture of N2 (78% by volume), O2 (21%), and argon (1%). Get solution
15. Estimate the average temperature of the air inside a hot-air balloon (see Figure 1.1). Assume that the total mass of the unfilled balloon and payload is 500 kg. What is the mass of the air inside the balloon? ... Get solution
16. The exponential atmosphere. (a) Consider a horizontal slab of air whose thickness (height) is dz. If this slab is at rest, the pressure holding it up from below must balance both the pressure from above and the weight of the slab. Use this fact to find an expression for dP/dz, the variation of pressure with altitude, in terms of the density of air.(b) Use the ideal gas law to write the density of air in terms of pressure, temperature, and the average mass m of the air molecules. (The information needed to calculate m is given in Problem.) Show, then, that the pressure obeys the differential equation...called the barometric equation.(c) Assuming that the temperature of the atmosphere is independent of height (not a great assumption but not terrible either), solve the barometric equation to obtain the pressure as a function of height: P(z) = P(0)e−mgz/kT. Show also that the density obeys a similar equation.(d) Estimate the pressure, in atmospheres, at the following locations: Ogden, Utah (4700 ft or 1430 m above sea level); Leadville, Colorado (10,150 ft, 3090 m) ; Mt. Whitney, California (14,500 ft, 4420 m); Mt. Everest, Nepal/Tibet (29,000 ft, 8850 m). (Assume that the pressure at sea level is 1 atm.)Problem: Calculate the mass of a mole of dry air, which is a mixture of N2 (78% by volume), O2 (21%), and argon (1%). Get solution
17. 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,...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 (N2): T (K)B (cm3 / mol)100−160200 −35300 −4.2400 9.0500 16.9600 21.3(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,...where a and b are constants that depend on the type of gas. Calculate the second and third virial coefficients (B and C) for a gas obeying the van der Waals equation, in terms of a and b. (Hint: The binomial expansion says that ... provided that |px| ≪ 1. Apply this approximation to the quantity [1 − (nb/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.) Get solution
18. Calculate the rms speed of a nitrogen molecule at room temperature. Get solution
19. Suppose you have a gas containing hydrogen molecules and oxygen molecules, in thermal equilibrium. Which molecules are moving faster, on average? By what factor? Get solution
20. 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, UF6, 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. Get solution
21. During a hailstorm, hailstones with an average mass of 2 g and a speed of 15 m/s strike a window pane at a 45° angle. The area of the window is 0.5 m2 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? Get solution
22. 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 Δt is ... where P is the pressure, m is the average molecular mass, and ... is the average x velocity of those molecules that collide with the wall.(b) It’s not easy to calculate ..., but a good enough approximation is...where the bar now represents an average over all molecules in the gas. Show that...(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...Solve this equation (assuming constant temperature) to obtain a formula of the form N(t) = N(0)e−t/τ, where τ is the “characteristic time” for N (and P) to drop by a factor of e.(d) Calculate the characteristic time for air at room temperature to escape from a 1-liter container punctured by a 1-mm2 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. Get solution
23. Calculate the total thermal energy in a liter of helium at room temperature and atmospheric pressure. Then repeat the calculation for a liter of air. Get solution
24. 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). Get solution
25. 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.) Get solution
26. 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? Get solution
27. 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. Get solution
28. 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. Get solution
29. A cup containing 200 g of water is sitting on your dining room table. After carefully measuring it s temperature to be 20°C, you leave the room. Returning ten minutes later, you measure its temperature again and find that it is now 25°C. What can you conclude about the amount of heat added to the water? (Hint: This is a trick question.) Get solution
30. Put a few spoonfuls of water into a bottle with a tight lid. Make sure everything is at room temperature, measuring the temperature of the water with a thermometer to make sure. Now close the bottle and shake it as hard as you can for several minutes. When you’re exhausted and ready to drop, shake it for several minutes more. Then measure the temperature again. Make a rough calculation of the expected temperature change, and compare. Get solution
31. 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.(a) Sketch a graph of pressure vs. volume for this process.(b) Calculate the work done on the gas during this process, assuming that there are no “other” types of work being done.(c) Calculate the change in the helium’s energy content during this process.(d) Calculate the amount of heat added to or removed from the helium during this process.(e) Describe what you might do to cause the pressure to rise as the helium expands. Get solution
32. By applying a pressure of 200 atm, you can compress water to 99% of its usual volume. Sketch this process (not necessarily to scale) on a PV diagram, and estimate the work required to compress a liter of water by this amount. Does the result surprise you? Get solution
33. An ideal gas is made to undergo the cyclic process shown in Figure. For each of the steps A, B, and C, determine whether each of the following is positive, negative, or zero: (a) the work done on the gas; (b) the change in the energy content of the gas; (c) the heat added to the gas. Then determine the sign of each of these three quantities for the whole cycle. What does this process accomplish?Figure: PV diagrams... Get solution
34. An ideal diatomic gas, in a cylinder with a movable piston, undergoes the rectangular cyclic process shown in Figure. Assume that the temperature is always such that rotational degrees of freedom are active, but vibrational modes are “frozen out.” Also assume that the only type of work done on the gas is quasistatic compression-expansion work.(a) For each of the four steps A through D, compute the work done on the gas, the heat added to the gas, and the change in the energy content of the gas. Express all answers in terms of P1, P2, V1 and V2. (Hint: Compute ΔU before Q, using the ideal gas law and the equipartition theorem.)(b) Describe in words what is physically being done during each of the four steps, for example, during step A, heat is added to the gas (from an external flame or something) while the piston is held fixed.(c) Compute the net work done on the gas, the net heat added to the gas, and the net change in the energy of the gas during the entire cycle. Are the results as you expected? Explain briefly.Figure: PV diagrams... Get solution
35. Derive equation 1 from equation 2.Equation 1:Vγ p = constantEquation 2:VTf/2 = constant. Get solution
36. 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 K, what is the temperature after compression? Get solution
37. In a Diesel engine, atmospheric air is quickly compressed to about 1/20 of its original volume. Estimate the temperature of the air after compression, and explain why a Diesel engine does not require spark plugs. Get solution
38. 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. Get solution
39. 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...where ρ 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 ΔP to a chunk of the material, and this increase results in a (negative) change in volume ΔV, then B is defined as the change in pressure divided by the magnitude of the fractional change in volume:...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? Get solution
40. In Problem you calculated the pressure of earth’s atmosphere as a function of altitude, assuming constant temperature. Ordinarily, however, the temperature of the bottommost 10–15 km of the atmosphere (called the troposphere) decreases with increasing altitude, due to heating from the ground (which is warmed by sunlight). If the temperature gradient |dT/dz| exceeds a certain critical value, convection will occur: Warm, low-density air will rise, while cool, high-density air sinks. The decrease of pressure with altitude causes a rising air mass to expand adiabatically and thus to cool. The condition for convection to occur is that the rising air mass must remain warmer than the surrounding air despite this adiabatic cooling.(a) Show that when an ideal gas expands adiabatically, the temperature and pressure are related by the differential equation...(b) Assume that dT/dz is just at the critical value for convection to begin, so that the vertical forces on a convecting air mass are always approximately in balance. Use the result of Problem 1 (b) to find a formula for dT/dz in this case. The result should be a constant, independent of temperature and pressure, which evaluates to approximately −10°C/km. This fundamental meteorological quantity is known as the dry adiabatic lapse rate.Problem 1:The exponential atmosphere. (a) Consider a horizontal slab of air whose thickness (height) is dz. If this slab is at rest, the pressure holding it up from below must balance both the pressure from above and the weight of the slab. Use this fact to find an expression for dP/dz, the variation of pressure with altitude, in terms of the density of air.(b) Use the ideal gas law to write the density of air in terms of pressure, temperature, and the average mass m of the air molecules. (The information needed to calculate m is given in Problem 2.) Show, then, that the pressure obeys the differential equation...called the barometric equation.(c) Assuming that the temperature of the atmosphere is independent of height (not a great assumption but not terrible either), solve the barometric equation to obtain the pressure as a function of height: P(z) = P(0)e−mgz/kT. Show also that the density obeys a similar equation.(d) Estimate the pressure, in atmospheres, at the following locations: Ogden, Utah (4700 ft or 1430 m above sea level); Leadville, Colorado (10,150 ft, 3090 m) ; Mt. Whitney, California (14,500 ft, 4420 m); Mt. Everest, Nepal/Tibet (29,000 ft, 8850 m). (Assume that the pressure at sea level is 1 atm.)Problem 2:Calculate the mass of a mole of dry air, which is a mixture of N2 (78% by volume), O2 (21%), and argon (1%). Get solution
41. 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 (100°C), then is quickly transferred into a Styrofoam cup containing 250 g of water at 20°C. After a minute or so, the temperature of the contents of the cup is 24°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 gained by the water?(b) How much heat is lost 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 g, what is its specific heat capacity? Get solution
42. The specific heat capacity of Albertson’s Rotini Tricolore is approximately 1.8 J /g∙°C. Suppose you toss 340 g of this pasta (at 25°C) into 1.5 liters of boiling water. What effect does this have on the temperature of the water (before there is time for the stove to provide more heat)? Get solution
43. Calculate the heat capacity of liquid water per molecule, in terms of k. Suppose (incorrectly) that all the thermal energy of water is stored in quadratic degrees of freedom. How many degrees of freedom would each molecule have to have? Get solution
44. At the back of this book is a tab1e of thermodynamic data for selected substances at room temperature. Browse through the CP values in this table, and check that you can account for most of them (approximately) using the equipartition theorem. Which values seem anomalous? Get solution
45. As an illustration of why it matters which variables you hold fixed when taking partial derivatives, consider the following mathematical example. Let w = xy and x = yz.(a) Write w purely in terms of x and z, and then purely in terms of y and z(b) Compute the partial derivatives...and show that they are not equal. (Hint: To compute... use a formula for w in terms of x and y, not z. Similarly, compute ... 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 and z), and show that it matters which variable is held fixed. Get solution
46. Measured heat capacities of solids and liquids are almost always at constant pressure, not constant volume. To see why, estimate the pressure needed to keep V fixed as T increases, as follows.(a) First imagine slightly increasing the temperature of a material at constant pressure. Write the change in volume, dV1, in terms of dT and the thermal expansion coefficient β introduced in Problem 1.(b) Now imagine slightly compressing the material, holding its temperature fixed. Write the change in volume for this process, dV2, in terms of dP and the isothermal compressibility κT defined as...(This is the reciprocal of the isothermal bulk modulus defined in Problem 2.)(c) Finally, imagine that you compress the material just enough in part (b) to offset the expansion in part (a). Then the ratio of dP to dT is equal to ... since there is no net change in volume. Express this partial derivative in terms of β and κT. Then express it more abstractly in terms of the partial derivatives used to define β and κT. For the second expression you should obtain...This result is actually a purely mathematical relation, true for any three quantities that are related in such a way that any two determine the third.(d) Compute β, κT and ... for an ideal gas, and check that the three expressions satisfy the identity you found in part (c).(e) For water at 25°C, β = 2.57 × 10−4 K−1 and κT = 4.52 × 10−10 Pa−1. Suppose you increase the temperature of some water from 20°C to 30°C. How much pressure must you apply to prevent it from expanding? Repeat the calculation for mercury, for which (at 25°C) β = 1.81 × 10−4 K−1 and κT = 4.04 × 10−11 Pa−l. Given the choice, would you rather measure the heat capacities of these substances at constant V or at constant P?Problem 1:When the temperature of liquid mercury increases by one degree Celsius (or one kelvin), its volume increases by one part in 5500. The fractional increase in volume per unit change in temperature (when the pressure is held fixed) is called the thermal expansion coefficient, β:...(where V is volume, T is temperature, and Δ signifies a change, which in this case should really be infinitesimal if β is to be well defined). So for mercury, β = 1/5500 K−1 = 1.81 × 10−4 K−1. (The exact value varies with temperature, but between 0°C and 200°C the variation is less than 1%.)(a) Get a mercury thermometer, estimate the size of the bulb at the bottom, and then estimate what the inside diameter of the tube has to be in order for the thermometer to work as required. Assume that the thermal expansion of the glass is negligible.(b) The thermal expansion coefficient of water varies significantly with temperature: It is 7.5 × 10−4 K−1 at 100°C, but decreases as the temperature is lowered until it becomes zero at 4°C. Below 4°C it is slightly negative, reaching a value of − 0.68 ×10−4 K−1 at 0°C. (This behavior is related to the fact that ice is less dense than water.) With this behavior in mind, imagine the process of a lake freezing over, and discuss in some detail how this process would be different if the thermal expansion coefficient of water were always positive.Problem 2: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...where ρ 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 ΔP to a chunk of the material, and this increase results in a (negative) change in volume ΔV, then B is defined as the change in pressure divided by the magnitude of the fractional change in volume:...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? Get solution
47. Your 200-g cup of tea is boiling-hot. About how much ice should you add to bring it down to a comfortable sipping temperature of 65°C? (Assume that the ice is initially at −15°C. The specific heat capacity of ice is 0.5 cal/g∙°C.) Get solution
48. 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/m2 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. Get solution
49. Consider the combustion of one mole of H2 with 1/2 mole of O2 under standard conditions, as discussed in the text. How much of the heat energy produced comes from a decrease in the internal energy of the system, and how much comes from work done by the collapsing atmosphere? (Treat the volume of the liquid water as negligible.) Get solution
50. Consider the combustion of one mole of methane gas:CH4(gas) + 2O2(gas) → CO2(gas) + 2H2O(gas).The system is at standard temperature (298 K) and pressure (105 Pa) both before and after the reaction.(a) First imagine the process of converting a mole of methane into its elemental consituents (graphite and hydrogen gas). Use the data at the back of this book to find ΔH for this process.(b) Now imagine forming a mole of CO2 and two moles of water vapor from their elemental constituents. Determine ΔH for this process.(c) What is ΔH for the actual reaction in which methane and oxygen form carbon dioxide and water vapor directly? Explain.(d) How much heat is given off during this reaction, assuming that no “other” forms of work are done?(e) What is the change in the system’s energy during this reaction? How would your answer differ if the H2O ended up as liquid water instead of vapor?(f) The sun has a mass of 2 × 1030 kg and gives off energy at a rate of. 3.9 × 1026 watts. If the source of the sun’s energy were ordinary combustion of a chemical fuel such as methane, about how long could it last? Get solution
51. Use the data at the back of this book to determine ΔH for the combustion of a mole of glucose,C6H12O6 + 6O2 → 6CO2 + 6H2O.This is the (net) reaction that provides most of the energy needs in our bodies. Get solution
52. The enthalpy of combustion of a gallon (3.8 liters) of gasoline is about 31,000 kcal. The enthalpy of combustion of an ounce (28 g) of corn flakes is about 100 kcal. Compare the cost of gasoline to the cost of corn flakes, per calorie. Get solution
53. Look up the enthalpy of formation of atomic hydrogen in the back of this book. This is the enthalpy change when a mole of atomic hydrogen is formed by dissociating 1/2 mole of molecular hydrogen (the more stable State of the element). From this number, determine the energy needed to dissociate a single H2 molecule, in electron-volts. Get solution
54. A 60-kg hiker wishes to climb to the summit of Mt. Ogden, an ascent of 5000 vertical feet (1500 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°C a reasonable temperature to assume, the latent heat of vaporization of water is 580 cal/g, 8% more than at 100°C) Get solution
55. 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:...Here each Ū 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 ...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 −GM2/R, times some constant of order 1.(e) Estimate the average temperature of the sun, whose mass is 2 × 1030 kg and whose radius is 7 × 108 m. Assume, for simplicity, that the sun is made entirely of protons and electrons. Get solution
56. Calculate the rate of heat conduction through a layer of still air that is 1 mm thick, with an area of 1 m2 , for a temperature difference of 20°C. Get solution
57. Home owners and builders discuss thermal conductivities in terms of the R value (R for resistance) of a material, defined as the thickness divided by the thermal conductivity:...(a) Calculate the R value of a 1/8-inch (3.2 mm) piece of plate glass, and then of a 1 mm layer of still air. Express both answers in SI units.(b) In the United States, R values of building materials are normally given in English units, °F∙ft2∙hr/Btu. A Btu, or British thermal unit, is the energy needed to raise the temperature of a pound of water by 1°F. Work out the conversion factor between the SI and English units for R values. Convert your answers from part (a) to English units.(c) Prove that for a compound layer of two different materials sandwiched together (such as air and glass, or brick and wood), the effective total R value is the sum of the individual R values.(d) Calculate the effective R value of a single piece of plate glass with a 1.0-mm layer of still air on each side. (The effective thickness of the air layer will depend on how much wind is blowing; 1 mm is of the right order of magnitude under mot conditions.) Using this effective R value, make a revised estimate of the heat loss through a 1-m2 single-pane window when the temperature in the room is 20°C higher than the outdoor temperature. Get solution
58. According to a standard reference table, the R value of a 3.5-inch-Lhick 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.) Get solution
59. Make a rough estimate of the total rate of conductive heat loss through the windows, walls, floor, and roof of a typical house in a cold climate. Then estimate the cost of replacing this lost energy over the course of a month. If possible, compare your estimate to a real utility bill. (Utility companies measure electricity by the kilowatt-hour, a unit equal to 3.6 MJ. In the United States, natural gas is billed in therms, where 1 therm = 105 Btu. Utility rates vary by region; I currently pay about 7 cents per kilowatt-hour for electricity and 50 cents per therm for natural gas.) Get solution
60. A frying pan is quickly heated on the stovetop to 200°C. It has an iron handle that is 20 cm long. Estimate how much time should pass before the end of the handle is too hot to grab with your bare hand. (Hint: The cross sectional area of the handle doesn’t matter. The density of iron is about 7.9 g/cm3 and its specific heat is 0.45 J/g∙° C). Get solution
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°C per kilometer of depth and the thermal conductivity of the rock is 2.5 W/m∙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 km.) Get solution
62. 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 Δx, derive the heat equation,...where K= κt/cρ, c is the specific heat of the material, and ρ 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...where T0 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. Get solution
63. At about what pressure would the mean free path of an air molecule at room temperature equal 10 cm, the size of a typical laboratory apparatus? Get solution
64. Make a rough estimate of the thermal conductivity of helium at room temperature. Discuss your result, explaining why it differs from the value for air. Get solution
65. Pretend that you live in the 19th century and don’t know the value of Avogadro’s number (or of Boltzmann’s constant or of the mass or size of any molecule). Show how you could make a rough estimate of Avogadro’s number from a measurement of the thermal conductivity of a gas, together with other measurements that are relatively easy. Get solution
66. In analogy with the thermal conductivity, derive an approximate formula for the viscosity of an ideal gas in terms of its density, mean free path, and average thermal speed. Show explicitly that the viscosity Is Independent of pressure and proportional to the square root of the temperature. Evaluate your formula numerically for air at room temperature and compare to the experimental value quoted In the text. Get solution
67. Make a rough estimate of how far food coloring (or sugar) will diffuse through water in one minute. Get solution
68. Suppose you open a bottle of perfume at one end of a room. Very roughly, how much time would pass before a person at the other end of the room could smell the perfume, if diffusion were the only transport mechanism? Do you think diffusion is the dominant transport mechanism in this situation? Get solution
69. Imagine a narrow pipe, filled with fluid, in which the concentration of a certain type of molecule varies only along the length of the pipe (in the x direction). By considering the flux of these particles from both directions into a short segment Δx, derive Fick’s second law,...Noting the similarity to the heat equation derived in Problem, discuss the implications of this equation in some detail.Problem: 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 Δx, derive the heat equation,...where K= κt/cρ, c is the specific heat of the material, and ρ 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...where T0 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. Get solution
70. In analogy with the thermal conductivity, derive an approximate formula for the diffusion coefficient of an ideal gas in terms of the mean free path and the average thermal speed. Evaluate your formula numerically for air at room temperature and atmospheric pressure, and compare to the experimental value quoted in the text. How does D depend on T, at fixed pressure? Get solution
2. The Rankine temperature scale (abbreviated °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? Get solution
3. Determine the kelvin temperature for each of the following:(a) human body temperature;(b) the boiling point of water (at the standard pressure of 1 atm);(c) the coldest day you can remember;(d) the boiling point of liquid nitrogen (−196°C);(e) the melting point of lead (327°C). Get solution
4. 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. Get solution
5. When you’re sick with a fever and you take your temperature with a thermometer, approximately what is the relaxation time? Get solution
6. 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. Get solution
7. When the temperature of liquid mercury increases by one degree Celsius (or one kelvin), its volume increases by one part in 5500. The fractional increase in volume per unit change in temperature (when the pressure is held fixed) is called the thermal expansion coefficient, β:...(where V is volume, T is temperature, and Δ signifies a change, which in this case should really be infinitesimal if β is to be well defined). So for mercury, β = 1/5500 K−1 = 1.81 × 10−4 K−1. (The exact value varies with temperature, but between 0°C and 200°C the variation is less than 1%.)(a) Get a mercury thermometer, estimate the size of the bulb at the bottom, and then estimate what the inside diameter of the tube has to be in order for the thermometer to work as required. Assume that the thermal expansion of the glass is negligible.(b) The thermal expansion coefficient of water varies significantly with temperature: It is 7.5 × 10−4 K−1 at 100°C, but decreases as the temperature is lowered until it becomes zero at 4°C. Below 4°C it is slightly negative, reaching a value of − 0.68 ×10−4 K−1 at 0°C. (This behavior is related to the fact that ice is less dense than water.) With this behavior in mind, imagine the process of a lake freezing over, and discuss in some detail how this process would be different if the thermal expansion coefficient of water were always positive. Get solution
8. For a solid, we also define the linear thermal expansion coefficient, α, as the fractional increase in length per degree:...(a) For steel, α is 1.1 × 10−5 K−1. Estimate the total variation in length of a 1-km steel bridge between a cold winter night and a hot summer day.(b) The dial thermometer in Figure uses a coiled metal strip made of two different metals laminated together. Explain how this works.(c) Prove that the volume thermal expansion coefficient of a solid is equal to the sum of its linear expansion coefficients in the three directions: β = αx + αy + αz. (So for an isotropic solid, which expands the same in all directions, β = 3α.)Figure: A selection of thermometers. In the center are two liquid-in-glass thermometers, which measure the expansion of mercury (for higher temperatures) and alcohol (for lower temperatures). The dial thermometer to the right measures the turning of a coil of metal, while the bulb apparatus behind it measures the pressure of a fixed volume of gas. The digital thermometer at left-rear uses a thermocouple—a junction of two metals—which generates a small temperature dependent voltage. At left-front is a set of three potter’s cones, which melt and droop at specified clay-firing temperatures.... Get solution
10. Estimate the number of air molecules in an average-sized room. Get solution
11. 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. Get solution
12. Calculate the average volume per molecule for an ideal gas at room temperature and atmospheric pressure. Then take the cube root to get an estimate of the average distance between molecules. How does this distance compare to the size of a small molecule like N2 or H2O? Get solution
13. A mole is approximately the number of protons in a gram of protons. The mass of a neutron is about the same as the mass of a proton, while the mass of an electron is usually negligible in comparison, so if you know the total number of protons and neutrons in a molecule (i.e., its “atomic mass”), you know the approximate mass (in grams) of a mole of these molecules. Referring to the periodic table at the back of this book, find the mass of a mole of each of the following: water, nitrogen (N2), lead, quartz (SiO2). Get solution
14. Calculate the mass of a mole of dry air, which is a mixture of N2 (78% by volume), O2 (21%), and argon (1%). Get solution
15. Estimate the average temperature of the air inside a hot-air balloon (see Figure 1.1). Assume that the total mass of the unfilled balloon and payload is 500 kg. What is the mass of the air inside the balloon? ... Get solution
16. The exponential atmosphere. (a) Consider a horizontal slab of air whose thickness (height) is dz. If this slab is at rest, the pressure holding it up from below must balance both the pressure from above and the weight of the slab. Use this fact to find an expression for dP/dz, the variation of pressure with altitude, in terms of the density of air.(b) Use the ideal gas law to write the density of air in terms of pressure, temperature, and the average mass m of the air molecules. (The information needed to calculate m is given in Problem.) Show, then, that the pressure obeys the differential equation...called the barometric equation.(c) Assuming that the temperature of the atmosphere is independent of height (not a great assumption but not terrible either), solve the barometric equation to obtain the pressure as a function of height: P(z) = P(0)e−mgz/kT. Show also that the density obeys a similar equation.(d) Estimate the pressure, in atmospheres, at the following locations: Ogden, Utah (4700 ft or 1430 m above sea level); Leadville, Colorado (10,150 ft, 3090 m) ; Mt. Whitney, California (14,500 ft, 4420 m); Mt. Everest, Nepal/Tibet (29,000 ft, 8850 m). (Assume that the pressure at sea level is 1 atm.)Problem: Calculate the mass of a mole of dry air, which is a mixture of N2 (78% by volume), O2 (21%), and argon (1%). Get solution
17. 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,...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 (N2): T (K)B (cm3 / mol)100−160200 −35300 −4.2400 9.0500 16.9600 21.3(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,...where a and b are constants that depend on the type of gas. Calculate the second and third virial coefficients (B and C) for a gas obeying the van der Waals equation, in terms of a and b. (Hint: The binomial expansion says that ... provided that |px| ≪ 1. Apply this approximation to the quantity [1 − (nb/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.) Get solution
18. Calculate the rms speed of a nitrogen molecule at room temperature. Get solution
19. Suppose you have a gas containing hydrogen molecules and oxygen molecules, in thermal equilibrium. Which molecules are moving faster, on average? By what factor? Get solution
20. 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, UF6, 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. Get solution
21. During a hailstorm, hailstones with an average mass of 2 g and a speed of 15 m/s strike a window pane at a 45° angle. The area of the window is 0.5 m2 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? Get solution
22. 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 Δt is ... where P is the pressure, m is the average molecular mass, and ... is the average x velocity of those molecules that collide with the wall.(b) It’s not easy to calculate ..., but a good enough approximation is...where the bar now represents an average over all molecules in the gas. Show that...(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...Solve this equation (assuming constant temperature) to obtain a formula of the form N(t) = N(0)e−t/τ, where τ is the “characteristic time” for N (and P) to drop by a factor of e.(d) Calculate the characteristic time for air at room temperature to escape from a 1-liter container punctured by a 1-mm2 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. Get solution
23. Calculate the total thermal energy in a liter of helium at room temperature and atmospheric pressure. Then repeat the calculation for a liter of air. Get solution
24. 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). Get solution
25. 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.) Get solution
26. 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? Get solution
27. 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. Get solution
28. 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. Get solution
29. A cup containing 200 g of water is sitting on your dining room table. After carefully measuring it s temperature to be 20°C, you leave the room. Returning ten minutes later, you measure its temperature again and find that it is now 25°C. What can you conclude about the amount of heat added to the water? (Hint: This is a trick question.) Get solution
30. Put a few spoonfuls of water into a bottle with a tight lid. Make sure everything is at room temperature, measuring the temperature of the water with a thermometer to make sure. Now close the bottle and shake it as hard as you can for several minutes. When you’re exhausted and ready to drop, shake it for several minutes more. Then measure the temperature again. Make a rough calculation of the expected temperature change, and compare. Get solution
31. 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.(a) Sketch a graph of pressure vs. volume for this process.(b) Calculate the work done on the gas during this process, assuming that there are no “other” types of work being done.(c) Calculate the change in the helium’s energy content during this process.(d) Calculate the amount of heat added to or removed from the helium during this process.(e) Describe what you might do to cause the pressure to rise as the helium expands. Get solution
32. By applying a pressure of 200 atm, you can compress water to 99% of its usual volume. Sketch this process (not necessarily to scale) on a PV diagram, and estimate the work required to compress a liter of water by this amount. Does the result surprise you? Get solution
33. An ideal gas is made to undergo the cyclic process shown in Figure. For each of the steps A, B, and C, determine whether each of the following is positive, negative, or zero: (a) the work done on the gas; (b) the change in the energy content of the gas; (c) the heat added to the gas. Then determine the sign of each of these three quantities for the whole cycle. What does this process accomplish?Figure: PV diagrams... Get solution
34. An ideal diatomic gas, in a cylinder with a movable piston, undergoes the rectangular cyclic process shown in Figure. Assume that the temperature is always such that rotational degrees of freedom are active, but vibrational modes are “frozen out.” Also assume that the only type of work done on the gas is quasistatic compression-expansion work.(a) For each of the four steps A through D, compute the work done on the gas, the heat added to the gas, and the change in the energy content of the gas. Express all answers in terms of P1, P2, V1 and V2. (Hint: Compute ΔU before Q, using the ideal gas law and the equipartition theorem.)(b) Describe in words what is physically being done during each of the four steps, for example, during step A, heat is added to the gas (from an external flame or something) while the piston is held fixed.(c) Compute the net work done on the gas, the net heat added to the gas, and the net change in the energy of the gas during the entire cycle. Are the results as you expected? Explain briefly.Figure: PV diagrams... Get solution
35. Derive equation 1 from equation 2.Equation 1:Vγ p = constantEquation 2:VTf/2 = constant. Get solution
36. 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 K, what is the temperature after compression? Get solution
37. In a Diesel engine, atmospheric air is quickly compressed to about 1/20 of its original volume. Estimate the temperature of the air after compression, and explain why a Diesel engine does not require spark plugs. Get solution
38. 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. Get solution
39. 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...where ρ 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 ΔP to a chunk of the material, and this increase results in a (negative) change in volume ΔV, then B is defined as the change in pressure divided by the magnitude of the fractional change in volume:...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? Get solution
40. In Problem you calculated the pressure of earth’s atmosphere as a function of altitude, assuming constant temperature. Ordinarily, however, the temperature of the bottommost 10–15 km of the atmosphere (called the troposphere) decreases with increasing altitude, due to heating from the ground (which is warmed by sunlight). If the temperature gradient |dT/dz| exceeds a certain critical value, convection will occur: Warm, low-density air will rise, while cool, high-density air sinks. The decrease of pressure with altitude causes a rising air mass to expand adiabatically and thus to cool. The condition for convection to occur is that the rising air mass must remain warmer than the surrounding air despite this adiabatic cooling.(a) Show that when an ideal gas expands adiabatically, the temperature and pressure are related by the differential equation...(b) Assume that dT/dz is just at the critical value for convection to begin, so that the vertical forces on a convecting air mass are always approximately in balance. Use the result of Problem 1 (b) to find a formula for dT/dz in this case. The result should be a constant, independent of temperature and pressure, which evaluates to approximately −10°C/km. This fundamental meteorological quantity is known as the dry adiabatic lapse rate.Problem 1:The exponential atmosphere. (a) Consider a horizontal slab of air whose thickness (height) is dz. If this slab is at rest, the pressure holding it up from below must balance both the pressure from above and the weight of the slab. Use this fact to find an expression for dP/dz, the variation of pressure with altitude, in terms of the density of air.(b) Use the ideal gas law to write the density of air in terms of pressure, temperature, and the average mass m of the air molecules. (The information needed to calculate m is given in Problem 2.) Show, then, that the pressure obeys the differential equation...called the barometric equation.(c) Assuming that the temperature of the atmosphere is independent of height (not a great assumption but not terrible either), solve the barometric equation to obtain the pressure as a function of height: P(z) = P(0)e−mgz/kT. Show also that the density obeys a similar equation.(d) Estimate the pressure, in atmospheres, at the following locations: Ogden, Utah (4700 ft or 1430 m above sea level); Leadville, Colorado (10,150 ft, 3090 m) ; Mt. Whitney, California (14,500 ft, 4420 m); Mt. Everest, Nepal/Tibet (29,000 ft, 8850 m). (Assume that the pressure at sea level is 1 atm.)Problem 2:Calculate the mass of a mole of dry air, which is a mixture of N2 (78% by volume), O2 (21%), and argon (1%). Get solution
41. 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 (100°C), then is quickly transferred into a Styrofoam cup containing 250 g of water at 20°C. After a minute or so, the temperature of the contents of the cup is 24°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 gained by the water?(b) How much heat is lost 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 g, what is its specific heat capacity? Get solution
42. The specific heat capacity of Albertson’s Rotini Tricolore is approximately 1.8 J /g∙°C. Suppose you toss 340 g of this pasta (at 25°C) into 1.5 liters of boiling water. What effect does this have on the temperature of the water (before there is time for the stove to provide more heat)? Get solution
43. Calculate the heat capacity of liquid water per molecule, in terms of k. Suppose (incorrectly) that all the thermal energy of water is stored in quadratic degrees of freedom. How many degrees of freedom would each molecule have to have? Get solution
44. At the back of this book is a tab1e of thermodynamic data for selected substances at room temperature. Browse through the CP values in this table, and check that you can account for most of them (approximately) using the equipartition theorem. Which values seem anomalous? Get solution
45. As an illustration of why it matters which variables you hold fixed when taking partial derivatives, consider the following mathematical example. Let w = xy and x = yz.(a) Write w purely in terms of x and z, and then purely in terms of y and z(b) Compute the partial derivatives...and show that they are not equal. (Hint: To compute... use a formula for w in terms of x and y, not z. Similarly, compute ... 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 and z), and show that it matters which variable is held fixed. Get solution
46. Measured heat capacities of solids and liquids are almost always at constant pressure, not constant volume. To see why, estimate the pressure needed to keep V fixed as T increases, as follows.(a) First imagine slightly increasing the temperature of a material at constant pressure. Write the change in volume, dV1, in terms of dT and the thermal expansion coefficient β introduced in Problem 1.(b) Now imagine slightly compressing the material, holding its temperature fixed. Write the change in volume for this process, dV2, in terms of dP and the isothermal compressibility κT defined as...(This is the reciprocal of the isothermal bulk modulus defined in Problem 2.)(c) Finally, imagine that you compress the material just enough in part (b) to offset the expansion in part (a). Then the ratio of dP to dT is equal to ... since there is no net change in volume. Express this partial derivative in terms of β and κT. Then express it more abstractly in terms of the partial derivatives used to define β and κT. For the second expression you should obtain...This result is actually a purely mathematical relation, true for any three quantities that are related in such a way that any two determine the third.(d) Compute β, κT and ... for an ideal gas, and check that the three expressions satisfy the identity you found in part (c).(e) For water at 25°C, β = 2.57 × 10−4 K−1 and κT = 4.52 × 10−10 Pa−1. Suppose you increase the temperature of some water from 20°C to 30°C. How much pressure must you apply to prevent it from expanding? Repeat the calculation for mercury, for which (at 25°C) β = 1.81 × 10−4 K−1 and κT = 4.04 × 10−11 Pa−l. Given the choice, would you rather measure the heat capacities of these substances at constant V or at constant P?Problem 1:When the temperature of liquid mercury increases by one degree Celsius (or one kelvin), its volume increases by one part in 5500. The fractional increase in volume per unit change in temperature (when the pressure is held fixed) is called the thermal expansion coefficient, β:...(where V is volume, T is temperature, and Δ signifies a change, which in this case should really be infinitesimal if β is to be well defined). So for mercury, β = 1/5500 K−1 = 1.81 × 10−4 K−1. (The exact value varies with temperature, but between 0°C and 200°C the variation is less than 1%.)(a) Get a mercury thermometer, estimate the size of the bulb at the bottom, and then estimate what the inside diameter of the tube has to be in order for the thermometer to work as required. Assume that the thermal expansion of the glass is negligible.(b) The thermal expansion coefficient of water varies significantly with temperature: It is 7.5 × 10−4 K−1 at 100°C, but decreases as the temperature is lowered until it becomes zero at 4°C. Below 4°C it is slightly negative, reaching a value of − 0.68 ×10−4 K−1 at 0°C. (This behavior is related to the fact that ice is less dense than water.) With this behavior in mind, imagine the process of a lake freezing over, and discuss in some detail how this process would be different if the thermal expansion coefficient of water were always positive.Problem 2: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...where ρ 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 ΔP to a chunk of the material, and this increase results in a (negative) change in volume ΔV, then B is defined as the change in pressure divided by the magnitude of the fractional change in volume:...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? Get solution
47. Your 200-g cup of tea is boiling-hot. About how much ice should you add to bring it down to a comfortable sipping temperature of 65°C? (Assume that the ice is initially at −15°C. The specific heat capacity of ice is 0.5 cal/g∙°C.) Get solution
48. 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/m2 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. Get solution
49. Consider the combustion of one mole of H2 with 1/2 mole of O2 under standard conditions, as discussed in the text. How much of the heat energy produced comes from a decrease in the internal energy of the system, and how much comes from work done by the collapsing atmosphere? (Treat the volume of the liquid water as negligible.) Get solution
50. Consider the combustion of one mole of methane gas:CH4(gas) + 2O2(gas) → CO2(gas) + 2H2O(gas).The system is at standard temperature (298 K) and pressure (105 Pa) both before and after the reaction.(a) First imagine the process of converting a mole of methane into its elemental consituents (graphite and hydrogen gas). Use the data at the back of this book to find ΔH for this process.(b) Now imagine forming a mole of CO2 and two moles of water vapor from their elemental constituents. Determine ΔH for this process.(c) What is ΔH for the actual reaction in which methane and oxygen form carbon dioxide and water vapor directly? Explain.(d) How much heat is given off during this reaction, assuming that no “other” forms of work are done?(e) What is the change in the system’s energy during this reaction? How would your answer differ if the H2O ended up as liquid water instead of vapor?(f) The sun has a mass of 2 × 1030 kg and gives off energy at a rate of. 3.9 × 1026 watts. If the source of the sun’s energy were ordinary combustion of a chemical fuel such as methane, about how long could it last? Get solution
51. Use the data at the back of this book to determine ΔH for the combustion of a mole of glucose,C6H12O6 + 6O2 → 6CO2 + 6H2O.This is the (net) reaction that provides most of the energy needs in our bodies. Get solution
52. The enthalpy of combustion of a gallon (3.8 liters) of gasoline is about 31,000 kcal. The enthalpy of combustion of an ounce (28 g) of corn flakes is about 100 kcal. Compare the cost of gasoline to the cost of corn flakes, per calorie. Get solution
53. Look up the enthalpy of formation of atomic hydrogen in the back of this book. This is the enthalpy change when a mole of atomic hydrogen is formed by dissociating 1/2 mole of molecular hydrogen (the more stable State of the element). From this number, determine the energy needed to dissociate a single H2 molecule, in electron-volts. Get solution
54. A 60-kg hiker wishes to climb to the summit of Mt. Ogden, an ascent of 5000 vertical feet (1500 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°C a reasonable temperature to assume, the latent heat of vaporization of water is 580 cal/g, 8% more than at 100°C) Get solution
55. 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:...Here each Ū 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 ...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 −GM2/R, times some constant of order 1.(e) Estimate the average temperature of the sun, whose mass is 2 × 1030 kg and whose radius is 7 × 108 m. Assume, for simplicity, that the sun is made entirely of protons and electrons. Get solution
56. Calculate the rate of heat conduction through a layer of still air that is 1 mm thick, with an area of 1 m2 , for a temperature difference of 20°C. Get solution
57. Home owners and builders discuss thermal conductivities in terms of the R value (R for resistance) of a material, defined as the thickness divided by the thermal conductivity:...(a) Calculate the R value of a 1/8-inch (3.2 mm) piece of plate glass, and then of a 1 mm layer of still air. Express both answers in SI units.(b) In the United States, R values of building materials are normally given in English units, °F∙ft2∙hr/Btu. A Btu, or British thermal unit, is the energy needed to raise the temperature of a pound of water by 1°F. Work out the conversion factor between the SI and English units for R values. Convert your answers from part (a) to English units.(c) Prove that for a compound layer of two different materials sandwiched together (such as air and glass, or brick and wood), the effective total R value is the sum of the individual R values.(d) Calculate the effective R value of a single piece of plate glass with a 1.0-mm layer of still air on each side. (The effective thickness of the air layer will depend on how much wind is blowing; 1 mm is of the right order of magnitude under mot conditions.) Using this effective R value, make a revised estimate of the heat loss through a 1-m2 single-pane window when the temperature in the room is 20°C higher than the outdoor temperature. Get solution
58. According to a standard reference table, the R value of a 3.5-inch-Lhick 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.) Get solution
59. Make a rough estimate of the total rate of conductive heat loss through the windows, walls, floor, and roof of a typical house in a cold climate. Then estimate the cost of replacing this lost energy over the course of a month. If possible, compare your estimate to a real utility bill. (Utility companies measure electricity by the kilowatt-hour, a unit equal to 3.6 MJ. In the United States, natural gas is billed in therms, where 1 therm = 105 Btu. Utility rates vary by region; I currently pay about 7 cents per kilowatt-hour for electricity and 50 cents per therm for natural gas.) Get solution
60. A frying pan is quickly heated on the stovetop to 200°C. It has an iron handle that is 20 cm long. Estimate how much time should pass before the end of the handle is too hot to grab with your bare hand. (Hint: The cross sectional area of the handle doesn’t matter. The density of iron is about 7.9 g/cm3 and its specific heat is 0.45 J/g∙° C). Get solution
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°C per kilometer of depth and the thermal conductivity of the rock is 2.5 W/m∙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 km.) Get solution
62. 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 Δx, derive the heat equation,...where K= κt/cρ, c is the specific heat of the material, and ρ 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...where T0 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. Get solution
63. At about what pressure would the mean free path of an air molecule at room temperature equal 10 cm, the size of a typical laboratory apparatus? Get solution
64. Make a rough estimate of the thermal conductivity of helium at room temperature. Discuss your result, explaining why it differs from the value for air. Get solution
65. Pretend that you live in the 19th century and don’t know the value of Avogadro’s number (or of Boltzmann’s constant or of the mass or size of any molecule). Show how you could make a rough estimate of Avogadro’s number from a measurement of the thermal conductivity of a gas, together with other measurements that are relatively easy. Get solution
66. In analogy with the thermal conductivity, derive an approximate formula for the viscosity of an ideal gas in terms of its density, mean free path, and average thermal speed. Show explicitly that the viscosity Is Independent of pressure and proportional to the square root of the temperature. Evaluate your formula numerically for air at room temperature and compare to the experimental value quoted In the text. Get solution
67. Make a rough estimate of how far food coloring (or sugar) will diffuse through water in one minute. Get solution
68. Suppose you open a bottle of perfume at one end of a room. Very roughly, how much time would pass before a person at the other end of the room could smell the perfume, if diffusion were the only transport mechanism? Do you think diffusion is the dominant transport mechanism in this situation? Get solution
69. Imagine a narrow pipe, filled with fluid, in which the concentration of a certain type of molecule varies only along the length of the pipe (in the x direction). By considering the flux of these particles from both directions into a short segment Δx, derive Fick’s second law,...Noting the similarity to the heat equation derived in Problem, discuss the implications of this equation in some detail.Problem: 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 Δx, derive the heat equation,...where K= κt/cρ, c is the specific heat of the material, and ρ 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...where T0 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. Get solution
70. In analogy with the thermal conductivity, derive an approximate formula for the diffusion coefficient of an ideal gas in terms of the mean free path and the average thermal speed. Evaluate your formula numerically for air at room temperature and atmospheric pressure, and compare to the experimental value quoted in the text. How does D depend on T, at fixed pressure? Get solution
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