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Chapter 19 Entropy Chapter 19 Summary Concepts Quantitative tools

Entropy of a closed system (Sections 19.1, 19.2, 19.3, 19.5)

The probability of an event is the fraction of times it occurs in a large number of repetitions. The macrostate of a system is that state described by its large-scale properties (on a scale much larger than molecular size), such as volume, pressure, and temperature. The basic state of a system is that state described in terms of the properties of the particles that make up the system, such as their position and velocity. If the interaction between different parts of a system randomizes the distribution of energy in the system, each particle in the system tends to have an equal share of that energy (equipartition of energy), and the particles tend to distribute uniformly in the available volume (equipartition of space).

If a closed volume of space containing N distinguishable particles is divided into M distinguishable equal-sized compartments, the number of basic states Ω in the system is

Ω = MN (19.1) The entropy S of a system is a unitless quantity that is a measure of the number of basic states in the system and is defined as

S ≡ lnΩ (19.4) If the volume of a closed (constant-energy) gas containing N particles changes from Vi to Vf, the entropy change of the gas is

ln f

i

VS N

V⎛ ⎞

Δ = ⎜ ⎟⎝ ⎠

(19.8)

The second law of thermodynamics (Sections 19.4, 19.5) The equilibrium state of a system is the macrostate having the largest number of basic states and is therefore the most probable macrostate. At equilibrium, the number of basic states reaches its maximum value Ωmax, and the entropy of the system is a maximum. Second law of thermodynamics (entropy law): A closed system always evolves so as to maximize the number of basic states Ω. When this number has reached a maximum, the system is in equilibrium. For a system that is not closed, the entropy

For a closed system evolving toward equilibrium, the entropy increases:

ΔS > 0 (19.5) For a closed system at equilibrium, Ω = Ωmax and the entropy does not change:

ΔS = 0 (19.6) These two equations, collectively called the entropy law, are a mathematical expression of the second law of thermodynamics. If a system consists of two subsystems A and B, the number of basic states for the system is

Ω = ΩAΩB (19.9)

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can increase, decrease, or stay the same. An ideal gas consists of a very large number of particles that occupy a very small fraction of the volume of their container and interact with one another only during collisions that randomize the energy distribution and the spatial distribution of the particles.

and the entropy of the system is S = SA + SB (19.10)

Entropy and energy (Section 19.6) In a monatomic ideal gas, internal structure of the atoms can be ignored. For this reason, the thermal energy Eth of the gas is the incoherent kinetic energy of all the atoms. If two gases are in thermal equilibrium, their absolute temperatures are the same. In this condition, thermal energy is equipartitioned over the system and the entropy of the system is a maximum.

The root-mean-square speed vrms of a gas is the square root of the average of the squares of the speeds of all the atoms of the gas:

( )2rms avv v= (19.21)

If a gas contains N atoms each of mass m, the average kinetic energy of an atom is

( )2 21 12 2

thav rmsav

EK m v mvN

≡ = = (19.27)

If the thermal energy of an ideal monatomic gas changes from Eth,i to Eth,f, the resulting entropy change is

,

,

3 ln2

th f

th i

ES N

E⎛ ⎞

Δ = ⎜ ⎟⎜ ⎟⎝ ⎠ (19.33)

The absolute temperature T of a gas, measured in kelvins, is related to the rate at which the entropy of the gas changes with respect to its thermal energy:

1

B th

dSk T dE

≡ (19.38)

where kB is the Boltzmann constant: 231.380 10 J/KBk

−= × (19.39)

Properties of a monatomic ideal gas (Sections 19.7, 19.8) The pressure in an ideal monatomic gas

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containing N atoms at an absolute temperature T occupying a volume V is

23

thEPV

= (19.48)

and the thermal energy of the gas is 32th BE Nk T= (19.50)

The ideal gas law applied to the gas is

BNP k TV

= (19.51)

The average kinetic energy of its atoms is 21 3

2 2av rms BK mv k T= = (19.52)

where m is the mass of each atom, and their root-mean-square speed is

3 Brms

k Tvm

= (19.53)

If the gas is in equilibrium, its entropy is 32ln constantS N T V

⎛ ⎞= +⎜ ⎟

⎝ ⎠ (19.60)

If the gas changes from an equilibrium state having Ti and Vi to one having Tf and Vf, its entropy change is

3 ln ln2

f f

i i

T VS N N

T V⎛ ⎞ ⎛ ⎞

Δ = +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

(19.61)

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Review questions Section 19.1 States 1. What do the conservation principles imply about the time evolution of a system? 2. In Example 19.1, what is the distinction between the kinetic energy associated with the swing of the pendulum & the kinetic energy associated with the particles of the gas in the container enclosing the pendulum? 3. What kind of motion is Brownian motion? 4. What does macroscopic mean? 5. What does the probability of an event mean? 6. How does a macroscopic state differ from a basic state, for a system consisting of a very large number of particles? 7. What does the recurrence time of a system mean? Section 19.2 Equipartition of Energy 8. How is the total energy shared among the randomly interacting parts of a system? Section 19.3 Equipartition of Space 9. How do the molecules in a randomly interacting gas distribute themselves in space? Section 19.4 Evolution toward the Most Probable Macrostate 10. What distinguishes the macrostate corresponding to equilibrium? 11. What is the underlying reason for irreversibility? 12. From where do we get our sense of the direction of time? 13. In words, what is the second law of thermodynamics? 14. Does the second law of thermodynamics apply to all systems? Section 19.5 Dependence of Entropy on Volume 15. How easy is it to specify (i.e. measure) the atomic versus the macroscopic parameters of a state for any realistic, everyday system? 16. What are the conditions that define an ideal gas? 17. What is the statistical definition for the entropy of a system? 18. Give a statement of the second law of thermodynamics in terms of entropy. 19. What is the relation between the entropy of a combined system & the entropies of its constituent systems? 20. How do changes in the entropy, for a system of an isolated ideal gas in equilibrium, depend on its volume? Section 19.6 Dependence of Entropy on Energy 21. What is the meaning of the root-mean-square value of a quantity? 22. How do changes in the entropy of a given volume of an ideal monatomic gas depend on the thermal energy of the gas? 23. How is the absolute temperature of an ideal gas defined in terms of its entropy & thermal energy? 24. If two different ideal gases are in thermal equilibrium, how are their entropies & their absolute temperatures related?

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Section 19.7 Properties of a Monatomic Ideal Gas 25. What two atomic parameters determine the pressure of an ideal monatomic gas? [Note: volume & temperature are not atomic parameters, although number density is.] 26. State the ideal gas law in words? 27. Is the ideal gas law ever a good description of the behavior of real gases, & if so, when? 28. What parameters determine the rms-speed for an equilibrium state of an ideal gas?

Section 19.8 Entropy of a Monatomic Ideal Gas 29. In words, how does the change in entropy of an ideal monatomic gas, between two equilibrium states in general, depend on the macroscopic quantities which describe the states, i.e. the pressure, volume, absolute temperature & number of molecules? 30. How does the change in entropy in the preceding question depend on the way the gas changes from the initial to the final equilibrium state?

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Developing a feel Calculate or estimate the following quantities: 1. The number of basic states of 1 kilobit of computer memory. [A] 2. The number of basic states of one page of text. [B,C] 3. How much time it would take for one person typing letters randomly to produce the first ten letters of Hamlet. [D,E] 4. The population of Manhattan (in New York City) if the Earth’s population is equipartitioned. [F,G,H] 5. The entropy decrease if all the air molecules in your bedroom moved so as to occupy only the top half of the room. [I,J] 6. The entropy change when a helium party balloon deflates. [K,L,M] 7. The entropy increase when inflating a bicycle tire by discharging a 15 g 2CO cylinder into it. [N,O,P] 8. The entropy increase when you leave your 1-liter soda bottle uncapped and it goes flat (i.e., all the 2CO escapes). [Q,R,S] 9. The entropy increase when you increase the temperature of your bedroom by 1 degree (assume that air is an ideal monatomic gas). [J,T] 10. Entropy increase of the air in your car tires as it heats up while you drive (assume that air is a monatomic ideal gas). [U,V,W] 11. RMS speed of air molecules at room temperature. [T,X] 12. RMS speed of helium atoms just above its boiling point. [Y,Z] 13. RMS speed of hydrogen atoms in the center of the sun. [AA,BB] 14. What is the energy of the atmosphere (assuming that it is composed of a monatomic ideal gas)? [CC,DD,EE,FF] 15. The entropy change of the atmosphere over the United States from 2:00 PM to 2:00 AM. [GG,HH,II,JJ] Hints: A. How many basic states does one bit have? B. How many basic states does one character have? C. How many characters per page? D. How many characters per second do you type? E. What is the probability of a ten character random letter sequence to match a specific ten-letter sequence? F. What is the Earth’s population? G. What is the land area of the Earth? H. What is the area of Manhattan? I. What is the volume of a typical bedroom? J. How many air molecules are in the room? K. What is the volume of a helium party balloon? L. How many helium atoms are in the balloon? M. What is the effective volume of the atmosphere? N. What is the volume of the 2CO cylinder?

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O. What is the volume of the tire? P. How many 2CO molecules are in the cylinder? Q. What is the volume of the atmosphere? R. What is the mass of 2CO in the soda bottle? S. How many 2CO molecules are in the soda bottle? T. What is the temperature of your room? U. What is the temperature change? V. What is the volume of the air in your car tires? W. How many molecules of air in your tires? X. What is the mass of an air molecule? Y. What is the boiling temperature of helium? Z. What is the mass of a helium atom? AA. What is the temperature of the center of the sun? BB. What is the mass of a hydrogen atom? CC. What is the area of the Earth? DD. What is the mass of the atmosphere? EE. How many air molecules are in the atmosphere? FF. What is the average temperature of the atmosphere? GG. What is the area of the United States? HH. What is the mass of the atmosphere over the United States? II. How many air molecules are over the United States? JJ. What is their temperature change? Key (all values approximate): A. 2 B. 26 C. 32 10× D. 5 E. 1 in 10 1426 10≈ F. 97 10× G.

14 210 m H. 7 26 10 m× I. 24 3m J. 310 moles or 266 10× molecules K. 2 310 m− L. 233 10× M. 18 34 10 m× N. 3 6 33cm 3 10 m−= × O. 3 310 m− P. 232 10× Q. 18 310 m R. 5 g S. 225 10× T. 300 K U. 10 K V. 2 36 10 m−× W. 245 10× X. 0.03 kg / 265 10 kgAN

−= × Y. 4 K Z. 277 10−× kg AA. 72 10× K BB. 272 10−× kg CC. 14 24 10 m× DD. 184 10× kg EE. 438 10×

FF. 23 10× K GG. 13 210 m HH. 1710 kg II. 422 10× JJ. 10 K

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Worked and guided problems TK These examples involve material from this chapter but are not associated with any particular section. Typically, an example that is worked out in detail is followed immediately by an example whose solution you should work out by following the guidelines provided.

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Questions and problems Section 19.1 States 1. A standard deck of playing cards contains 52 cards. Cards are either numbered (2-10) or given a character (Jack, Queen, King, or Ace). There are four copies (or suits) of each card. What is the probability of randomly picking from the deck (a) a card with the number two on it, and (b) a card that has a character on it? 2. What is the probability that the sums of the values on three die thrown will equal 4? Assume the dice are cubic with the numbers 1-6 on different faces. 3. A pendulum is in a container that holds only two other particles (of air for instance). If the pendulum initially has six quantized units of energy, and the two other particles have none, what is the probability that all energy will leave the pendulum and be absorbed by the particles? 4. Suppose for the purposes of a calculable estimate that particles can only move in one of six directions: up, down, left, right, forward, and backward. If six particles move at random such that all six collide and emerge with a new random direction every 610− seconds, how much time in a full day would all six particles be moving in the same direction? 5. Two standard dice are thrown. What is the probability that the sum of the values on the two dice will be seven?

6. A pendulum is released from an elevation of 1 m and allowed to swing. This is done in a room filled with 211.50 10× nitrogen molecules with an average velocity of 550 m/s. If the pendulum has the same amount of energy as the surrounding nitrogen, what is the pendulum’s mass? 7. Suppose a box contains enough particles such that there are 135.00 10× basic states. If the recurrence time is 365 days, how much time passes on average between collisions?

8. A coin is being tossed and the outcome is either heads (H) or tails (T). (a) Which of the following sequences of throws is more likely: HHHHH or HTHTH, and (b) is it more likely that the number of times the coin turns up “heads” will be 5 or 3? 9. A manufacturer of motorcycles sells three different types of motorcycles, each of which are available in either yellow, red, glossy black or matte black. Each of the motorcycles is offered with either a 600 cc motor or an 850 cc motor. (a) How many different configurations are available from the manufacturer? (b) If each motorcycle is available in 26 different options packages, how many different configurations are available from the manufacturer? 10. A bag contains 10 cola flavored jellybeans, 12 green apple flavored jellybeans, 3 peach flavored jellybeans and 20 blueberry-flavored jellybeans. You reach into the bag

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and randomly grab a jellybean, what is the probability that you pick (a) a cola flavored jellybean, (b) a peach flavored jellybean, (c) a blueberry flavored jellybean? 11. A musical instrument has valves that can be either opened or closed to produce an effectively longer or shorter horn, which produces different notes. How many different effective lengths can be produced by a horn with (a) two valves, (b) three valves? (c) six valves? (d) Some trumpet players do something called “half valve work” where they play a valve that is neither fully open nor fully closed. If this adds a third setting to the open or closed valve, how many effective lengths can be produced with a trumpet with three valves by a player doing half-valve work? 12. An adsorbing filter allows gas molecules to stick to locations on the surface of the material. Once a gas molecule sticks to that location, the location is filled. The material can no longer remove gas molecules once all of its locations are filled. A square nanometer of this adsorbing material has 6 separate adsorbing locations, each of which will hold a single molecule. In the immediate volume above the area, there are 100 gas molecules, each with a slightly different energy, thus each molecule is unique. How many ways can these 100 molecules adsorb onto the 6 locations? Section 19.2 Equipartition of Energy 13. A system contains three particles and three units of energy. Find the probability of all energy being found on one particle. 14. There is a system that contains four particles and five units of energy. What is the probability that all five units of energy will be transferred to any one particle? 15. Consider a room held at a temperature T. The air in the room is made up largely of nitrogen molecules ( -26

nitrogenm =4.652×10 kg ) and oxygen molecules

( -26oxygenm =5.3154×10 kg ). The room has been held at a steady temperature and allowed

to equilibrate for a long time. What is the ratio of the average speed of nitrogen molecules to the average speed of oxygen molecules? 16. Consider the same 0.1 kg pendulum from Example 19.3. The number of Nitrogen molecules in the box is increased from 2310 to 2410 . If other variables (initial temperature, volume of the box, average energy transferred per collision) remain the same, how long would the pendulum take to stop? 17. In a system of three particles with 9 units of energy shared between them, which is more likely to be found at a given time: particle A has 2 units of energy, or particle A has 3 units of energy? 18. If one mole of nitrogen molecules has a total kinetic energy of 498 J, what is the most likely speed of each molecule?

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19. An asteroid weighing 64.5 10× kg and initially moving at a speed of 1055 m/s enters Earth’s atmosphere. As it drifts through the atmosphere it collides with 810 moles of nitrogen molecules which had an average speed of 850 m/s before the asteroid entered the picture. If the energy is distributed evenly, by how much was the speed of each nitrogen molecule increased by the time the asteroid reached a terminal speed of 35 m/s? 20. A volume contains a mole of ideal monatomic gas, mass per particle 261.33 10 kg−× per molecule. A tenth of a mole of a different ideal monatomic gas, mass per particle

255.32 10 kg−× , is added to the volume. What is the final average speed of the first gas if the second gas has a speed of 123 m/s? 21. A volume holds 50 monatomic A-particles, and has a total energy of 120 units. 100 monatomic B-particles with a total energy of 180 units are added to this. After a long time, (a) how many units of energy does an A-particle have? (b) How many units of energy does a B-particle have? 22. A 1 kg ball is dropped inside of a box from a height of 1 m, as shown in Figure 19.22. The box contains one mole of nitrogen molecules, each with a mass of 264.652 10 kg−× and moving with an average speed of 100 m/s. The ball rebounds perfectly elastically against the walls of the box so that its energy is absorbed only by the gas molecules. (a) What is the average energy of each gas molecule before the ball is dropped? (b) After a long time the ball appears to have no movement. What is the new average speed of the molecules at this point?

Figure 19.22

23. A spherical balloon of 0.5 m diameter contains a mole of helium gas. It is left in the sun for ten minutes. During that time, the sun radiates the balloon with an average of 1120 watts/m2 of power as shown. During this time, 32% of the total energy is reflected off the surface of the balloon, and 67.5% of the total energy is either radiated back out of the balloon as heat or passes through the balloon and gas without being absorbed. Before

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being warmed by the sun, the average speed of the helium molecules is 367 m/s. What is the average molecular speed after an hour of being in the sun? (Assume constant balloon diameter from beginning to end, and a mass of each helium molecule to be

261.33 10 kg−× .) Section 19.3 Equipartition of Space 24. A cube is divided into octants (numbered 1-8 in no important order). Inside the cube there are 3 particles. How many basic states are there that yield a macrostate in which (a) all three particles are in octant number 1, and (b)exactly 2 particles are in octant number 8? 25. Consider a partitioned box like that shown in Figure 9.9, with the partition removed. Assume that all particles in the box have an average speed such that they may cross from one side of the box to the other every 2.50 seconds, on average. Note that this does not mean they uniformly switch sides; it means that they move randomly in such a way that they enter a new random distribution every 2.50 seconds. Find the probability for all particles to be in the left half of the box, and the average amount of time required for this to happen for the number of particles equaling (a) 2, (b) 10, (c) 100. 26. CR Example 19.5 deals with a few particles in a box. Any restrictions based on the size of the box have been ignored (for example, how many particles can fit into one quadrant of the box). If one has only a few particles, or one has large box, such considerations would not affect the probabilities. What would change if the box were very small or the number of particles were very large. Assume that multiple particles cannot occupy the same space. More specifically, consider the case in which the box is so small that each quadrant of the box can hold a maximum of 30 particles, and a total of 114 particles are placed inside the box. What can you say about the basic states or macrostates of the box now? What is now the most probable number of particles in the upper left quadrant? What is the average number of particles in the upper left quadrant? Hint: instead of considering where the particles are located, consider where they are not. 27. Suppose a box takes up 1/64 of the room around it. The box is opened, releasing two particles into the room. (a) What is the probability that at any subsequent time both particles will return to the box at once? (b) What is the probability that at any subsequent time the two particles will occupy the same 1/64 of the room? 28. Three particles are released in a box divided into four quarters. How many basic states are there if the particles are (a) indistinguishable, and (b) distinguishable? 29. A room is divided into octants and four distinguishable particles move freely within the room. Let the basic states be given by specifying the octant in which each particle islocated. (a) Give an example of a macrostate for this system. (b) How many basic states are there? (c) The door to this room is opened allowing the particles to occupy an adjacent, identical room, also divided into octants. Having doubled the amount of space that can be occupied, by what factor has the number of basic states increased?

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30. Six identical particles are in a box divided into four quadrants. For a certain experiment, you need to have a certain number of particles in the upper left quadrant. You observe that in a given day, this only occurs for 51.43 minutes. How many particles are required to be in the upper left quadrant for the experiment? 31. A cube of 1 m3 contains N particle(s). Find the probability that all of the particle(s) will find themselves contained within a smaller internal volume of the cube V if (a)

1, 1 2N V= = , (b) 1, 1 4N V= = , (c) 1, 1 100N V= = , (d) 2, 1 2N V= = , (e) 4, 1 2N V= = (f) 100, 1 100N V= = , (g) 1,000, 1 100N V= = . 32. A space probe is designed to measure the particle density of interstellar gas by enclosing a cubic meter of space and measuring the number of atoms that pass through an onboard detector. The detector encloses a volume of space contained within a cylinder 1.5 cm in diameter and 0.5 cm long. In one area of space, the density of interstellar gas is assumed to be 1 atom per cm3. Out of a hundred tests, how many times would the sensor register the presence of an atom contained within the sensor volume? Section 19.4 Evolution toward the Most Probable Macrostate 33. Table 19.2 displays the number of basic states for various distributions of 10 units of energy to 20 particles. How much more likely is it for 7 units of energy to be found in partition A than for 3 units of energy to be found in partition A? 34. Suppose a box is partitioned in half as in figure 19.3. Let there be 25 distinguishable particles on the left side, and 20 such particles on the right side. Initially, there are 9 units of energy in the box, and all are on particles in the right side. The system is closed, but the two sides can exchange energy through collisions with the partition. If one unit of energy can be exchanged during one collision, what is the smallest number of collisions that must occur before the system reaches equilibrium? 35. A box is partitioned into two chambers. On the right half of the chamber are 3 distinguishable particles, and on the left half are 2 such particles. There are five units of energy to be distributed among all the particles. Initially, all five units of energy are on the left side. The particles can exchange energy through collisions with the partition, but the particles cannot move to the opposite side. Calculate the total number of basic states (a) initially, and (b) once the system has reached equilibrium. 36. How many ways are there to divide ten indistinguishable units of energy between fourteen distinguishable particles? Hint: use table 19.2. 37. Reproduce table 19.2 replacing the total number of energy units with 4, and letting side A of the partition contain only two particles, and side B contain 2 particles also. Check to see if the equipartition of energy is favorable even with such a small number of particles. 38. A circular box is divided into three equal regions (A, B, and C) by a membrane that allows the transfer of energy but not the transfer of particles between regions. Each

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region shares a wall with the other two (like three sections of a pie). Each region of the box contains 13 particles. Suppose each collision with a wall transfers one unit of energy. Initially all 39 units of energy in the system are in region A. Is it possible that the system has evolved to its most probable state after exactly (a) 18 collisions, (b) 27 collisions, or (c) 29 collisions? 39. Suppose a deck of cards (52 distinct cards) are being blown around a room by a very powerful fan. The left and right halves of the room have been marked off and labeled. At any given point, how much more likely is it to find the same number of cards on the left and right sides of the room, than to find (a) all cards on the left or (b) 25 cards on the left and 27 cards on the right? 40. A container holds 1 mole of ideal gas at 72 K. One-half mole of ideal gas at 126 K is put into the same container. What is the equilibrium temperature? 41. A system has the following macrostates and basic states:

(a) In which macrostate does the system or systems find equilibrium? (b) Which system or systems are least likely to occur? (c) How many basic states are possible? (d) What is the probability of finding the system in macrostate 2? 42. Two systems are in equilibrium and independent from each other, but with equal probability of equilibrium. System 1 has 3422 times more basic states than System 2. System 2 has an equilibrium macrostate with 489 basic states. How many basic states are in the equilibrium state of System 1? 43. A system with three macrostates has an equilibrium macrostate 2E that is six times more likely to occur than its two least likely macrostates 1 3and E E , both of which have only one basic state each. (a) What is the probability that 2E will occur? (b) What is the probability that 1 3and E E will occur? (c) What is the sum of the probabilities for all states? Section 19.5 Dependence of Entropy on Volume 44. Suppose you are dropping five marbles randomly onto a chessboard (8 squares by 8 squares). If the five marbles are all different, and the basic states are determined by the squares on which each of the marbles stops, what is the total number of basic states possible? Assume a square can be occupied by multiple marbles.

Macrostate Corresponding basic states

1 1 2 4 3 6 4 4 5 1

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45. Find the total number of basic states of five particles being placed in a box with 100 equal compartments (a) if multiple particles can occupy the same compartment, and (b) if only one particle can be in each compartment. 46. Suppose you have a partitioned box with ten particles on each side of the partition. On the left you have very small particles that occupy a volume LδV . On the right, there are larger particles that take up more space per particle R LδV =8δV . If the box is 1 meter long, and the partition starts out pushed very far to the right, what will be the position of the partition when it reaches equilibrium? Hint: consider whether the results in this chapter assume that all particles occupy equal volumes. It may be useful to reexamine equations 19.3, 19.7, and see how they affect subsequent results. 47. You have 80 particles moving randomly in an isolated container with exactly 80 quantized units of energy. Which is less likely: finding all 80 units of energy on a single particle, or finding exactly one unit of energy on each particle? 48. A 1.0-meter box has a sliding partition initially fixed in place 75.0 cm from the left side of the box. 60 particles are placed on the right side of the partition, and 10 particles are placed in the left. If the partition is then allowed to move freely, what will be the change in entropy after the system reaches equilibrium? 49. A container has a volume of 1.00 cubic meter. Many bubbles of radius 1.50 cm are to be added to the container. Later, many bubbles of radius 2.00 cm are to added (once the smaller bubbles are removed). If the same number of bubbles are added in both cases, which case has the greater entropy, and by what factor? Treat the cubic region exactly large enough to enclose a bubble as the “chamber” that a bubble can occupy or not occupy. 50. Suppose 50 distinguishable particles have a volume of 9 310 m− , and are initially in a box with a volume of 31.00 cm . If the volume of the box is doubled (a) what by what factor does the number of basic states change, and (b) by what factor has the entropy increased? 51. A certain engine process allows the expansion of gas particles in a piston from a volume of 30.01 m to a volume of 30.100 m at a constant temperature. During this process, the entropy increases by 186.91 10× . How many particles were in the piston? 52. A change in volume occupied by a set of distinguishable particles increases the number of basic states by a factor of 6633, and increases the entropy by a factor of 1.10. How many distinguishable particles were there? 53. Consider two egg crates, one with spaces for a dozen eggs and one with spaces for half-a-dozen eggs. Both crates are filled with 6 eggs, all of different colors. What is the ratio of total basic states in the dozen crate vs. the half-dozen crate?

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54. Please rank the following equilibrium systems in terms of lowest to highest entropy. A. Your choice of any of one million different colors to assign to each of the estimated 70 sextillion ( )2170 10× stars in the Universe. B. One mole of gas, which can adsorb to any of 83 sites on a piece of nano-adsorbant material much smaller than the period at the end of this sentence. C. 0.8 moles of gas, which can adsorb to any of 106 sites on a piece of similar nano-adsorbant material. D. 100 trillion cells in the human body that can each be assigned with one of three levels of relative cellular health. E. 3 quadrillion ( )153 10× microbes in the average human body which can be assigned a description of either alive or dead. 55. An industrial safety device contains two separate volumes of gas as shown in Figure 19.55. Initially, 3 moles and 2 moles are each contained in their own volume of 3V , but upon actuation, each volume of gas is released into its own expansion chamber of volumes V. What is the total change in the entropy of the system?

Figure 19.55

Section 19.6 Dependence of Entropy on Energy 56. Four particles have the following velocities in (x,y,z) component notation:

1v =(-6,5,1) m/sr , 2v =(4,5,-2) m/sr , 3v =(7,0,8) m/sr , and 4v =(-4,-9,-6) m/sr . (a) What is the average velocity of these particles? (b) What is the root mean squared speed of these particles? (c) Why would equation 19.25 be unlikely to give the correct answer here? 57. A closed system contains 20.0 g of water vapor, and 1.00 g of diatomic oxygen. The water molecules initially have a rmsv of 400 m/s, and the oxygen molecules initially have a rmsv of 900 m/s. Once the system has equilibrated, what is the rmsv of the water? 58. A box is divided into quadrants of equal size, numbered 1-4. Each quadrant contains gaseous hexane ( 6 14C H ). The partition that separates the quadrants allows energy to be exchanged, but not particles. However, there is a tiny hole in the partition between quadrant 1 and 2 that allows gas particles to be exchanged, also. The initial mass of gas

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and rmsv are as follows: 1Q -3.00 g at 400 m/s, 2Q - 5.50 g at 500 m/s, 3Q - 2.00 g at 420 m/s, and 4Q - 6.75 g at 445 m/s. After the system sits for a long time, what is the final

rmsv of quadrant 2? 59. While watching a crowded market place filled with people, you calculate that the average of the square of people’s velocity to the North is 0.145 m/s. What are the people’s root mean squared speed? 60. What is the root mean squared speed of particles with the following velocities in (written in component form): (4,6,2) m/s, (8,-3,8) m/s, (7,1,6) m/s, and (8,9,5) m/s? 61. Suppose the absolute temperature is 400 K. If the rate of change of entropy with respect to energy is now doubled, what is the new absolute temperature? 62. Suppose a mole of gas initially has 75.0 J of thermal energy. If this energy is increased by 25.0 J, what will be the change in entropy? 63. A theory predicts a reaction which produces ideal gas molecules with the following speeds: 6.2 m/s, 7.4 m/s, 7.4 m/s, 7.8 m/s, 8.3 m/s, 12.6 m/s, 20.1 m/s, 20.1 m/s. (a) What is the average speed of the molecules? (b) What is the rms speed of the molecules? 64. An empty 2 liter soft drink bottle is filled with a hundredth of a mole of ideal monatomic gas at atmospheric pressure. (a) What is the average translational kinetic energy of each molecule in the bottle? (b) Compare with the kinetic energy of a single bacteria of mass 141 10 kg−× moving at a speed of 71 10 m/s−× and with a small garden slug, mass 1 g, slowly moving along at a speed of 0.01 m/s. 65. An engineered gas has thermal equilibrium empirically defined as

2/15eng th th

7 ln2

S N E NE= + . What is B1 k T for this gas?

66. A velocity selector in an air pollution control device measures the rms speed from an industrial process. If the rms speed exceeds a given limit, then the output is shut down to prevent the release of dangerous elements into the air. (Consider them to be ideal and monatomic.) The safety system is desired to trigger when the average density of the particles exceeds 5600 kg/m3. (a) At what threshold should the safety operator set the velocity selector if the gas output has a temperature of 936°C, and a maximum particle diameter of 10 nm? (b) As the particles rise and exit the smokestack, the 10 nm diameter particles coalesce to an average diameter of 1 mµ and cool to 190°C. What is the rms speed of the particles as they exit the smokestack? (Ignore viscosity and drag.) Section 19.7 Properties of a Monatomic Ideal Gas 67. A common room temperature is 24 degrees C. At that temperature, what is the average speed of helium? The mass of helium is -27

heliumm =6.645×10 kg .

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68. A particle accelerator applies a force to particles initially at rest that have a mass of

271.6727 10−× kg. This force is applied along the entire 100-meter length of the accelerator, which the particles traverse in 0.0179 s. The particles collide with a target and distribute their kinetic energy, elastically. The target is in thermal contact with the gas around it, which can be treated like a monatomic ideal gas. After the accelerator has been running for a long time, what is the temperature of the air? 69. Very dilute, cool gases A and B are separated by a partition, but allowed to exchange energy. After a long time, careful measurements show that the rmsv of gas A is six times greater than the rmsv of gas B. What is the relationship between the masses of the two types of gas molecules? 70. While working in a lab, you need to work with a certain dangerous gas. The gas is corrosive, meaning that with each collision between the gas and its container there is a certain probability (about a one in three million chance) that one gas molecule will react with one molecule of the container and weaken a tiny portion of the container wall. 4.00 g of this gas is stored in a spherical container with a volume of 6.45 liters, and walls that are1.00 cm (about 710 molecules) thick. The molecular weight of the gas is

-265.755×10 kg . The temperature of the gas is always close to room temperature, and certainly never rises above 30 degrees C. Your lab manager says that the gas container will be in the lab for your project for the next three weeks. Is this safe? Treat the gas as ideal. 71. What is the root mean square speed of helium atoms in a star where the temperature is 18,000 K? 72. An ideal gas with a density of 30.024 kg/m has an average kinetic energy of

217.5 10−× J per particle. If particles of the gas are colliding with a wall that has an area of 30.01 m , what is the average change in momentum per unit time the gas?

73. Suppose a piston compresses a gas so quickly that it has no time to exchange thermal energy with its surroundings (temperature is constant). The gas is compressed to one fifth its original volume. If the gas was initially at a pressure of 101,325 Pa, what was the final pressure? 74. What is the rms speed of a monatomic particle of mass 266.64 10 kg−× at a temperature of 273.15 K? 75. A mass of ideal monatomic gas molecules in a container with an rms speed of 42 m/s, decays into smaller molecules of equal total mass but with twice as many molecules and a temperature 1.2 times higher than the original temperature. What is the new rms speed of the smaller molecules?

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76. Neon atoms at a temperature of 260 K pass through a fan that gives each mole of neon gas an additional kinetic energy of 16 J of energy. What is the average temperature of an average neon atom immediately after coming through the fan? (b) If you found an increase in temperature in the previous part, how do you explain the fact that fans are usually used to make people cooler rather than hotter? 77. You wish to create a detector of dangerous radon gas using some improvised equipment available at your local hardware store. After some consultation with your physics book you create a sensor using the cooling core from an old refrigerator capable of cooling a sample volume of room air to 255 K, a detector tube of length 5 cm, a vacuum pump to evacuate the pipe, and a circuit which can time the flight of particles which enter the evacuated pipe. You intend to inject samples of the air into the evacuated pipe and measure their time of flight and rms speed from one end of the tube to the other to find if there is potentially radon gas in the air. (a) To approximately how many milliseconds should you set the time-of-flight sensor if you want to detect the monatomic radon molecules? (Neglect collision with other particles in the pipe.) (b) Can you suggest any improvements that could be made to this device to improve the accuracy of detection? (c) Do you think you will need to calibrate your system if you have to make the more realistic assumption that inter-particle collisions cannot be neglected? If so, why? Section 19.8 Entropy of a Monatomic Ideal Gas 78. What is the change in entropy per particle of an ideal gas that is heated from room temperature (300 K) to 400 K? Assume thermal equilibrium in both the initial and final cases. 79. Nitrogen is heated in a sealed container such that its rmsv changes from 350 m/s to 540 m/s. By how much has the entropy been increased per nitrogen molecule? Use

-26nitrogenm =4.652×10 kg .

80. An ideal gas is enclosed in a cubic container of side length L. Changes are made to the container and system such that the length of each side is halved and the pressure is tripled. (a) What is the change in entropy per particle for the gas in the box? (b) Is this system closed? 81. A 21.2 kg helium tank has been left in a hot car for hours. The temperature outside is 35.0 degrees C. After using all the helium in the tank to fill one hundred birthday balloons for a relative, the tank weighs 20.8 kg. You notice the balloons are shrinking over time. Each balloon is roughly spherical, and started out with a radius of 20.0 cm. Their radii have reached about 10.0 cm. Some of the children are afraid the balloons will deflate completely. You do some rough estimates and decide that the inward force from the walls of the balloon was initially 50.0 kPa, and that it must increase linearly with the surface area of the balloon. How can you explain the shrinking of the balloons? Should you prepare the children for the balloons deflating completely? If the balloons will remain partially inflated, what will be their equilibrium radius? What approximations or

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assumptions are you making in your calculations? Are your assumptions likely to be correct? 82. A monatomic ideal gas has its volume halved, while its pressure is held constant. What is the change in entropy per atom? 83. If 185,000 monatomic ideal gas atoms are heated from 300 K up to 500 K in a sealed container with volume of 0.056 3m , what is the change in entropy of the gas? 84. Suppose 3.65 moles of monatomic ideal gas are heated from 289 K to 458 K, and the entropy remains constant. If the initial volume of the sample was 0.098 3m , by what factor did the pressure increase or decrease during this process? 85. A mole of monatomic ideal gas is in thermal equilibrium with a surrounding liquid nitrogen bath, 77.2 K. After some time, the liquid nitrogen boils away to a surrounding encasement, and the gas and the nitrogen are at equilibrium in room temperature, 297 K. (a) What is the change in entropy of the gas? (b) The gaseous nitrogen is then recondensed and the monatomic ideal gas returns to its original temperature. What is the new total entropy change for just the monatomic ideal gas? (c) Where in this process did entropy increase as required by the second law of thermodynamics? 86. A single particle of monatomic ideal gas is cooled from 100 K to 50 K. (a) Calculate the change in entropy. (b) Does this result violate the second law of thermodynamics? If not, why not? 87. At thermal equilibrium, the entropy doubles for 10 particles of monatomic ideal gas, which is heated to 634 K at constant volume. What was the original temperature of the particles? 88. A monatomic particle moves on the x-axis with an rms speed of 200 m/s as shown in Figure 19.88. It enters an area that gives it an additional 100 m/s velocity directed in the y-axis. What is the change in entropy of the particle?

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Figure 19.88

89. Two volumes of ideal monatomic gas are insulated from each other as shown in Figure 19.89. Volume 1 contains 1 mole at equilibrium temperature T1. Volume 2 contains 0.5 moles at equilibrium temperature T2. Both volumes are allowed to go to the same thermal equilibrium at 156 K, but their volumes are prevented from mixing with each other. They both undergo an identical entropy change. (a) Find T1 in terms of T2. (b) If 2 100KT = , what is T1?

Figure 19.89