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The Nature of Science and Physics

Kinematics

Introduction to One-Dimensional KinematicsDisplacementVectors, Scalars, and Coordinate SystemsTime, Velocity, and SpeedVideo: One-Dimensional KinematicsAccelerationMotion Equations for Constant Acceleration in One DimensionProblem-Solving Basics for One-Dimensional KinematicsFalling ObjectsGraphical Analysis of One-Dimensional Motion

Two-Dimensional Kinematics

Dynamics: Force and Newton's Laws of Motion

Introduction to Dynamics: Newton's Laws of MotionDevelopment of Force ConceptNewton's First Law of Motion: InertiaNewton's Second Law of Motion: Concept of a SystemNewton's Third Law of Motion: Symmetry in ForcesVideo: Newton's LawsNormal, Tension, and Other Examples of ForcesProblem-Solving StrategiesFurther Applications of Newton's Laws of MotionExtended Topic: The Four Basic Forces—An Introduction

Further Applications of Newton's Laws: Friction, Drag, and Elasticity

Uniform Circular Motion and Gravitation

Introduction to Uniform Circular Motion and GravitationRotation Angle and Angular VelocityCentripetal AccelerationCentripetal ForceFictitious Forces and Non-inertial Frames: The Coriolis ForceNewton's Universal Law of GravitationVideo: GravitationSatellites and Kepler's Laws: An Argument for Simplicity

Work, Energy, and Energy Resources

Introduction to Work, Energy, and Energy ResourcesWork: The Scientific DefinitionKinetic Energy and the Work-Energy TheoremGravitational Potential EnergyVideo: Potential and Kinetic EnergyConservative Forces and Potential EnergyNonconservative ForcesConservation of EnergyPowerWork, Energy, and Power in HumansWorld Energy Use

Linear Momentum and Collisions

Rotational Motion and Angular Momentum

Introduction to Rotational Motion and Angular MomentumAngular AccelerationKinematics of Rotational MotionVideo: Rotational MotionDynamics of Rotational Motion: Rotational InertiaRotational Kinetic Energy: Work and Energy RevisitedAngular Momentum and Its ConservationVideo: Angular MomentumCollisions of Extended Bodies in Two DimensionsGyroscopic Effects: Vector Aspects of Angular Momentum

Statics and Torque

Fluid Statics

Introduction to Fluid StaticsWhat Is a Fluid?DensityPressureVariation of Pressure with Depth in a FluidPascal's PrincipleGauge Pressure, Absolute Pressure, and Pressure MeasurementArchimedes' PrincipleVideo: BuoyancyCohesion and Adhesion in Liquids: Surface Tension and Capillary ActionPressures in the Body

Fluid Dynamics and Its Biological and Medical Applications

Introduction to Fluid Dynamics and Biological and Medical ApplicationsFlow Rate and Its Relation to VelocityBernoulli's EquationVideo: Fluid FlowThe Most General Applications of Bernoulli's EquationViscosity and Laminar Flow; Poiseuille's LawThe Onset of TurbulenceMotion of an Object in a Viscous FluidMolecular Transport Phenomena: Diffusion, Osmosis, and Related Processes

Temperature, Kinetic Theory, and the Gas Laws

Heat and Heat Transfer Methods

Thermodynamics

Introduction to ThermodynamicsThe First Law of ThermodynamicsThe First Law of Thermodynamics and Some Simple ProcessesIntroduction to the Second Law of Thermodynamics: Heat Engines and Their EfficiencyCarnot's Perfect Heat Engine: The Second Law of Thermodynamics RestatedApplications of Thermodynamics: Heat Pumps and RefrigeratorsEntropy and the Second Law of Thermodynamics: Disorder and the Unavailability of EnergyStatistical Interpretation of Entropy and the Second Law of Thermodynamics: The Underlying Explanation

Oscillatory Motion and Waves

Introduction to Oscillatory Motion and WavesHooke's Law: Stress and Strain RevisitedPeriod and Frequency in OscillationsSimple Harmonic Motion: A Special Periodic MotionVideo: Harmonic MotionThe Simple PendulumEnergy and the Simple Harmonic OscillatorUniform Circular Motion and Simple Harmonic MotionDamped Harmonic MotionForced Oscillations and ResonanceWavesSuperposition and InterferenceEnergy in Waves: Intensity

Physics of Hearing

- Identify probabilities in entropy.
- Analyze statistical probabilities in entropic systems.

The various ways of formulating the second law of thermodynamics tell what happens rather than why it happens. Why should heat transfer occur only from hot to cold? Why should energy become ever less available to do work? Why should the universe become increasingly disorderly? The answer is that it is a matter of overwhelming probability. Disorder is simply vastly more likely than order.

When you watch an emerging rain storm begin to wet the ground, you will notice that the drops fall in a disorganized manner both in time and in space. Some fall close together, some far apart, but they never fall in straight, orderly rows. It is not impossible for rain to fall in an orderly pattern, just highly unlikely, because there are many more disorderly ways than orderly ones. To illustrate this fact, we will examine some random processes, starting with coin tosses.

$\begin{array}{ll}5\text{ heads,}&0\text{ tails}\\4\text{ heads,}&1\text{ tail}\\3\text{ heads,}&2\text{ tails}\\2\text{ heads,}&3\text{ tails}\\1\text{ head,}&4\text{ tails}\\0\text{ head,}&5\text{ tails}\end{array}\\$

These are what we call macrostates. A Using this nomenclature, a system of 5 coins has the 6 possible macrostates just listed. Some macrostates are more likely to occur than others. For instance, there is only one way to get 5 heads, but there are several ways to get 3 heads and 2 tails, making the latter macrostate more probable. Table 1 lists of all the ways in which 5 coins can be tossed, taking into account the order in which heads and tails occur. Each sequence is called a

Table 1. 5-Coin Toss | ||
---|---|---|

Individual microstates | Number of microstates | |

5 heads, 0 tails | HHHHH | 1 |

4 heads, 1 tail | HHHHT, HHHTH, HHTHH, HTHHH, THHHH | 5 |

3 heads, 2 tails | HTHTH, THTHH, HTHHT, THHTH, THHHT HTHTH, THTHH, HTHHT, THHTH, THHHT | 10 |

2 heads, 3 tails | TTTHH, TTHHT, THHTT, HHTTT, TTHTH, THTHT, HTHTT, THTTH, HTTHT, HTTTH | 10 |

1 head, 4 tails | TTTTH, TTTHT, TTHTT, THTTT, HTTTT | 5 |

0 heads, 5 tails | TTTTT | 1 |

Total: 32 |

The two most orderly possibilities are 5 heads or 5 tails. (They are more structured than the others.) They are also the least likely, only 2 out of 32 possibilities. The most disorderly possibilities are 3 heads and 2 tails and its reverse. (They are the least structured.) The most disorderly possibilities are also the most likely, with 20 out of 32 possibilities for the 3 heads and 2 tails and its reverse. If we start with an orderly array like 5 heads and toss the coins, it is very likely that we will get a less orderly array as a result, since 30 out of the 32 possibilities are less orderly. So even if you start with an orderly state, there is a strong tendency to go from order to disorder, from low entropy to high entropy. The reverse can happen, but it is unlikely.

Table 2. 100-Coin Toss | ||
---|---|---|

Macrostate | Number of microstates | |

Heads | Tails | (W) |

100 | 0 | 1 |

99 | 1 | 1.0 × 10^{2 } |

95 | 5 | 7.5 × 10^{7 } |

90 | 10 | 1.7 × 10^{13 } |

75 | 25 | 2.4 × 10^{23 } |

60 | 40 | 1.4 × 10^{28 } |

55 | 45 | 6.1 × 10^{28 } |

51 | 49 | 9.9 × 10^{28 } |

50 | 50 | 1.0 × 10^{29 } |

49 | 51 | 9.9 × 10^{28 } |

45 | 55 | 6.1 × 10^{28 } |

40 | 60 | 1.4 × 10^{28 } |

25 | 75 | 2.4 × 10^{23 } |

10 | 90 | 1.7 × 10^{13 } |

5 | 95 | 7.5 × 10^{7 } |

1 | 99 | 1.0 × 10^{2 } |

0 | 100 | 1 |

Total: 1.27 × 10^{30} |

The most likely conditions (or macrostates) for a gas are those we see all the time—a random distribution of atoms in space with a Maxwell-Boltzmann distribution of speeds in random directions, as predicted by kinetic theory. This is the most disorderly and least structured condition we can imagine. In contrast, one type of very orderly and structured macrostate has all of the atoms in one corner of a container with identical velocities. There are very few ways to accomplish this (very few microstates corresponding to it), and so it is exceedingly unlikely ever to occur. (See Figure 2b.) Indeed, it is so unlikely that we have a law saying that it is impossible, which has never been observed to be violated—the second law of thermodynamics.

The disordered condition is one of high entropy, and the ordered one has low entropy. With a transfer of energy from another system, we could force all of the atoms into one corner and have a local decrease in entropy, but at the cost of an overall increase in entropy of the universe. If the atoms start out in one corner, they will quickly disperse and become uniformly distributed and will never return to the orderly original state (Figure 2b). Entropy will increase. With such a large sample of atoms, it is possible—but unimaginably unlikely—for entropy to decrease. Disorder is vastly more likely than order.

The arguments that disorder and high entropy are the most probable states are quite convincing. The great Austrian physicist Ludwig Boltzmann (1844–1906)—who, along with Maxwell, made so many contributions to kinetic theory—proved that the entropy of a system in a given state (a macrostate) can be written as

$\Delta{S}=\frac{Q}{T}\\$

, which we have used extensively.Thus the second law of thermodynamics is explained on a very basic level: entropy either remains the same or increases in every process. This phenomenon is due to the extraordinarily small probability of a decrease, based on the extraordinarily larger number of microstates in systems with greater entropy. Entropy

where the subscript i stands for the initial 60 heads and 40 tails state, and the subscript f for the final 50 heads and 50 tails state. Substituting the values for

$\begin{array}{lll}\Delta{S}&=&\left(1.38\times10^{-23}\text{ J/K}\right)\left[\ln\left(1.0\times10^{29}\right)-\ln\left(1.4\times{10}^{29}\right)\right]\\\text{ }&=&2.7\times10^{-23}\text{ J/K}\end{array}\\$

*Examine the situation to determine if entropy is involved.**Identify the system of interest and draw a labeled diagram of the system showing energy flow.**Identify exactly what needs to be determined in the problem (identify the unknowns).*A written list is useful.*Make a list of what is given or can be inferred from the problem as stated (identify the knowns).*You must carefully identify the heat transfer, if any, and the temperature at which the process takes place. It is also important to identify the initial and final states.*Solve the appropriate equation for the quantity to be determined (the unknown).*Note that the change in entropy can be determined between any states by calculating it for a reversible process.*Substitute the known value along with their units into the appropriate equation, and obtain numerical solutions complete with units.**To see if it is reasonable: Does it make sense?*For example, total entropy should increase for any real process or be constant for a reversible process. Disordered states should be more probable and have greater entropy than ordered states.

- Disorder is far more likely than order, which can be seen statistically.
- The entropy of a system in a given state (a macrostate) can be written as
*S*=*k*ln*W*,*k*= 1.38 × 10^{−23}J/K is Boltzmann’s constant, and ln*W*is the natural logarithm of the number of microstates*W*corresponding to the given macrostate.

- Explain why a building made of bricks has smaller entropy than the same bricks in a disorganized pile. Do this by considering the number of ways that each could be formed (the number of microstates in each macrostate).

- Using Table 2, verify the contention that if you toss 100 coins each second, you can expect to get 100 heads or 100 tails once in 2 × 10
^{22}years; calculate the time to two-digit accuracy. - What percent of the time will you get something in the range from 60 heads and 40 tails through 40 heads and 60 tails when tossing 100 coins? The total number of microstates in that range is 1.22 × 10
^{30}. (Consult Table 2.) - (a) If tossing 100 coins, how many ways (microstates) are there to get the three most likely macrostates of 49 heads and 51 tails, 50 heads and 50 tails, and 51 heads and 49 tails? (b) What percent of the total possibilities is this? (Consult Table 2.)
- (a) What is the change in entropy if you start with 100 coins in the 45 heads and 55 tails macrostate, toss them, and get 51 heads and 49 tails? (b) What if you get 75 heads and 25 tails? (c) How much more likely is 51 heads and 49 tails than 75 heads and 25 tails? (d) Does either outcome violate the second law of thermodynamics?
- (a) What is the change in entropy if you start with 10 coins in the 5 heads and 5 tails macrostate, toss them, and get 2 heads and 8 tails? (b) How much more likely is 5 heads and 5 tails than 2 heads and 8 tails? (Take the ratio of the number of microstates to find out.) (c) If you were betting on 2 heads and 8 tails would you accept odds of 252 to 45? Explain why or why not.

Table 3. 10-Coin Toss Macrostate Number of Microstates Heads Tails ( *W*)10 0 1 9 1 10 8 2 45 7 3 120 6 4 210 5 5 252 4 6 210 3 7 120 2 8 45 1 9 10 0 10 1 **Total: 1024** - (a) If you toss 10 coins, what percent of the time will you get the three most likely macrostates (6 heads and 4 tails, 5 heads and 5 tails, 4 heads and 6 tails)? (b) You can realistically toss 10 coins and count the number of heads and tails about twice a minute. At that rate, how long will it take on average to get either 10 heads and 0 tails or 0 heads and 10 tails?
- (a) Construct a table showing the macrostates and all of the individual microstates for tossing 6 coins. (Use Table 3 as a guide.) (b) How many macrostates are there? (c) What is the total number of microstates? (d) What percent chance is there of tossing 5 heads and 1 tail? (e) How much more likely are you to toss 3 heads and 3 tails than 5 heads and 1 tail? (Take the ratio of the number of microstates to find out.)
- In an air conditioner, 12.65 MJ of heat transfer occurs from a cold environment in 1.00 h. (a) What mass of ice melting would involve the same heat transfer? (b) How many hours of operation would be equivalent to melting 900 kg of ice? (c) If ice costs 20 cents per kg, do you think the air conditioner could be operated more cheaply than by simply using ice? Describe in detail how you evaluate the relative costs.

$\begin{array}{ll}\left(6.35\times {\text{10}}^{\text{29}}\text{s}\right)\left(\frac{\text{1 h}}{\text{3600 s}}\right)& \left(\frac{\text{1 d}}{\text{24 h}}\right)\left(\frac{\text{1 y}}{\text{365.25 d}}\right)\\ =& 2.0\times {\text{10}}^{\text{22}}\text{y}\end{array}\\$

3. (a) 3.0 × 10

5. (a) −2.38 × 10

7. (b) 7; (c) 64; (d) 9.38%; (e) 3.33 times more likely (20 to 6)