LECTURE 10

pdf

School

University of Ottawa *

*We aren’t endorsed by this school

Course

1321

Subject

Physics

Date

Oct 30, 2023

Type

pdf

Pages

Uploaded by ChancellorTitanium12263

PHY 1321/PHY1331 Principles of Physics I Fall 2023 Dr. Andrzej Czajkowski 95 LECTURE 10 ENTROPY SECOND LAW OF THERMODYNAMICS 3 DEMONSTRATIONS: Movies 6 Class Quiz Questions 4 Suggested Problems READING ASSIGNMENT: Chapter22

PHY 1321/PHY1331 Principles of Physics I Fall 2023 Dr. Andrzej Czajkowski 96 Entropy • Entropy, S , is a state variable related to the second law of thermodynamics • The importance of entropy grew with the development of statistical mechanics • A main result is isolated systems tend toward disorder and entropy is a natural measure of this disorder Entropy and the Second Law • Entropy is a measure of disorder. • The entropy of the Universe increases in all real processes. • This is another statement of the second law of thermodynamics. Entropy and Heat • The original formulation of entropy dealt with the transfer of energy by heat in a reversible process. • Let dQ r be the amount of energy transferred by heat when a system follows a reversible path. • The change in entropy, d S is: • The change in entropy depends only on the endpoints and is independent of the path followed • The entropy change for an irreversible process can be determined by calculating the change in entropy for a reversible process that connects the same initial and final points • dQr is measured along a reversible path, even if the system may have followed an irreversible path • The meaningful quantity is the change in entropy and not the entropy itself • For a finite process, • The change in entropy of a system going from one state to another has the same value for all paths connecting the two states • The finite change in entropy depends only on the properties of the initial and final equilibrium states Δ S for a Reversible Cycle This integral symbol indicates the integral is over a closed path This demonstrates that Δ S depends on only the initial and final states and not the path between the states • In general, the total entropy and therefore the total disorder always increases in an irreversible process T dQ dS r = ò = - = T dQ S S Δ S r i f ò = = - = 0 T dQ S S Δ S i f

PHY 1321/PHY1331 Principles of Physics I Fall 2023 Dr. Andrzej Czajkowski 97 The total entropy of an isolated system undergoes a change that cannot decrease. This is another statement of the second law of thermodynamics • If the process is irreversible, then the total entropy of an isolated system always increases – In a reversible process, the total entropy of an isolated system remains constant – • The change in entropy of the Universe must be greater than zero for an irreversible process and equal to zero for a reversible process Heat Death of the Universe • Ultimately, the entropy of the Universe should reach a maximum value • At this value, the Universe will be in a state of uniform temperature and density • All physical, chemical, and biological processes will cease – The state of perfect disorder implies that no energy is available for doing work – This state is called the heat death of the Universe Δ S for a Quasi-Static, Reversible Process • Assume an ideal gas undergoes a quasi-static, reversible process • Its initial state has T i and V i • Its final state has T f and V f • The change in entropy is Δ S in Thermal Conduction Process • • • • The cold reservoir absorbs Q and its entropy changes by Q/T c • At the same time, the hot reservoir loses Q and its entropy changes by -Q/T h • Since T h > T c , the increase in entropy in the cold reservoir is greater than the decrease in entropy in the hot reservoir • Therefore, ΔS U > 0 – For the system and the Universe i f i f V f i r V V nRln T T ln nC T dQ Δ S + = = ò

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

PHY 1321/PHY1331 Principles of Physics I Fall 2023 Dr. Andrzej Czajkowski 98 Δ S in a Free Expansion • Consider an adiabatic free expansion • Q = 0 but cannot be used since that is for an irreversible process • For an isothermal process, this becomes • Since Vf > Vi , Δ S is positive • This indicates that both the entropy and the disorder of the gas increase as a result of the irreversible adiabatic expansion Δ S in Calorimetric Processes • The process is irreversible because the system goes through a series of non-equilibrium states • Assuming the specific heats remain constant and no mixing takes place: • • – – If mixing takes place, this result applies only to identical substances Δ S will be positive and the entropy of the Universe increases Microstates vs. Macrostates. Modern Statistical Mechanics • A microstate is a particular configuration of the individual constituents of the system • A macrostate is a description of the conditions from a macroscopic point of view – It makes use of macroscopic variables such as pressure, density, and temperature for gases THERE ARE MANY MICROSTATES REALIZING EACH MACROSTATE • For a given macrostate, a number of microstates are possible • It is assumed that all microstates are equally probable • When all possible macrostates are examined, it is found that macrostates associated with disorder have far more microstates than those associated with order Microstates vs. Macrostates, Probabilities • The probability of a system moving in time from an ordered macrostate to a disordered macrostate is far greater than the probability of the reverse – There are more microstates in a disordered macrostate • If we consider a system and its surroundings to include the Universe, the Universe is always moving toward a macrostate corresponding to greater disorder • We can treat entropy from a microscopic viewpoint through statistical analysis of molecular motions • A connection between entropy and the number of microstates ( W ) for a given macrostate is: S = k B ln W The more microstates that correspond to a given macrostate, the greater the entropy of that microstate. • This shows that entropy is a measure of disorder 2 f 2 2 1 f 1 1 T T 2 2 T T 1 1 T T 2 T T 1 f i T T ln c m T T ln c m T dT c m T dT c m T dQ T dQ T dQ Δ S f f 1 f f 1 + = + = + = = ò ò ò ò ò 2 2 i f f i r V V nRln T dQ Δ S = = ò

PHY 1321/PHY1331 Principles of Physics I Fall 2023 Dr. Andrzej Czajkowski 99 Entropy, Marble Example • Suppose you have a bag with 50 red marbles and 50 green marbles • You draw a marble, record its color, return it to the bag, and draw another • Continue until four marbles have been drawn • What are the possible macrostates, and what are their probabilities? Entropy, Marble Example, Results MACROSTATE POSSIBLE MICROSTATES TOTAL NUMBER OF MICROSTATES 4R RRRR 1 1G,3R RRRG,RRGR,RGRR,GRRR 4 2G,2R RRGG,GRRG,RGRG,RGGR,GRGR,GGRR 6 3G,1R GGGR,GGRG,GRGG,RGGG 4 4G GGGG 1 • The most ordered microstates are the least likely • The most disordered microstate is the most likely Entropy Dice Example: Consider 2 dice throw. There is 11 possible outcomes: How many different ways one can obtain each result? EACH TOTAL IS A MACROSTATE (THE NUMBERS OF COMBINATIONS TO OBTAIN THIS TOTAL IS A NUMBER OF MICROSTATES ) Total 1 - 0 combination Total 2 - 1 combination (1,1) Total 3 - 2 combinations (1,2); (2,1) Total 4 - 3 combinations (1,3); (2,2);(3,1) Total 5 - 4 combinations (1,4); (2,3);(3,2);(4,1) Total 6 - 5 combinations (1,5); (2,4);(3,3);(4,2);(5,1) Total 7 - 6 combinations (1,6); (2,5);(3,4);(4,3);(5,2);(6,1) Total 8 - 5 combinations (2,6); (3,5);(4,4);(5,3);(6,2) Total 9 - 4 combinations (3,6); (4,5); (5,4);(6,3) Total 10 - 3 combinations (4,6); (5,5); (6,4) Total 11 - 2 combinations (5,6); (6,5) Total 12 - 1 combination (6,6) If we consider the total number thrown to be a macrostate then we see right away that 7 is the most probable macrostate. 2 and 12 are the least probable macrostates. 1, 13 and larger numbers are impossible macrostates Entropy, Molecule Example • One molecule in a two-sided container has a 1-in-2 chance of being on the left side • Two molecules have a 1-in-4 chance of being on the left side at the same time • Three molecules have a 1-in-8 chance of being on the left side at the same time • Consider 100 molecules in the container • The probability of all of the molecules to congregate in the half (left) of the container (½) 100 • The probability of separating 50 fast molecules on one side and 50 slow molecules on the other side is (½) 100 • If we have one mole of gas, this is found to be extremely improbable

PHY 1321/PHY1331 Principles of Physics I Fall 2023 Dr. Andrzej Czajkowski 100 Entropy and the arrow of time The “natural direction of irreversible processes results in the perceived “arrow of time.” Observing the irreversible processes run in the opposite direction would appear to the observer as running backward in time. Such “local time inversions” are not strictly speaking forbidden but they are very unlikely to happen. Boltzmann famously predicted that if the universe was old enough and large enough such local time inversions could and in fact should be observed . ENTROPY AND THE SECOND LAW OF THERMODYNAMICS This far we discussed three different formulations of the Second Law of Thermodynamics: The first two were expressing Second Law in terms of efficiencies of heat engines and refrigerators. Second Law of Thermodynamics forbids existence of heat engines with 100 % efficiency. It also forbids existence of the refrigerators what pump heat from colder to hotter reservoir at no expense to the user. BTW: If we had machine that does that we could construct the Perpetuum Mobile of Second Kind(*). The third way to express the Second Law of Thermodynamics is related to concept of entropy: It states that in any real process the total change of entropy is never negative Δ࠵? ≥ 0 In other word, entropy stays constant or always increases as result of thermodynamic process. Many of you also know that the increasing entropy is associated with increasing disorder in nature, so it is the second law that tells us how systems will evolve in nature. This statement is frequently associated with the entropy as defined in thermodynamics: Δ࠵? = & ࠵?࠵? ࠵? ! " Unfortunately, the above definition of entropy does not really imply increased disorder as result of the increased entropy.

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

PHY 1321/PHY1331 Principles of Physics I Fall 2023 Dr. Andrzej Czajkowski 101 To see link between the disorder and entropy we need to work with another definition, the one proposed by Ludwig Boltzmann ࠵? = ࠵?࠵?࠵?࠵? k is Boltzmann constant, W is the total number of the microstates realizing certain macrostate. To understand this definition, one needs to discuss microstates vs. macrostates, as it is done in Statistical Mechanics. Concept is not very difficult: Macro-state is the state of the whole system (let’s say gas) characterized by various macro parameters (T, P, V). Micro-state is the unique state of gas in which we know all the particular positions and their velocities. I hope we can all can appreciate the astronomical number of microstates corresponding to each macro-state of the system. W in the Boltzmann formula for S is the total sum of all the possible microstates that could be created with constraints imposed by the MACROSTATE ( P,V and thus T and thus Energy!) W is a huge number!! And entropy is simply scaled (natural logarithm: ln ) measure of this number! NOW, LET’S THINK ABOUT THE MEANING OF SECOND LAW SECOND LAW: Δ࠵? = ࠵? ! − ࠵? " = ࠵?࠵?࠵?࠵? ! − ࠵?࠵?࠵?࠵? ! = ࠵?࠵?࠵? ࠵? ! ࠵? " ≥ 0 Systems evolve from Low S (Low W) to High S ( high W) SYSTEMS EVOLVE FROM STATES CHARACTERIZED BY LOW W TO STATES CHARTACTERIZED BY HIGH W!

PHY 1321/PHY1331 Principles of Physics I Fall 2023 Dr. Andrzej Czajkowski 102 This defines the “natural order” of processes which we perceive as ARROW OF TIME. It is truly amazing result! EXAMPLE: Let’s think about an w mole of an ideal gas with total energy of 1000J. This energy could be distributed evenly between all of the gas constituents. It could be given to just 1 molecule of gas ( while all the others have 0 energy) – there would be N A combinations of such situation ( as the energy travels from one molecule to another) It could also be given to 2 molecules picked from N A with N A (N A -1) combinations of such situation and so on. If we transfer all of the energy to small section of gas – say to 1000 molecules in the whole mole, we create a state with very small W compared to the state where all molecules have small portion of the total energy. Or, if we create the state in which all molecules are occupying a small fraction of the volume it will not stay there, simply because in such state its W is less than in the state when it occupies the whole volume! To better count the microstates of the gas one could use the 6N dimensional phase space in which each point corresponds to the particular set of values of all coordinates of the has molecules (3N) and all coordinates of their velocities (3N) System evolution in time would correspond to motion of such point in this 6N dimensional space. It could be shown that the total volume in such 6N dimensional space that is available to the system is limited—this would is another way to find W.

PHY 1321/PHY1331 Principles of Physics I Fall 2023 Dr. Andrzej Czajkowski 103 Perpetuum Mobile of Second Kind If there is a heat pump that takes the heat from the cold reservoir and dumps it in the hot reservoir without our input energy (work),then we start this pump and simultaneously we start the heat engine ( of limited efficiency) to work between these two reservoirs. Heat engine will take some heat from hot reservoir, perform work for us, and dump some of it into the cold reservoir. Normally it would go on as long as the long the hot reservoir is at higher T than cold reservoir. Performing work by an engine would bring up to the cold reservoir to the same temperature as the hot reservoir. However, since our perfect heat pump cools the cold reservoir while warming up the hot reservoir, this will never happen! As result, we have machine that will provide work for free forever! INITIAL STATE SMALL VOLUME IN THE 6N-DIMENSIONAL PHASE SPACE: LOW W FINAL STATE LARGE VOLUME IN THE 6N-DIMENSIONAL PHASE SPACE: HIGH W

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

PHY 1321/PHY1331 Principles of Physics I Fall 2023 Dr. Andrzej Czajkowski 104 SUGGESTED PROBLEMS 1Demonstrate that the change of entropy in the quasi-static revoersble process is indeed given by : 2 A 1.00-mol sample of H 2 gas is contained in the left-hand side of the container shown in Figure P22.45, which has equal volumes left and right. The right-hand side is evacuated. When the valve is opened, the gas streams into the right-hand side. What is the final entropy change of the gas? Does the temperature of the gas change? 3. A 1.00-mol sample of a diatomic ideal gas, initially having pressure P and volume V , expands so as to have pressure 2 P and volume 2 V . Determine the entropy change of the gas in the process. 4. A 2.00-L container has a center partition that divides it into two equal parts, as shown in Figure P22.46. The left side contains H 2 gas, and the right side contains O 2 gas. Both gases are at room temperature and at atmospheric pressure. The partition is removed and the gases are allowed to mix. What is the entropy increase of the system? i f i f V f i r V V nRln T T ln nC T dQ Δ S + = = ò

PHY 1321/PHY1331 Principles of Physics I Fall 2023 Dr. Andrzej Czajkowski 105 EXTRA READING FOR SERIOUS STUDENTS ONLY: Once the Entropy of the system is established one may use the Maxwell’s Relations to obtain other Thermodynamic Potentials: S - The Entropy U - The internal energy dU = TdS - pdV H - The Enthalpy dH = TdS + Vdp A - The Helmholtz’ Free Energy dA = -SdT - Vdp G - The Gibbs’ Free Energy dG = -Sdt + Vdp State variables: V – volume; P – pressure; T - temperature The four most common Maxwell relations are the equalities of the second derivatives of each of the four thermodynamic potentials, with respect to their thermal natural variable (temperature T or entropy S ) and their mechanical natural variable (pressure p or volume V ) where the potentials as functions of their natural thermal and mechanical variables are: There are more relations to be obtained, The enthalpy and free energy are very important in discussion of chemical in bio-chemical processes! You will see more of these as you progress through the senior courses. V S S p V T ÷ ø ö ç è æ ¶ ¶ - = ÷ ø ö ç è æ ¶ ¶ p S S V p T ÷ ø ö ç è æ ¶ ¶ + = ÷ ÷ ø ö ç ç è æ ¶ ¶ V T T p V S ÷ ø ö ç è æ ¶ ¶ + = ÷ ø ö ç è æ ¶ ¶ p V T V p S ÷ ø ö ç è æ ¶ ¶ - = ÷ ÷ ø ö ç ç è æ ¶ ¶

PHY 1321/PHY1331 Principles of Physics I Fall 2023 Dr. Andrzej Czajkowski 106 THERMODYNAMICS SUMMARY Probability of finding the speed of a particle in the range (v;v+dv )is: Integrals: ΔEint = Q + W pV=nRT Change ΔEint W Q ΔS P = const nC v ΔT -p(V f -V i ) nC p ΔT V = const nC v ΔT 0 nC v ΔT T = const 0 Q = 0 nC v ΔT 0 0 ΔL = αLΔT ΔS = βSΔT ΔV = γVΔT P = e σ A T 4; σ =5.67x 10 -8 W/(K 4 m 2 ) Q = mcΔT Q = Lm c(water) = 4186 J/(kg C); c(ice) = 2090 J/(kg C); c(steam) = 2010J/(kg C) L(melting) = 3.33x10 5 J/kg L (vaporization) = 2.26x10 6 J/kg dv e v kT m dv v P kT mv 2 2 2 3 2 2 1 4 ) ( - ú û ù ê ë é = p p 2 1 2 ú û ù ê ë é = m kT v MP 2 1 3 ú û ù ê ë é = m kT v rms 2 1 8 ú û ù ê ë é = m kT v avg p > < = 2 3 1 v p r V Nm = r a dx e ax p 2 1 0 2 = ò + ¥ - a dx xe ax 2 1 0 2 = ò + ¥ - 3 0 2 4 1 2 a dx e x ax p = ò + ¥ - 2 0 3 2 1 2 a dx e x ax = ò + ¥ - 5 0 4 8 3 2 a dx e x ax p = ò + ¥ - 15 1 4 0 3 p = - ò + ¥ dx e x x ò = D T dQ S i f p T T nC ln i f V T T nC ln i f V V nRT ln - i f V V nRT ln i f V V nR ln ( ) i i f f V p V p - - 1 1 g . const pV = g V P C C = g R C C v p = - H C H L H CRN T T Q Q Q Q W - = - = = Î 1 it for pay we what want we what COP = dx dT kA P =

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version