Document Sample

Lecture 4 Thermal Equilibrium, Temperature, and Entropy Recall that, since the systems 1 and 2 are independent of each other, the total multiplicity for ﬁxed (E, V, N ) is just: Ω(E, V, N ) = Ω(E, V, N, α) α = Ω(E, V, N, E1 , V1 , N1 ) (4.1) E1 ,V1 ,N1 = Ω1 (E1 , V1 , N1 )Ω2 (E2 , V2 , N2 ) E1 ,V1 ,N1 where the summations are performed over all possible macrostates of the combined system. If we let N1 and V1 be ﬁxed, then what does the total multiplicity look like as a function of E1 ? Ω (E ) Ω (E ) 1 1 1 1 0 E* E 1 1 Figure 4.1: Resultant multiplicity for two systems brought in contact which achieve thermal ∗ equlibrium at E1 • Note that this is a “sharp” function for large systems (refer to Prob. 2.21 in Tutorial 1) and only a small number of macrostates dominate the statistics of the combined system. ∗ • The most probable macrostate (conﬁguration) occurs at E1 which corresponds to ther- mal equilibrium. This is the macrostate which has the greatest multiplicity. • Since the ﬂuctuations are small, we can replace our averages, X over all macrostates by that of an average only over the most probable conﬁguration. This replacement should have minimal error. 4.1 Thermal Equilibrium Let us return to Eq. 4.1 but, as mentioned above, ﬁx N1 , V1 and, since E, V, N, V2 , N2 were also ﬁxed, thus E1 is the only independent variable. Thus, we have Ω(E) = Ω1 (E1 )Ω2 (E2 ) = Ω1 (E1 )Ω2 (E − E1 ) (4.2) E1 E1 ∗ By our Third Postulate, we expect that the most probable macrostate corresponding to E1 is such that, dΩ(E) ∂Ω1 dE1 ∂Ω2 dE2 = Ω2 + Ω1 = 01 dE ∗ E1 ∂E1 V1 ,N1 dE1 ∂E2 V2 ,N2 dE1 dE2 Recall that E2 = E − E1 and, thus, dE1 = −1, therefore, dividing by Ω1 Ω2 we ﬁnd, 1 ∂Ω1 1 ∂Ω2 = Ω1 ∂E1 V1 ,N1 Ω2 ∂E2 V2 ,N2 ∂ ln Ω1 ∂ ln Ω2 = ∂E1 V1 ,N1 ∂E2 V2 ,N2 We deﬁne a quantity σ, called the entropy as: the measure of the disorder of a system in a given macrosta σ(E, V, N ) ≡ ln Ω(E, V, N ) (4.3) which is just a pure number (dimensionless). Using this deﬁnition, we arrive at the funda- mental condition for thermal equilibrium for two systems in thermal contact, ∂σ1 ∂σ2 = (4.4) ∂E1 V1 ,N1 ∂E2 V2 ,N2 • At thermal equilibrium, there is no substantial heat transfer (although there will be small temperature ﬂuctuations). 1 When performing thermodynamic derivatives, such as these, we must be careful to clearly label the macrostate parameters, such as N1 , V1 , which are held ﬁxed while diﬀerentiating • If there is no heat transfer, then the two systems must be at the same temperature, i.e., τ1 = τ2 . • From Eq. 4.4, this implies that τ is a function of ∂σ/∂E. In particular, we deﬁne the fundamental temperautre, τ , as 1 ∂σ = . (4.5) τ ∂E V,N • Now the “standard” entropy S, which you are most likely familiar with, is Bolzmann’s Deﬁnitions, S = kB σ = kB ln Ω(E, V, N ) (4.6) where kB = 1.381 × 10−23 J/K is Boltzmann’s contant. • With this deﬁnition, we can express the “standard” temperature as, 1 ∂S = (4.7) T ∂E V,N Some observations about Entropy: 1. Entropy is additive, which can be seen from the deﬁnition of the total multiplicity for two systems in contact, Ω(E, V, N ) = Ω1 (E1 , V1 , N1 )Ω2 (E2 , V2 , N2 ) ∴, S(E, V, N ) = S1 (E1 , V1 , N1 ) + S2 (E2 , V2 , N2 ). This implies that entropy is an extensive quantity. Deﬁnitions: Extensive Quantity: proportional to the size of the system → double the size, double the quantity. Intensive Quantity: independent of the size of the system e.g. tem- perature, pressure. 2. Entropy is increased when constraints are removed: ↓ Constraints →↑ no. of accessible microstates of the system →↑ no. of degrees of freedom →↑ S (a) (b) Figure 4.2: By removing the partition, a signiﬁcant constraint on the system is lifted allowing for a greater number of accessible microstates and, hence, an increase in entropy Picture: Note: Even though we said at the beginning of the course that a conﬁguration like (a) (but without the barrier) was not impossible to observe, although highly improbable, we would never see the system go from this maximum probable conﬁguration (b) to the initial one (a) (without the barrier). That is, this is an irreversible process, even though the equations of motion are reversible in time. The Second Law of Thermodynamics or The Law of Increase of Entropy (Mul- tiplicity): The entropy of a closed system either remains constant or increases when a constraint is removed. Consider our two systems in thermal contact and evaluate the “rate of change of entropy”: dS ∂S dE1 ∂S dE2 = + dt ∂E1 dt ∂E2 dt or Clausius’ Principle: dS 1 1 dE1 = − >0 (4.8) dt T1 T2 dt If T1 < T2 , dE1 > 0, since heat ﬂows from the subsystem at higher temperature to that at dt lower temperature. Picture: T1 < T2 1 2 Heat Flow E1 E2 E S(E) Entropy Energy E2 E1 t t Figure 4.3: A demonstration of Clausius’ Principle: since T1 < T2 , heat will ﬂow from 2 → 1 and the entropy will increase until the two systems are at the same temperature and the entropy reaches a constant value (maximum threshold for a given total energy E) Let us apply these to a system composed of two Einstein solids in thermal contact. Example 4.1 Two Einstein Solids in Thermal Contact • Consider two Einstein solids, A and B, where NA = 4 and NB = 5 and they contain a total of 10 units of energy, qtotal = qA + qB = 10. • When brought into thermal contact, the two systems will exchange energy until they reach thermal equilibrium → which corresponds to the macrostate (of the combined sys- tem) which has the greatest multiplicity and, hence, the greatest probability of occuring • Calculate the multiplicities for all of the possible macrostates of the combined system? Which macrostate has the greatest probability of occuring? Table 4.1: Diﬀerent possible macrostates and multiplicities for a system of two Einstein solids, one containing four oscillators, the other having ﬁve oscillators, and both sharing a total of ten units of energy qA ΩA qB ΩB Ωtotal = ΩA ΩB 0 1 10 1001 1001 1 4 9 715 2860 2 10 8 495 4950 3 20 7 330 6600 4 35 6 210 7350 5 56 5 126 7056 6 84 4 70 5880 7 120 3 35 4200 8 165 2 15 2475 9 220 1 5 1100 10 286 0 1 286 8000 6000 Ωtot 4000 2000 0 0 2 4 6 8 10 qA Figure 4.4: Resultant multiplicity for two Einstein solids brought in contact which achieve thermal equlibrium at qA = 4

DOCUMENT INFO

Shared By:

Categories:

Tags:
Lecture Notes, thermal analysis, Mechanical properties, Michael Nielsen, Drude theory, free electron gas, Course Introduction, Thermal Conductivity, thermal properties, informal seminar series

Stats:

views: | 9 |

posted: | 3/22/2010 |

language: | English |

pages: | 6 |

OTHER DOCS BY akgame

How are you planning on using Docstoc?
BUSINESS
PERSONAL

By registering with docstoc.com you agree to our
privacy policy and
terms of service, and to receive content and offer notifications.

Docstoc is the premier online destination to start and grow small businesses. It hosts the best quality and widest selection of professional documents (over 20 million) and resources including expert videos, articles and productivity tools to make every small business better.

Search or Browse for any specific document or resource you need for your business. Or explore our curated resources for Starting a Business, Growing a Business or for Professional Development.

Feel free to Contact Us with any questions you might have.