STATISTICAL MECHANICS

Transcription

1 STATISTICAL MECHAICS PD Dr. Christian Holm PART Entropy and Temperature

2 Fundamental assumption a closed system is equally likely to be in any of its accessible micro- states, and all accessible micro- states are assumed to be equally probable. A closed system will have constant enery E, a constant number of particles, constant volume V, and constant values of all external parameters that may influence the system ravity, electric fields, etc. A micro-state is accessible, if its properties are compatible with the system parameters.

3 Probability The probability to find the system in this state is Ps /, if s is accessible, and Ps 0 otherwise The total probability to find the system in some state is unity P s s Averae value of a quantity X is then iven by s X X s P s An ensemble of systems is composed of very many systems here systems, all constructed alike. For a closed system s X X s/ Ensemble averae

4 Example: Construction of an ensemble 5 > 53 total states, for s/ we have 5! , 0 3!! 3

5 Thermal contact between two systems o particle transfer

6 The most probable confiuration: The enery of the combined system is s s s -mbs s -mbs The total number of particles is The enery can be redistributed in many ways between the two systems such that const. A confiuration is defined as the set of all states with specified values of s and s The total number of states in one confiuration is iven by, s, s

7 The number of accessible states of the combined systems and is then iven by / s, s, s s s / where s s s The confiuration where is maximal is called the most probable confiuration, sˆ, s sˆ

8 The most probable confiuration dominates properties, ˆ, s ˆ All possible confiurations

9 Accessible state Most probable state Wall Equilibrium state does not depend on time Microscopic laws reversible in time Macroscopic laws not arrow of time!

10 Thermal Equilibrium,,, The larest term in the sum will overn the properties const. 0 d d 0 ; d d d 0

11 This may be rewritten as lo lo, lo, Which property is equal in thermal equilibrium?? The temperature T! Define a pure number s, called the entropy, via

12 The Temperature The derivative of the entropy is used to define the temperature k B T, V k B.38x 0.38x joules/kelvin ers/kelvin Boltzmann constant the fundamental temperature dimension of enery and thereforeτ τ k B T

13 Comment τ We could also invert this to write τ which has a different meanin τ τ, whereas τ τ, Τ can be expressed as a function of different independent variables. These can vary in real experiments or theoretical questions

14 The Entropy loarithm of the number of states accessible to the system In classical thermodynamics the entropy S is defined as: T S and therefore S kb We will call S the conventional entropy The entropy of two independent systems is the sum of the separate entropies consequence of that is the product of and and the lo definition in S >

15 Other properties of entropy: additivity: lo * lo lo S is an extensive variable like, V, Independence of the path way in which thermal equilibrium was established. 3 the larer the number of accessible states, the larer is the entropy.

16 [ ] ] [ ] [ ] [ i f i f i f δ δ What happens to the entropy if heat flows from warm to cold?

17 For τ > τ the r.h.s. is positive, so entropy increases when enery flows from the hot system to the cold system. The final system is in a more probable condition [ ] ] [ δ δ δ δ δ τ τ is in principle measurable

18 Example: entropy increase on heat flow 0 copper 350K contact Let a 0 specimen of copper at a temperature of 350K be placed in thermal contact with an identical specimen at a temperature 90K. The specific heat in this temperature rane is approx J - K - const.. What is the final temperature 0 copper 90K. What is, the enery transferred to reach equilibrium 3. What is the chane of entropy of the two specimens when a transfer of 0. J has taken place?

19 . The final temperature is T f K 30K. The transferred enery is J K 30K. 7J 3. The chane of entropy is iven by S T T

20 The entropy of the first body is decreased by S K 0.J 4 JK The entropy of the second body is increased by J S K 0. 4 The total entropy increases by S JK 4 4 S JK In fundamental units JK k B The number of accessible states increases by e e JK 0 36

21 Law of increase of entropy We can easily show that the total entropy always increases when two systems are brouht into thermal contact.,,, One of the terms is initial,, initial Since all terms are positive this proofs that entropy increases when a constraint is removed: Before:, const. After: const

22 final How is equilibrium reached? initial lo max lo t 0

23 Ways to increase entropy For a lare system we will never not once in the life of the universe 0 8 s observe sinificant differences between the actual value of the entropy and the value of the entropy of the most probable confiuration

24 Laws of thermodynamics When thermodynamics is introduced in class, normally four postulates are used, called the laws of thermodynamics. These laws are all contained in the statistical formulation of thermal physics: Zeroth law: If two systems are in themal equilibrium with a third system, they must be in thermal equilibrium with each other. This follows from: imply and In other words,. lo lo ; lo lo 3 3 τ τ τ τ τ τ

25 First law: Heat is a form of enery: This is basically the statement of the conservation of enery more later in the course Second law: If a closed system is in a confiuration that is not the equilibrium confiuration, the most probable consequence will be that the entropy of the system will increase monotonically in successive instants of time. More popular: o perpetual motion of the second kind: ot possible to extract heat from one part of the environment and deliver it to another Third law: The entropy of a system approaches a constant value as the temperature approaches zero ernst law If the round state has a well defined multiplicity 0, then 0lo 0 as τ -> 0. Be aware of lasses