Thermodynamics versus Statistical Mechanics

1. Thermodynamics versus Statistical Mechanics • Both disciplines are very general, and look for description of macroscopic (many-body) systems in equilibrium • There are extensions (not rigorously founded yet) to non-equilibrium processes in both • But thermodynamics does not give definite quantitative answers about properties of materials, only relations between properties • Statistical Mechanics gives predictions for material properties • Thermodynamics provides a framework and a language to discuss macroscopic bodies without resorting to microscopic behaviour • Thermodynamics is not strictly necessary, as it can be inferred from Statistical Mechanics

2. 1. Review of Thermodynamic and Statistical Mechanics This is a short review 1.1. Thermodynamic variables • We will discuss a simple system: • one component (pure) system • no electric charge or electric or magnetic polarisation • bulk (i.e. far from any surface) • The system will be characterised macroscopically • by 3 variables: • N, number of particles (Nmnumber of moles) • V, volume • E, internal energy only sometimes in this case system is isolated

3. Types of thermodynamic variables: • Extensive: proportional to system size N, V, E(simple system) • Intensive: independent of system size p, T, m(simple system) Not all variables are independent. The equations of state relate the variables: f (p,N,V,T) = 0 For example, for an ideal gas or No. of particles No. of moles Gas constant 8.3143 J K-1 mol-1 Boltzmann constant 1.3805x10-23 J K-1

4. Any 3 variables can do. Some may be more convenient than others. For example, experimentally it is more useful to consider T, instead of E (which cannot be measured easily) in this case system interchanges energy with surroundings in this case system is isolated Thermodynamic limit:

5. 1.2 Laws of Thermodynamics in an isolated system Thermodynamics is based on three laws proportional to system size 1. First law of thermodynamics Energy, E, is a conserved and extensive quantity change in energy involved in infinitesimal process amount of heat transferred to the system mechanical work done on the system (hidden) SYSTEM (explicit)

6. 1.2 Laws of Thermodynamics Thermodynamics is based on three laws inexact differentials W & Q do not exist (not state functions) exact differential E does exist (it is a state function) 1. First law of thermodynamics Energy, E, is a conserved and extensive quantity change in energy involved in infinitesimal process amount of heat transferred to the system mechanical work done on the system (hidden) SYSTEM (explicit)

7. Thermodynamic (or macroscopic) work independent of system size are conjugate variables (intensive, extensive)

8. surroundings In mechanics: where and F is a conservative force SYSTEM (hidden) (explicit) In fact dE = dWtot= dW + dQ Only the part of dWtot related to macroscopic variables can be computed (since we can identify a displacement). The part related to microscopic variables cannot be computed macroscopically and is separated out from dWtotas dQ

9. surroundings SYSTEM (hidden) (explicit) In fact dE = dWtot= dW + dQ Only the part of dWtot related to macroscopic variables can be computed (since we can identify a displacement). The part related to microscopic variables cannot be computed macroscopically and is separated out from dWtotas dQ

10. system’s pressure = F / A F = external force A volume change in slow compression • mechanical work (through macroscopic variable V): gas if • heat transfer (through microscopic variables): molecules in base of container get kinetic energy from fire, and transfer energy to gas through conduction (molecular collisions)

11. the system adsorbs heat from reservoir 1 the system transfers heat to reservoir 2 the system performs work HEAT ENGINE

12. Equilibrium state A state where there is no change in the variables of the system (only statistical mechanics gives a meaningful, statistical definition) Thermodynamic process A change in the state of the system from one equilibrium state to another It can viewed as a trajectory in a thermodynamic surface defined by the equation of state For example, for an ideal gas final state reversible path initial state specific volume

13. quasistatic process • a process that takes place so slowly that equilibrium can be assumed at all times. No perfect quasistatic processes exist in the real world • reversible process • a process such that variables can be reversed and the system would follow the same path back, with no change in system or surroundings. The system is always very close to equilibrium the wall separating the two parts is slightly non-adiabatic (slow flow of heat from left to right) T1 > T2 A quasistatic process is not necessarily reversible • irreversible process • unidirectional process: once it happens, it cannot be reversed spontaneously

14. Calculation of work in a process The work done on the system on going from state A to state B is work done by the system - One has to know the equation of state p = p (v,T) of the substance work done by the system along the cycle In a cycle DE = 0 but Therefore: the heat adsorbed by the system is equal to the work done by the system on the environment

15. Types of processes • Isochoric: there is no volume change isobaric isochoric • Isobaric: no change in pressure is the enthalpy. Also: important in chemistry and biophysics where most processes are at constant pressure (1 atm)

16. Isothermal: no change in temperature, i.e. dT = 0 • For an ideal gas adiabatic isothermal (ideal gas) • Adiabatic: no heat transfer, i.e.dQ = 0 Adiabatic heating If the system contracts adiabatically W>0 and E increases (for an ideal gas this means T increases: the gas gets hotter) Adiabatic cooling If the system expands adiabatically W<0 and E decreases (for an ideal gas and many systems this means T decreases: the gas gets cooler)

17. Adiabatic cooling Dp<0 DE<0 for an ideal gas and many other systems this means DT<0 isotherm isotherm work done by the system VA VB

18. 2. Second law of thermodynamics There is an extensive quantity, S, called entropy, which is a state function and with the property that In an isolated system (E=const.), an adiabatic process from state A to B is such that In an infinitesimal process The equality holds for reversible processes; if process is irreversible, the inequality holds

19. Example of irreversible process isolated system the internal wall is removed ideal gas expanded gas V/2 V/2 V Arrow of time DS can be easily calculated using statistical mechanics entropy of ideal gas in volume V entropy of ideal gas in volume V/2

20. The entropy of an ideal gas is • entropy before: • entropy after: • entropy change: The inverse process involves DS<0 and is in principle prohibited

21. (N,V,E) The existence of S is the price to pay for not following the hidden degrees of freedom. It is a genuine thermodynamic (non-mechanical) quantity time evolution from non-equilibrium state • at equilibrium it is a function • it is a monotonic function of E An adiabatic process involves changes in hidden microscopic variables at fixed (N,V,E). In such a process maximum

22. S is a thermodynamic potential: all thermodynamic quantities can be derived from it (much in the same way as in mechanics, where the force is derived from the energy): equations of state Since S increases monotonically with E, it can be inverted to give E = E(N,V,S) equations of state entropy representation of thermodynamics energy representation of thermodynamics

23. Equivalent (more utilitarian) statements of 2nd law Historically they reflect the early understanding of the problem Kelvin: There exists no thermodynamic process whose sole effect is to extract heat from a system and to convert it entirely into work (the system releases some heat) Clausius: No process exists in which the sole effect is that heat flows from a reservoir at a given temperature to a reservoir at a higher temperature (work has to be done on the system) Lord Kelvin Clausius As a corollary: the most efficient heat engine operating between two reservoirs at temperatures T1 and T2 is the Carnot engine Carnot

24. S is connected to the energy transfer through hidden degrees of freedom, i.e. to dQ. In a process the entropy change of the system is alternative statement of 2nd law reversible process where irreversible process If dQ > 0 (heat from environment to system) dS > 0 In a finite process from A to B: For reversible processes T-1 is an integrating factor, since DS only depends on A and B, not on the trajectory

25. The name entropy was given by Clausius in 1865 to a state function whose variation is given by dQ/T along a reversible process A clearer explanation of entropy was given Clausius by Boltzmann in terms of probability arguments in 1877 and then by Gibbs a few years later: It can be shown that this S corresponds to the thermodynamic S Gibbs Wahrscheindlichkeit (probability) where pi is the probability of the system being in a microstate i CONNECTION WITH ORDER More order means less states available If all microstates are equally probable (as is the case if E = const.) then pi =1/W, where W is the number of microstate of the same energy E, and Boltzmann

26. Does S always increase? Yes. But beware of environment...In general, for an open system: < entropy change of environment > entropy change due to internal processes entropy change due to interaction with environment entropy change of systemmay be positive or negative e.g. living beings...

27. Processes can be discussed profitably using the entropy concept. For a reversible process: • If the reversible process is isothermal: S increases if the system absorbs heat, otherwise S decreases • If the reversible process is adiabatic: Reversible isothermal processes are isentropic But in irreversible ones the entropy may change

28. (depends on the trajectory) In a finite process: In a cycle: work done by the system in the cycle Q Q DS = 0 heat absorbed by system

29. CARNOT CYCLE Isothermal process. Heat Q1 is absorbed T1 Adiabatic process. No heat Adiabatic process. No heat Q=-W T2 Change of entropy: Isothermal process. Heat Q2 is released it is impossible to perform a cycle with W 0 and Q2 = 0

30. Efficiency of a Carnot heat engine: Carnot theorem: The efficiency of a cyclic Carnot heat engine only depends on the operating temperatures (not on material) By measuring the efficiency of a real engine, a temperature T2 can be determined with respect to a reference temperature T1