 # Thermodynamics versus Statistical Mechanics

Jan 14, 2016

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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 - PowerPoint PPT Presentation

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

• 1. Review of Thermodynamic and Statistical MechanicsThis is a short review1.1. Thermodynamic variablesWe 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 macroscopicallyby 3 variables: N, number of particles (Nm number of moles) V, volume E, internal energyonly sometimesin this case system is isolated

• Types of thermodynamic variables: Extensive: proportional to system size Intensive: independent of system size Not all variables are independent. The equations of state relate the variables: f (p,N,V,T) = 0For example, for an ideal gasorN, V, E (simple system)p, T, m (simple system)Boltzmann constant1.3805x10-23 J K-1No. of particlesGas constant8.3143 J K-1 mol-1No. of moles

• 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)Thermodynamic limit: in this case system is isolatedin this case system interchanges energy with surroundings

• 1.2 Laws of ThermodynamicsThermodynamics is based on three laws1. First law of thermodynamicsSYSTEMEnergy, E, is a conserved and extensive quantity(hidden)(explicit)change in energy involved in infinitesimal processmechanical work done on the systemamount of heat transferred to the systemproportional to system sizein an isolated system

• 1.2 Laws of ThermodynamicsThermodynamics is based on three laws1. First law of thermodynamicsSYSTEMEnergy, E, is a conserved and extensive quantity(hidden)(explicit)change in energy involved in infinitesimal processmechanical work done on the systemamount of heat transferred to the systeminexact differentialsW & Q do not exist (not state functions)exact differentialE does exist (it is a state function)

• Thermodynamic (or macroscopic) workare conjugate variables (intensive, extensive)independent of system size

xiintensive variablem-p-H...Xiextensive variableNVM...xidXimdN-pdV-HdM...

• (explicit)(hidden)SYSTEMsurroundingsIn fact dE = dWtot= dW + dQOnly 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 dWtot as dQIn mechanics:

where

and F is a conservative force

• (explicit)(hidden)SYSTEMsurroundingsIn fact dE = dWtot= dW + dQOnly 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 dWtot as dQ

• Asystems pressure = F / AF = external forcevolume change in slow compression mechanical work (through macroscopic variable V): heat transfer (through microscopic variables):molecules in base of container get kinetic energy from fire, and transfer energy to gas through conduction (molecular collisions)gasif

• the system performs workthe system adsorbs heat from reservoir 1the system transfers heat to reservoir 2HEAT ENGINE

• Equilibrium stateA state where there is no change in the variables of the system(only statistical mechanics gives a meaningful, statistical definition)A change in the state of the system from one equilibrium state to anotherThermodynamic processspecific volumeIt can viewed as a trajectory in a thermodynamic surface defined by the equation of stateFor example, for an ideal gasinitial statefinal statereversible path

• quasistatic processa process that takes place so slowly that equilibrium can be assumed at all times. No perfect quasistatic processes exist in the real world irreversible processunidirectional process: once it happens, it cannot be reversed spontaneously reversible processa 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 equilibriumA quasistatic process is not necessarily reversiblethe wall separating the two parts is slightly non-adiabatic (slow flow of heat from left to right)T1 > T2

• Calculation of work in a processThe work done on the system on going from state A to state B isOne has to know the equation of state p = p (v,T) of the substanceIn a cycle DE = 0 but -work done by the systemwork done by the system along the cycleTherefore:the heat adsorbed by the system is equal to the work done by the system on the environment

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

• Isothermal: no change in temperature, i.e. dT = 0 For an ideal gas (ideal gas) Adiabatic coolingIf the system expands adiabatically W0 and E increases(for an ideal gas this means T increases: the gas gets hotter) isothermal Adiabatic: no heat transfer, i.e. dQ = 0adiabatic

• isothermisothermAdiabatic coolingwork done by the systemDp
• 2. Second law of thermodynamicsThere is an extensive quantity, S, called entropy, which is a state function and with the property thatIn an isolated system (E=const.), an adiabatic process from state A to B is such thatIn an infinitesimal processThe equality holds for reversible processes; if process is irreversible, the inequality holds

• DS can be easily calculated using statistical mechanicsthe internal wall is removedideal gasexpanded gasExample of irreversible process entropy of ideal gas in volume Ventropy of ideal gas in volume V/2V/2VV/2Arrow of timeisolated system

• The entropy of an ideal gas is entropy before: entropy after: entropy change:The inverse process involves DS
• at equilibrium it is a function

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

• 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):Since S increases monotonically with E, it can be inverted to give E = E(N,V,S)entropy representation of thermodynamicsenergy representation of thermodynamicsequations of stateequations of state

• Equivalent (more utilitarian) statements of 2nd lawKelvin: 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)As a corollary: the most efficient heat engine operating between two reservoirs at temperatures T1 and T2 is the Carnot engine 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)ClausiusLord KelvinCarnotHistorically they reflect the early understanding of the problem

• 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 iswherereversible processirreversible processIf dQ > 0 (heat from environment to system) dS > 0In 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 trajectoryalternative statement of 2nd law

• The name entropy was given by Clausius in 1865 to a state function whose variation is given by dQ/T along a reversible processwhere pi is the probability of the system being in a microstate iIf 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, andIt can be shown that this S corresponds to the thermodynamic Sby Boltzmann in terms of probability arguments in 1877 and then by Gibbs a few years later:CONNECTION WITH ORDERMore order means less states availableWahrscheindlichkeit(probability)GibbsA clearer explanation of entropy was givenClausiusBoltzmann

• Does S always increase? Yes. But beware of environment...In general, for an open system:entropy change due to internal processesentropy change due to interaction with environment>
• 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 decreasesReversible isothermal processes are isentropicBut in irreversible ones the entropy may change If the reversible process is adiabatic:

• In a finite process:(depends on the trajectory)In a cycle:work done by the system in the cycleQheat absorbed by systemQDS = 0

• CARNOT CYCLEQ=-WIsothermal process. Heat Q1 is absorbed