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Thermodynamics versus Statistical Mechanics 1.Both disciplines are very general, and look for description of macroscopic (many-body) systems in equilibrium 2.There are extensions (not rigorously founded yet) to non- equilibrium processes in both 3.But thermodynamics does not give definite quantitative answers about properties of materials, only relations between properties 4.Statistical Mechanics gives predictions for material properties 5.Thermodynamics provides a framework and a language to discuss macroscopic bodies without resorting to microscopic behaviour 6.Thermodynamics is not strictly necessary, as it can be inferred from Statistical Mechanics

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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 (N m number of moles) V, volume E, internal energy only sometimes in this case system is isolated

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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) = 0 For example, for an ideal gas or N, V, E (simple system) p, T, (simple system) Boltzmann constant 1.3805 x 10 -23 J K -1 No. of particles Gas constant 8.3143 J K -1 mol -1 No. of moles

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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 isolated in this case system interchanges energy with surroundings

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

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

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Thermodynamic (or macroscopic) work are conjugate variables (intensive, extensive) x i intensive variable -p-H... X i extensive variable NVM... x i dX i dN -pdV-HdM... independent of system size

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(explicit) (hidden) SYSTEM surroundings In fact dE = dW tot = W + Q Only the part of dW tot 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 dW tot as Q In mechanics: where and F is a conservative force

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(explicit) (hidden) SYSTEM surroundings In fact dE = dW tot = W + Q Only the part of dW tot 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 dW tot as Q

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A system’s pressure = F / A F = external force volume 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) gas if

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

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Equilibrium state A 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 another Thermodynamic process specific volume It can viewed as a trajectory in a thermodynamic surface defined by the equation of state For example, for an ideal gas initial state final state reversible path

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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 irreversible process unidirectional process: once it happens, it cannot be reversed spontaneously 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 A quasistatic process is not necessarily reversible the wall separating the two parts is slightly non-adiabatic (slow flow of heat from left to right) T 1 > T 2

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

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

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Isothermal: no change in temperature, i.e. dT = 0 For an ideal gas (ideal gas) 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) Adiabatic heating If the system contracts adiabatically W>0 and E increases (for an ideal gas this means T increases: the gas gets hotter) isothermal Adiabatic: no heat transfer, i.e. Q = 0 adiabatic

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isotherm Adiabatic cooling work done by the system p E<0 for an ideal gas and many other systems this means T<0 VAVA VBVB

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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

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

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

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at equilibrium it is a function it is a monotonic function of 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 An adiabatic process involves changes in hidden microscopic variables at fixed (N,V,E). In such a process maximum (N,V,E) time evolution from non-equilibrium state

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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 thermodynamics energy representation of thermodynamics equations of state

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Equivalent (more utilitarian) statements of 2 nd law 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) As a corollary: the most efficient heat engine operating between two reservoirs at temperatures T 1 and T 2 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) Clausius Lord Kelvin Carnot Historically they reflect the early understanding of the problem

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S is connected to the energy transfer through hidden degrees of freedom, i.e. to Q. In a process the entropy change of the system is where reversible process irreversible process If Q > 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 S only depends on A and B, not on the trajectory alternative statement of 2 nd law

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

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

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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 Reversible isothermal processes are isentropic But in irreversible ones the entropy may change If the reversible process is adiabatic:

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

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CARNOT CYCLE Q=-W Isothermal process. Heat Q 1 is absorbed Adiabatic process. No heat Isothermal process. Heat Q 2 is released Change of entropy: T1T1 T2T2 Adiabatic process. No heat it is impossible to perform a cycle with W 0 and Q 2 = 0

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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 T 2 can be determined with respect to a reference temperature T 1

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