01._thestatisticalbasisofthermodynamics [compatibility mode]

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 1. The Statistical Basis of Thermodynamics 1. The Macroscopi c & the Mi cr oscopi c St ates 2. Con tac t between Sta tist ics & Th ermodyna mi cs: Physical Significance of the Number (  N ,V ,  E ) 3. Fur the r Con tac t bet wee n Sta tist ics & Ther modyn amics 4. Th e Cla ss ic al Id ea l Gas 5. The Entr opy of Mixing & th e Gibbs Paradox 6. The “Correct” Enu me rat ion of the Mic rostates

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Thermodynamics Chap 1 Notes

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  • 1. The Statistical Basis of Thermodynamics

    1. The Macroscopic & the Microscopic States

    2. Contact between Statistics & Thermodynamics:

    Physical Significance of the Number (N,V,E)

    3. Further Contact between Statistics & Thermodynamics

    4. The Classical Ideal Gas

    5. The Entropy of Mixing & the Gibbs Paradox

    6. The Correct Enumeration of the Microstates

  • 1.1. The Macroscopic & the Microscopic States

    System of N identical particles in volume V, with

    , ,N

    N V finiteV

    (Thermodynamic limit )

    E.g., Non-interacting particles:

    1

    N

    ii

    N n

    1

    N

    i ii

    E n

    i = single particle energiesni = # of pcles with energy i

    A macrostate is specified by parameters ( N, V, E, ... ).

    Postulate of equal a priori probabilities:All microstates satisfying the macrostate parameters are equally likely to occur.

    , , ,N V E = # of all microstates that give rise to the macrostate (extensive) parameters N, V, E, ... .

    Let

  • 1.2. Contact between Statistics & Thermodynamics: Physical Significance of the Number (N,V,E)

    Consider 2 systems A1 & A2 in thermal contact with each other,

    i.e., partition is fixed, impermeable but heat conducting.

    ( Nj , Vj & E(0) = E1 + E2 are fixed )

    A1( N1 , V1 , E1 )

    A2( N2 , V2 , E2 )

    01 2

    0 0

    1 1 2 2 E E EE E E

    Equilibrium is achieved if E1 ( with E2 = E(0) E1 ) maximizes

    (0) :

    0

    1

    0E

    1 1 2 22 2 1 1

    1 2

    E EE E

    E E

    2 2 2 21 2

    E E

    E E

    (0) denotes properties of the composite system

  • 1 1 2 22 2 1 1

    1 2

    0E E

    E EE E

    1 1 2 21 1 1 2 2 2

    1 1E E

    E E E E

    1 1 2 21 2

    ln lnE E

    E E

    ln EE

    Let 2 systems are in thermal equilibrium

    if they have the same .

    Thermodynamics :,

    1

    N V

    S

    E T

    Planck :

    lnS k Boltzmann :

    lnS k

    1

    k T

    k = Boltzmann constant

    3rd law

    0th law ( thermal eqm.)

  • 1.3. Further Contact between Statistics & Thermodynamics

    For an impermeable but movable & heat conducting partition,

    Nj , V(0) = V1 +V2 & E

    (0) = E1 + E2 are fixed.

    Equilibrium is achieved, i.e., (0) is maximized, if

    0

    1

    0E

    0

    1

    0V

    and

    i.e., both system have the same values of &,

    ln

    N EV

    1st law: dE T dS P dV dN

    ,N E

    SP T

    V

    P

    k T

    chemical potential

    ~ mech. eqm.

  • For a permeable, movable & heat conducting partition,

    N(0) = N1 + N2 , V(0) = V1 +V2 & E

    (0) = E1 + E2 are fixed.

    Equilibrium is achieved, i.e., (0) is maximized, if

    0

    1

    0E

    0

    1

    0V

    i.e., Both system have the same values of , , &

    ,

    ln

    V EN

    1st law: dE T dS P dV dN

    ,V E

    ST

    N

    k T

    0

    1

    0N

    ~ chemical eqm.

  • Summary

    Connection between statistical mechanics & thermodynamics is

    lnS k

    Once is written in terms of the independent thermodynamical variables,

    all other thermodynamic quantities can be obtained via the Maxwell relations.

  • UInternal Energy

    SV , X

    P, YT

    HEnthalpy

    GGibbs free energy

    FHelmholtz free energy

    V

    UT

    S

    Mnemonics for the Maxwell Relations

    P

    GS

    T

    U

    S P

    V T

    G

    T V

    P S

    dU TdS P dV Y dX 1

    z x y

    x y z

    y z x

    dH TdS Vd P Y dX

    varvar

    varF W

    2

    S V

    PT

    V S

    U

    S V

    Good Physicists Have Studied Under Very Fine Teachers

    = U ( P) V Y X

    = F ( P) V Y X = H TS

    = U TS

    = U(V,S,X)

  • 1.4. The Classical Ideal Gas

    Non-interacting, classical ( distinguishable), point particles: , , NN E V V

    lnS k N V const

    ,N E

    S P

    V T

    P V n R T

    Nk

    V

    Cf n R

    kN

    A

    R

    N 231.38 10 /J K

    23

    8.31 /

    6.02 10 /

    J mol K

    mol

    58.62 10 /eV K

    const here means indep. of V.

  • Quantum (Obeying Schrodinger Eq) Free Particles

    Let these particles be confined within a cube of edge L.

    Dirichlet boundary conditions: 0 at walls ( where x,y,z = 0,L ).

    sin sin sinyx znn n

    A x y zL L L

    1,2,3, ; , ,in i x y z

    Neumann boundary conditions: n 0 at walls.

    cos cos cosyx znn n

    A x y zL L L

    ,1,2,3,0in

    22 2 2

    2 2 2, ,2 2

    x y z x y z

    kn n n n n n

    m m L

    1-particle energy :

    2

    2 2 2

    2/38x y z

    hn n n

    m V 3V L

  • 2

    2 2 2

    2/38x y z

    hn n n

    m V

    * 2 2 2x y zn n n

    2*

    2/38

    h

    m V

    i.e.

    Let

    ( * is a positive integer )

    1, ,N E V # of { nx, ny, nz } satisfying2/3

    * 2 2 2

    2

    8x y z

    m Vn n n

    h

    , ,N E V # of { nix, niy, niz } satisfying

    For N non-interacting particles

    2

    2 2 2

    2/318

    N

    i x i y i zi

    hE n n n

    m V

    2*

    2/38

    hE

    m V

    * 2 2 21

    N

    i x i y i zi

    E n n n

    3

    2

    1

    N

    rr

    n

    2/3

    * 2 2 2

    21

    8N

    i x i y i zi

    m V EE n n n

    h

    2/3, , ,N E V N V E

  • 2/3, , ,N E V N V E 2/3, , ,S N E V S N V E

    For reversible adiabatic processes, S & N are kept constant.

    2/3V E const

    2/3

    1/3,

    20

    3 N S

    EdV V dE

    V

    ,

    2

    3N S

    E E

    V V

    ,N S

    EP

    V

    2

    3

    EP

    V

    ,f N S

    Valid for both classical & quantum statistics

    (adiabatic processes)

    5/3PV const

  • Better behaved quantity is ( N,E,V),

    defined as the # of lattice points with non-negative coordinates & lying within

    the volume bounded by the surface of a sphere, centered at the origin, and

    with radius

    Counting States: Distinguishable Particles

    State labels { nix, niy, niz } form a lattice in the 3N-D n-space.

    ( N,E,V) = # of lattice points with non-negative coordinates & lying on the

    surface of a sphere, centered at the origin, and with radius

    *R

    fluctuates wildly even for small E changes unless N >>1.

    2/3*

    2

    8m V ER

    h

  • * * * 3/211 4

    1,8 3

    N

    As R , the lattice points become a continuum.

    * 3/2

    6

    Better approximations:

    Number of points on the x-y, y-z, z-x planes is

    Since these points are shared by 2 neighboring sectors, the

    volume integral counts each as half a point.

    * * 3/2 *13

    6 8

    Dirichlet B.C.

    (exclude all nj = 0 points )

    * * 3/2 *13

    6 8

    Neumann B.C.

    (include all nj = 0 points )

    *134

    ( Density of states in n-space is 1. )

  • Volume of an n-D sphere of radius R is/2

    !2

    nn

    sphV Rn

    ( see App.C )

    Volume of points with non-negative coordinates

    1

    2

    n

    sphV V

    ( Take non-negative-half of every dimension )

    3 3 /2

    * * 3 /21,32

    !2

    N NNN E E

    N

    2/3

    *

    2

    8, , ,

    mV EN V E N E

    h

    3 /2

    3

    2

    3!

    2

    NNm EV

    Nh

    ! 1n n

  • 3 /2

    3

    2, ,

    3!

    2

    NNm EV

    N V ENh

    Stirlings formula: ln ! lnn n n n for n >>1

    3/2

    3

    3ln , , ln 2 ln !

    2

    V NN V E N m E

    h

    3/2

    3

    4 3ln , , ln

    3 2

    V m E NN V E N

    h N

    3

    23 3 3ln ! ln

    2 2 2

    N N NN

    Let (N,V,E) = # of states lying between E & E+ .

    , ,

    , ,N V E

    N V EE

    3, ,

    2

    NN V E

    E

    3/2

    3

    4 3ln , , ln

    3 2

    V m E NN V E N

    h N

    ln , ,N V E ln , ,N V E

  • 3/2

    3

    4 3ln , , ln

    3 2

    V m E NN V E N

    h N

    3/2

    3

    4 3, , ln ln

    3 2

    V m E N kS N V E k N k

    h N

    2

    2/3

    3 2, , exp 1

    4 3

    h N SE S V N

    m V N k

    3

    2E N k T

    3

    2n R T

    ,N V

    ET

    S

    2

    3E

    N k

    ,

    V

    N V

    EC

    T

    3

    2N k

    3

    2n R

    ,N S

    EP

    V

    2

    3E

    V

    2

    3P V E N k T n R T

    ,

    P

    N P

    HC

    T

    ,N P

    E PV

    T

    VC n R

    5

    2n R

    5

    3P

    V

    C

    C

  • 3/2

    3

    4 3, , ln

    3 2

    V m E N kS N V E N k

    h N

    3

    2E N k T

    Isothermal processes ( N, T = const ) : E const

    ,N E

    SS V

    V

    N kV

    V lnN k V

    ln lnf i f iS S N k V V lnf

    i

    VN k

    V

    Adiabatic processes ( N, S = const ) :

    3/2V E const 3/2V T const

    P V n R T 5/2P V const P V

    1V T

    Alternatively, ,N SdE P dV 2

    3

    EdV

    V also leads to 3/2V E const

    5

    3

  • 1.5. The Entropy of Mixing & the Gibbs Paradox

    3/2

    3

    4 3ln

    3 2

    V m E N kS N k

    h N

    This S is not extensive, i.e., , , , ,S N V E S N V E

    Mixing of 2 ideal gases 1 & 2 (at fixed T ) :

    3

    3ln

    2before itotal i i

    i i i

    VS S k N

    2

    3 4ln 1 ln

    2 3

    m EN k V N k

    h N

    3/2

    2

    2 3ln

    2

    m k TS N k V

    h

    3

    2E N k T

    3

    3ln

    2aftertotal mixed i

    i i

    VS S k N

    i

    i

    V V

    3

    3ln

    2

    VN k

    22

    mkT

    Thermal wavelength

  • Entropy of mixing of gases :

    after beforetotal totalmixingS S S ln 0ii i

    Vk N

    V

    Gibbs paradox :

    For the mixing of different parts of the same gas in equilibrium (Ni / Vi = N / V ,

    i = ), the formula still applies & we also have S > 0, which is unacceptable.

    i

    i

    N N

    V V

    3

    3ln

    2before itotal i i

    i i i

    VS S k N

    3

    3ln

    2aftertotal i

    i i

    VS k N

    Irreversible process: S > 0 is expected.

  • For the mixing of different parts of the same gas in eqm., ,i

    i

    i

    N N

    V V

    lnimixi i

    VS k N

    V ln lni i

    i

    k N N k N N

    ln ln 0mixed i i ii

    S k N N S k N N

    Thus, Gibbs paradox is resolved using Gibbs recipe :

    3/2

    3

    4 3, , l 1n

    3 2

    V m ES N V E N k N

    h NNk

    Sackur-Tetrode eq.

    S is now extensive, i.e., , , , ,S N V E S N V E

    ln ln 0mixed i i i ii

    S k N N N S k N N N orln ln !N N N N

    ln !S S k N

    3

    3ln

    2before itotal i

    i

    VS k N

    33

    ln2

    aftertotal

    VS k N

    lnii i

    Nk N

    N

  • Revised Formulae

    3/2

    3

    4 3, , l 1n

    3 2

    V m ES N V E N k N

    h NNk

    2/323 2

    , , e2

    p4 3

    x 13

    h N SNE S V N

    m V N k

    extensive

    In general, relations derived using the previous definition of S

    are not modified if they do not involve explicit expression of S.

    ,V S

    E

    N

    2 21

    33

    E S

    N N k

    intensive

    Gibbs recipe is cancelled by removing all terms in red.

    ,N V

    ET

    S

    2

    3

    E

    N k

    ,N S

    EP

    V

    2

    3

    E

    V

  • 23

    ET

    N k

    2

    3

    EP

    V

    2 2

    33

    E

    N

    E E S

    N N N k

    1PE ST

    NV

    3/2

    3

    4 3l 1n

    3 2

    V m ES N k N k

    h NN

    3

    3ln

    21k

    N

    VN

    3

    2 2

    3 3

    3 21 ln 1

    2 3 N

    Vk T

    3lnk TV

    N

    21

    3

    2

    3

    E S

    N N k

    3ln 1k TN

    NV

    A G P V N P V N N kT

    21

    3

    2

    3

    E S

    N N k

    22

    mkT

  • 1.6. The Correct Enumeration of the Microstates

    Elementary particles are all indistinguishable.

    In the distribution of N particles such that ni particles occupy the i state,

    !

    !D

    ii

    N

    n

    for distinguishable particles

    1 for indistinguishable particles

    In the classical (high T ) limit, 0in i !D N

    !D

    N

    Gibbs recipe corresponds to