Module 2

Statistical Thermodynamics: Molecules to Machines

Statistical Thermodynamics: Molecules to MachinesVenkat Viswanathan May 20, 2015Module 2: Classical ThermodynamicsLearning Objectives Introduction of the basic concepts that underlie classical thermodynamics, which gives the governing laws for macroscopic matter. First two laws of thermodynamics. Thermodynamics driving forces that arise from the second law of thermodynamics. Classical thermodynamics applied to phase coexistence.Key Concepts:First and Second Laws of Thermodynamics, thermodynamic driving forces, ensembles, extensive and intensive variables, Legendre transform, Euler equation, Phase equilibrium.First Law of ThermodynamicsThe first law of thermodynamics, also known as the conservation of energy principle, provides the basis for studying the relationships among the various forms of energy and energy interactions. The law states that although energy assumes many forms, the total quantity of energy is constant, and when energy disappears in one form it appears simultaneously in other forms.The energy of a system is measured by its internal energy U , and is defined as the net sum of the kinetic and potential energies of all the molecules in a macroscopic material. The internal energy is an extensive property, meaning that the internal energy depends linearly on the size of the system.According to the First Law of Thermodynamics any change in the internal energy occurs due to the addition of heat Q and work W , according to the relation:dU = Q + W(1)The difference in the notation of the change of internal energy (dU ) from the addition of heat (Q) or work (W ), lies in the fact that internal energy is a state variable, while heat and work are path dependent. However, once heat and work are applied, they are indistinguishable on a macroscale.Second Law of ThermodynamicsThe origins of the second law lies in a statement that it is impossible for any self-acting process or machine to produce net flow of heat from a region of low temperature. The law states that an isolated system always proceeds from an ordered to a more disordered state, or else it undergoes no change at all.The level of order from a system is measured by the entropy S, and according to the second law the entropy of the universe tends to a maximum (Rudolf Claussius, 1865). This is stated mathematically as:(dS)U ,V ,N 0(2)Where the subscripts U , V and N indicates constant internal energy, volume and number of particles, respectively.Thermodynamics Driving ForcesA thermodynamic system can be defined according to their extensive variables, i.e., variables that vary linearly with the system size. For example, we define the internal energy U to be a function of extensive vari-

Figure 1: The total internal energy of a system containing two subsystems, is the sum of the internal energies of each subsystemables S, V , and N = (N1, N2, ..., Nr ), where r is the number of chemical species in the system. Therefore, we have U = U (S, V , N ). Differential changes in the extensive variables leads to a differential change in the internal energy, such that:dU =

. U . SHAPE \* MERGEFORMAT

S

dS+V ,N1,...,Nr


V

dV+S,N1,...,Nr


Ni

dNi(3)S,V ,Nj=iThe partial derivatives are intensive variables, i.e. variables that do not depended on the size of the system. We define these as:. U . SHAPE \* MERGEFORMAT

S

TV ,N1,...,Nr


V

pS,N1,...,Nr


Ni

i(4)S,V ,Nj=iWhere T is temperature, p is pressure, and i is chemical potential of species i.The change in internal energy is written as:rdU = TdS pdV + . idNi(5)i=1Since U is a state variable, we can choose any path to find a differential change. Equation (5) states that any change in the internal energy of the system, is the result of a quasi-static heat flux (TdS), mechanical work (pdV ) and chemical work (idNi) to the system.Equivalently, we can define the system according to their entropyS = S(U , V , N ), resulting in a differential change:1pr idS =

dU +T

dV .dNi(6)i=1Which is useful to discuss the second law of thermodynamics. Consider a closed, isolated system which cannot exchange heat, work,or particles with its surroundings (Fig. 2). We initially prepare the system such that part of the system (subsystem 1) has parameter values U (1), V (1), and N (1), and analogous for the rest of the system (subsystem2).

Figure 2: Closed, Isolated System containing two subsystemMaking the statement that the entropy is additive, i.e. S(1) + S(2) =S), we can write:1p(1)

r (1)dS =

dU (1) +

dV (1) . i dN (1)+T (1)1

T (1)p(2)

i=1 r

T (1)i(2)dU (2) +

dV (2) . i dN (2)

(7)T (2)

T (2)

i=1

T (2)iSince the total system is closed and isolated, we can write dU (2) =dU (1), dV (2) = dV (1), and dN (2) = dN (1).iiTherefore, we can write:dS =

. 1 SHAPE \* MERGEFORMAT

T (1)

1 T (2)

.dU (1) +

. p(1) SHAPE \* MERGEFORMAT

T (1)

p(2) . T (2)

dV (1)r . (1).i

(2) . i

(1)i=1

T (1) T (2)

dNi(8)According to the second law of thermodynamics, any spontaneous process requires dS 0, thus striving for the maximum value at equilibrium. The condition at equilibrium satisfy the condition dS = 0 for arbitrary small changes dU (1), dV (1), and dN (1): Thermal equilibrium occurs when T (1) = T (2) Mechanical equilibrium occurs when p(1) = p(2) Chemical equilibrium (no reactions) occurs when (1) = (2)

Figure 3: Phase behavior of polystyrene-poly (2-vinylpyridine) diblock co-polymer. Transmission elec- tron micrographs show lamellar phase (fraction polystyrene f = 0.4), gyroid phase (f = 0.38), and cylindrical phase (f = 0.35). The experimental phase diagram is shown in the lower rightiipanel.Notice that although thermal and mechanical equilibrium implies spatial homogeneity of T and p, the condition of spatial homogeneity i should not be interpreted as homogeneous concentration or density of species i (vapor-liquid equilibrium in a box or a structured fluid like ablock copolymer - Fig. 31).1 Mark F. Schulz, Ashish K. Khand-Consider the case p(1) = p(2), (1)

(2)

pur, Frank S. Bates, Kristoffer Almdal,i= i , and the partition between subsystems is rigid and impermeable (dV (1) = dN (1) = 0). If T (1) > T (2) the condition dS 0 is met only if dU (1) < 0 If T (1) < T (2) the condition dS 0 is met only if dU (1) > 0 Heat spontaneously flows from high temperature to low.

Kell Mortensen, Damian A. Hajduk, and Sol M. Gruner. Phase Behavior of Polystyrene Poly(2-vinylpyridine) Di- block Copolymers. Macromolecules, 29(8):28572867, January 1996. ISSN0024-9297In the case T (1) = T (2), (1) = (2), and the partition betweensubsystems is movable but impermeable (dN (1) = 0). If p(1) > p(2) the condition dS 0 is met only if dV (1) > 0 If p(1) < p(2) the condition dS 0 is met only if dV (1) < 0 Inhomogeneities in pressure leads to the spontaneous expansion of high pressure regions into low pressure regions.In the case T (1) = T (2), p(1) = p(2):

Figure 4: A schematic of the underlying molecular mechanism for heat conduction in a crystal. The atoms in a crystal fluctuate due to thermal energy; the(1)(2)

(1)

higher the temperature, the larger the If i> ithe condition dS 0 is met only if dNi< 0

fluctuations. At short time t1, a crystal has a region of high temperature, If (1)

(2)

(1)

indicated by the red atoms. The fluc-i< ithe condition dS 0 is met only if dNi> 0 Matter flows from high chemical potential to low (not always high concentration to low).

tuations of the hot atoms (red) leads to the neighboring cold atoms (blue) heating up due to interactions. As time progresses from t1 through t4, the hot spot dissipates and cools down, leading to an increase in the neighboring temperature. The process of heat transport from high temperature to low temperature due to molecular fluctuations is conduction.Thermodynamic EnsemblesThe governing thermodynamic potential of the closed, isolated system is the entropy S(U , V , N ), i.e. the control or canonical variables are U , V , and N . However, the closed, isolated system is only one possible thermodynamic system. In many instances, it is desirable to have different control variables, where we replace one (or more) extensive variable with their conjugate intensive variable (Fig. 6).For convenience, we start from the equation of state U (S, V , N ),which is equivalent to S(U , V , N ). The extensive variables and their conjugate intensive variables are related through the relationshipsS T =


S

V ,N1,...,Nr

Figure 5: Four different ensembles and their canonical variablesV p =


V

S,N1,...,NrNi i =


Ni

S,V ,Nj=i

(9)A thermodynamic ensemble is defined by the canonical variables of the system. Changing from one ensemble to another amounts to shifting from one thermodynamic potential to another that depends on the new canonical variables. Replacing an extensive variable for its conjugate intensive variable is effectively replacing the control variable with the slope of the control variable.The mathematical method of performing this change of variables is called a Legendre Transform. The method is defined according to the following steps: A function y = f (x) is defined as a list of x, y, pairs, i.e. the plot ofy(x) graphs (x1, y1), (x2, y2), (x3, y3),... Equivalently, it is possible to express the same function in terms of the tangent slope c(x) and the corresponding y-intercepts, i.e. the points (c1, b1), (c2, b2), (c3, b3),... For a change dx in the x-variable, the change dy is given by:. dy .dy =

dx = c(x)dx(10)dx From this point on the curve, we can extend a line back to x = 0 to find the y-intercept b(x), such thaty(x) = c(x)x + b(x) b(x) = y(x) c(x)x(11) The resulting function for the series of intercepts versus the slopes (i.e. b = b(c)) contains the same information as y = y(x). A change in b(c) is given by db = dy cdx xdc = xdc.Example. Consider the function y = (x 2)2. We use a Legendre Transform to shift the variable from x to the slope c. The slope of the function is:cc = 2(x 2) x = 2 + 2(12)The Legendre transform gives us a new function b(c), given by:c2c2c2b = y cx =

4 2 2c = 4 2c(13)We note that this Legendre transform correctly relates back to the original function by noting thatdb = cdc2 2 = x(14)All of the thermodynamic potentials that define specific ensembles are Legendre transforms of the original potential U = U (S, V , N ): Transform from S to T for the closed, isothermal ensemble gives the Helmholtz free energy:. U .

Figure 6: Plot of the function y(x) = (x 2)2 and its corresponding Legen-24F = U

SHAPE \* MERGEFORMAT

S V ,N

S = U TS(15)which has a differential change:rdF = dU TdS SdT = SdT pdV + . idNi(16)i=1 Transform from S to T and V to p for the closed, isothermal, isobaric ensemble give the Gibbs free energy:G = U


S

V ,N

. U .SV

S,N

V = U TS + pV(17)which has a differential change:dG = dU TdS SdT + pdV + V dpr= SdT + V dp + . idNi(18)i=1Euler EquationThe internal energy is a homogeneous, first-order function, which requires that the internal energy must increase linearly with the size of the system. Mathematically, this is written as:U (S, V , N ) = U (S, V , N )(19)where is an arbitrary scaling factor for the system size. If we take the derivative of this equation with respect to , we arrive at:U (S, V , N ) = U (S, V , N )(20)The left-hand side is written as:. U (S, V , N ) . SHAPE \* MERGEFORMAT

(S)

V ,N

(S) SHAPE \* MERGEFORMAT

. U (S, V , N ) .+(V )

S,N

(V ) +r . U (S, V , N ) .(Ni)r

TS pViNii=1

(Ni)

S,V ,Nj=i

i=1

(21)Therefore, the total Legendre Transform equals zero, which is the Euler equation:rU = TS pV + . iNi(22)i=1Phase EquilibriumConsider a single component system that obeys the van der Waals equation of state:p = NRT

aN 2RTa

(23)V Nb

V 2 = v b v2where v = V /N is the molar volume.Although a and b are considered empirical constant, there are micro- scopic justifications for the van der Waals constants: The constant b represents the hard-core excluded volume of the molecules in the system. The constant a determines the strength of the two-body attractive interaction between molecules in the systemFor simplicity, we write the dimensionless equation of state:p =

1v 1

a v2

(24)

Figure 7: Closed, isothermal systemwhere p

= pb , v

= v , and a

= a , thus reducing the number of

containing a van der Waals fluid inparameters to just a to capture the temperature dependence.Consider the closed, isothermal system containing a van der Waals fluid in vapor-liquid equilibrium (Fig. 7). Our goals are to find the limits of stability of each phase, the conditions where the phases coexist, and the properties of the two phases in coexistence. The simple example of phase coexistence demonstrates the essential issues at work in more complex, multi-component, multiphase systems.

vapor-liquid equilibriumLimits of StabilityThermodynamic stability requires that T = 1 . v .

> 0, and thevp Tlimit of stability of a single pure phase occurs when:. p . SHAPE \* MERGEFORMAT

v a

1=(v 1)2

2a+ v3

= 0(25)This gives the cubic equation:v3 + 2a .v2 2v + 1. = 0(26) Equation (26) has three solutions: The first solution is less than one (v < 1) and is unphysical since the pressure p diverges as v 1+. The two solutions greater than v = 1 identify the limits of stability of the vapor and liquid phases.The two solutions are approach each other with decreasing a , until that meet at the critical point:

Figure 8: Plot of p versus v for a = 3.679. The Maxwell construction for coexistence states that vapor-liquid coexistence occurs when the areas A1 = A2. The dashed segment of curve indicates conditions where a single phase is unstablevc = 3andPhase Coexistence

a c =

27(27)8The equilibrium conditions are (v) = (l) and p(v) = p(l) = pcoex, and the molar Helmholtz free energy is written as:Figure 9: Plot of f versus v for a =f = F = f +NRT

. f. SHAPE \* MERGEFORMAT

v a

dv = f0

pdv

3.679. The common-tangent construction for coexistence states that vapor- liquid coexistence occurs when a single . 1

a .dv = f

log (v1)a

(28)

line is simultaneously tangent to two= f0 +

v 1 v2

0

v

points on the v f curve. The dashed segment of curve indicates conditionswhere f0 is an unspecified constant of integration (reference state).The Euler equation for a one-component system is:N = U ST + pV = F + pV = G(29)or in dimensionless form, we have:

where a single phase is unstable v

== f + pv = f0 + log(v 1)

2a

(30)RTv 1vSetting the chemical potentials in vapor and liquid phases equal, is equivalent to: v (v)

Figure 10: Plot of versus v for a =3.375 = 27 (critical point, blue), a =3.527, a = 3.679, a = 3.831, and a = (v) (l) =

v (l)

p(v)dv + pcoex .v(v) v(l). = 0(31)

3.982 (red). The dashed segments indi- cate conditions where a single phase is unstable.This is rewritten as:pcoex .v(v) v(l). =

v (v)v (l)

p(v)dv

(32)As written, this leads us to the Maxwell construction for vapor-liquid phase coexistence, which provides a graphical method of identifying phase coexistence (Fig. 8).A second way to view the coexistence condition is to note:

Figure 11: Plot of versus p for a =3.679. Coexistence conditions are de- (v) = (l) f(v)

f(v) SHAPE \* MERGEFORMAT

v

v(v) = f(l)

f(l) SHAPE \* MERGEFORMAT

v

v(l)

termined by the point where the curve crosses itself along the vapor and liq-(v)

(l)

uid branches. The dashed segment ofp(v) = p(l) f = f

(33)

curve indicates conditions where a sin-v

v

gle phase is unstable.This development gives the common-tangent construction (Fig. 9, 10). The most direct method of visualizing the coexistence condition isplot versus p. Numerically calculating the equilibrium conditions iseasiest using the plot of versus p, since the conditions involve a single point of crossing (Fig. 11, 12).The conditions on the v p plane that correspond to the coexistence points form the binodal curves. The limit of stability of the pure vapor and pure liquid phases form the spinodal curves. These curve form the phase diagram, which predicts the conditions of coexistence of the vapor and liquid phases (Fig. 13).ReferencesMark F. Schulz, Ashish K. Khandpur, Frank S. Bates, Kristoffer Alm- dal, Kell Mortensen, Damian A. Hajduk, and Sol M. Gruner. Phase Behavior of Polystyrene Poly(2-vinylpyridine) Diblock Copolymers. Macromolecules, 29(8):28572867, January 1996. ISSN 0024-9297.

Figure 12: Plot of versus p for a =3.375 = 27 (critical point, blue), a = 3.527, a = 3.679, a = 3.831, and a = 3.982 (red). The dashed segments indi- cate conditions where a single phase is unstable.

Figure 13: Plot of p versus v for a =3.375 = 27 (critical point, blue), a = 3.527, a = 3.679, a = 3.831, and a = 3.982 (red). The solid black curves are the binodal curves (coexistence), and the dashed black curves are the spinodal curves (limit of metastability).T

T

i

i

i

i

dre transform b(c) = c 2c

.=+ .

RT

b

bRT

0

8

8

8

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Module 2