0% found this document useful (0 votes)
54 views29 pagesNche612 Chapter16
The document discusses statistical thermodynamics, focusing on the relationship between microscopic properties of matter and its bulk properties. It introduces key concepts such as the Boltzmann distribution and the partition function, which are essential for understanding the populations of molecular states at thermal equilibrium. The chapter emphasizes the significance of temperature in determining the most probable populations of states in a system composed of independent molecules.
Uploaded by
Mbali MdlaloseCopyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF or read online on Scribd
0% found this document useful (0 votes)
54 views29 pagesNche612 Chapter16
The document discusses statistical thermodynamics, focusing on the relationship between microscopic properties of matter and its bulk properties. It introduces key concepts such as the Boltzmann distribution and the partition function, which are essential for understanding the populations of molecular states at thermal equilibrium. The chapter emphasizes the significance of temperature in determining the most probable populations of states in a system composed of independent molecules.
Uploaded by
Mbali MdlaloseCopyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF or read online on Scribd
/ 29
The distribution of molecular
states
164 Configurations and weights
162 ‘The molecular partition
function
16.1-Impact on biochemistry:
‘The helix-coil transition in
polypeptides
The internal energy and
the entropy
163 Theinternal energy
164 The statistical entropy
‘The canonical partition function.
165 The canonical ensemble
166 The thermodynamic
information in the partition
function
167 Independent molecules
Checklist of key ideas
Further reading
Further information 16.1
The Boltzmann aston
Further bvormation 16.2:
The Botzmann formula
Further information 16:3
Temperatures below zero
Discussion questions
Exercises
Problems.
Statistical
thermodynamics 1:
the concepts
‘Statistical thermodynamics provides the Ink etween the microscopie propertias of matter
andits bulk properties. Two key ideas are intraduced inthis chapte. Th ists the Boltzmann
istrbution, which is used to predict the populations of states in systems. at thermal
‘equlorum. In this chapter we sae its derivation in terms ofthe stibution of particles over
available states. The derivation leads naturaly tothe inteduction of the partion function,
\which isthe central mathematical concept of this and the next chapter. We see how to
incorpret the partion function anc how to calculate iin a number of simple cases, We then
's8@ how to extract thermodynamic information from the parttion function. In the fal part
of the chapter, we generalize the discussion to inchide systems that are composed of
assombles of intoracting particles. Very slr equations are dovelapedito those in the frst
part of the chapter, but they are much more wisely appleabe.
‘The preceding chapters of this part of the text have shown how the energy levels
‘of molecules can be calculated, determined spectroscopically, and related to their
structures. The next major step isto see how a knowledge of these energy levels can
be used to account for the properties of matter in bulk. To do so, we now introduce
the concepts of statistical thermodynamics, the link between individual molecular
properties and bulk thermodynamic properties.
‘The crucial step in going from the quantum mechanics of individual molecules
to the thermodynamics of bulk samples isto recognize thatthe latter deals with the
average behaviour of large numbers of molecules. For example, the pressure of a gas
depends on the average force exerted by its molecules, and there is no need to specify
which molecules happen to be striking the wal at any instant. Nor is it necessary to
consider the fluctuations in the pressure as different numbers of molecules collide
with the wall at different moments. The fluctuations in pressure are very small com-
pared with the steady pressure: itis highly improbable that there willbe a sudden fll
in the number of collisions, ora sudden surge. Fluctuations in other thermodynamic
properties also occur, but for large numbers of particles they are negligible compared
to the mean values,
‘This chapter introduces statistical thermodynamics in two stages. The frst the
derivation of the Boltzmann distribution for individual particles, is of restricted
applicability, but thas the advantage of aking us directly toa result of central import-
ance in a straightforward and elementary way. We can use statistical thermodynamics,
‘once we have deduced the Boltzmann distribution. Then (in Section 16.5) we extend
the arguments to systems composed of interacting particles. 16.1 CONFIGURATIONS AND WEIGHTS 561
The distribution of molecular states
We consider a closed system composed of N'molecules. Although the total energy is
constant at £, itis not possible to be definite about how that energy is shared between
the molecules. Collisions result in the ceascless redistribution of energy not only
between the molecules but also among their different modes of motion. The closest
‘we can come toa description of the distribution of energy is to report the population,
of state, the average number of molecules that occupy it, and to say that on average
there are n, molecules in a state of energy & The populations of the states remain
almost constant, but the precise identities of the molecules in each state may change
atevery collision,
‘The problem we address in this section isthe calculation of the populations of states
for any type of molecule in any mode of motion at any temperature, The only restric
tion is that the molecules should be independent, in the sense that the total energy
of the system isa sum of their individual energies. We are discounting (at this stage)
the possibility that in a real system a contribution to the total energy may arise from
interactions between molecules, We also adopt the principle of equal a priori prob-
abilities, the assumption that all possibilities for the distribution of energy are equally
probable. A priori means in this context loosely ‘as far as one knows’, We have no reason
to presume otherwise than that, for a collection of molecules at thermal equilibrium,
vibrational states of a certain energy, for instance, are as likely to be populated as
rotational states of the same energy.
‘One very important conclusion that will emerge from the following analysis is that
the populations of tates depend on a single parameter, the temperature ,Thatis, statist=
ical thermodynamics provides a molecular justification for the concept of tempera
ture and some insight into this crucially important quantity.
161 Configurations and weights
Any individual molecule may exist in states with energies é, €,.... We shall always,
take €, the lowest state, as the zero of energy (&,=0), and measure all other energies
relative to that state. To obtain the actual internal energy, U, we may have to add a
constant to the calculated energy of the system, For example, if we ate considering the
vibrational contribution tothe internal energy, then we must add the total zero-point
energy of any oscillators in the sample,
(@) Instantaneous configurations
‘Atany instant there will be ny molecules in the state with energy &,,m, with ¢,,and so
on. The specification of the set of populations ny. my... in the form {ng ys.» bisa
statement of the instantaneous configuration of the system. The instantaneous
configuration fluctuates with time because the populations change. We can picture a
large number of different instantaneous configurations. One, for example, might be
{N,0. ...}, corresponding to every molecule being in its ground state. Another
right be (N— 2,2,0,0,...}, in which two molecules are in the first excited state
The latter configuration is intrinsically more likely to be found than the former
because it can be achieved in more ways: {N0,0, ... } ean be achieved in only one
‘way, but {N—2,2,0,...} can be achieved in EN(N— 1) different ways (Fig. 16.1 see
Justification 16.1), At this stage in the argument, we are ignoring the requirement
thatthe total energy of the system should be constant (the second configuration has
a higher energy than the first). The constraint of total energy s imposed later in this
section,
Soe “or he ie
Ae At at ate
ail i
Fig. 161 Whereas 3 contiguration
15:00, ..}anbe achieved in only one
‘vay, a configuration (32.0, .. } canbe
achieved in the ten diferent ways shown
here, where the tinted blocks represent
diferent molecules, 362. 16 STATISTICAL THERMODYNAMICS |: THE CONCEPTS.
Fa.162 The 18 molecules shown here can
be distributed into four receptacles
(distinguished by the three vertical lines)
in 18! diferent ways, However, 3! ofthe
selections that put three molecules the
fist receptacle ate equivalent, 6! that put
six molecules into the second receptacle are
‘equivalent and so on. Hence the number
of distinguishable arrangements is
syste,
Comment 16.1
More formally, W'is called the
‘multinomial coefficient (see Appendix 2)
In eqn 16.1,x!, x factorial, denotes
x(x=1)(x=2) andy definition
ot
Equation 16.1 isa generalization of the formula W:
the configuration {N— 2,2... }
N=18
et 8 4
If, asa result of collisions, the system were to fluctuate between the configurations
(N.000,...} and {N ~2,2,0,... }, it would almost always be found in the second,
more likely state (especially if N were large). In other words, a system free to switch
between the two configurations would show properties characteristic almost exclus-
ively ofthe second configuration. A general configuration {rim ...}can be achieved
in W different ways, where Wis called the weight of the configuration. The weight of
the configuration {ng is given by the expression
NI
a] (asa)
4N(N~ 1), and reduces to it for
Justification 18.1 The weight ofa contiguration
First, consider the weight ofthe configuration {N'=22,00...).One candidate for
fremmotion to an upper state canbe elected in N we. There are N 1 candidates
for the second choice, so the total umber of choices i NUN = 1), Howeves, we
should not distinguish the choice fck il) rom the choice (il Jac) because they
lead tothe same configurations Therefore, nly half the choices endo ditingvish-
able configurations nd the total number af ditinguishabe choices ENUN— 1).
‘Now we generalize tis remark. Consider the number af ways of distributing
[balls into bins. The fs ball canbe selected in different ways, the next ball
in N~1 diferent ways forthe balls remaining, and eo on. Therefore, there are
NON = 1)...1 = Nt waye of selecting the balls for Gisibution over the bins
However, ifthere aren balls inthe bin labelled ey there would bem erent ways
in which the same bas could have been chosen (Fig. 162). Simul, there are
in say in which the n, balls inthe bin labelled, can be chose, and #0 om
Therefore, the tota number of distinguishable ways of distributing the balls so that
there aren in sm nbn 6, ot. reparles ofthe order in whic the balls were
chosen is M'iigin! which the conten of eqn 16.1
Mlustration 16.1 Calculating the weight of a istration
‘To calculate the number of ways of distributing 20 identical objects with the
arrangement 1,0, 3,5, 10, 1, we note that the configuration is {1,0,3,5,10,1) with,
[N'=20; therefore the weight is
20!
2 gsi xiot
Tosist01
Solf-test 16.1 Calculate the weight of the configuration in which 20 objects are
distributed in the arrangement 0, 1, 5, 0, 8,0,3, 2,0, 1 [4.19% 10" 16.1 CONFIGURATIONS AND WEIGHTS 563
It will urn out to be more convenient to deal with the natural logarithm of the
‘weight, In W, rather than with the weight itself. We shall therefore need the expression
™
In W= In.
)
In NI In(ng
In NI~ (ning tn my! tin ml)
=InN!-Don}
‘where in the first ine we have used In(x/y) =In x—In y and in the second In xy =In x
+n y. One reason for introducing ln W is that itis easier to make approximations. In
particular, we can simplify the factorials by using Stirling's approximation in the form
Inxt=xInx-x (16.2)
‘Then the approximate expression forth weight
In W= (Win N=N) Yom lan,=n)=NlnN-SEnlnn, (063)
"The final form ofeqn 16.3 is derived by noting thatthe sum ofn, is equal to N, so the
second and fourth terms in the second expression cancel,
(b) The Boltzmann distribution
‘We have seen that the configuration (N'— 2,2,0,....] dominates (N, hand it
should be easy to believe that there may be other configurations that have a much
greater weight than both, We shall se, in fact, that there is a configuration with so
‘great a weight that it overwhelms all the rest in importance to such an extent that the
system will almost always be found init. The properties ofthe system will therefore be
characteristic of that particular dominating configuration. This dominating config-
uration can be found by looking forthe values of, that lead toa maximum value of W.
Because Wisa function of all the m, we can do this search by varying the n, and look-
ing forthe values that correspond to dW’=0 (just asin the search for the maximum of
any function), o equivalently a maximum value of In W. However, there ate two
». The exception isthe term with £5 = 0
(or the go terms at zero energy if the ground state is g,-fold degenerate), because then
&,/KI'= 0 whatever the temperature, including zero. As there is only one surviving
term when T'=0, and its value is g,, it follows that
Jim a=85 (16.10)
‘Thatis, at T=0, the partition function is equal to the degeneracy of the ground state
‘Now consider the case when T is so high that for each term in the sum 6/kT = 0
Because €*= 1 when x~0, each term in the sum now contributes 1. IUfollows that the
suum is equal to the number of molecular states, which in general is infinite
lim q (sn)
In some ideatized cases, the molecule may have only inte number of states then the
upper limit of qs equal tothe numberof states, For example, if we were considering
only the spin energy eves of «radical in magnetic field, then there would be only
two states (m, = +4). The partition function for such a system can therefore be
expected to rise towards 22s Tis increased towards infinity.
‘Wesce that the molecular partition function gives an indication ofthe numberof states
that are hermaly accesible toa molecule atthe temperature ofthe system. At T=0, only
the ground level is accesible and q= gy. At very high temperatures, vrcually all states
are aceasible, and qs corresponding large
Example 16.2 Evaluating the partion function for a uniform ladder of energy levels
Evaluate the partition function for a molecule with an infinite number of equally
spaced nondegenerate energy levels (Fig 16.3). These levels can be thought of asthe
vibrational energy levels ofa diatomic molecule inthe harmonic approximation.
‘Method We expect the partition function to increase from 1 at T=0 and approach
infinity as To s, To evaluate eqn 16.8 explicitly, note that
1
Lextette a
Inx
Answer Ifthe separation of neighbouring levels is ¢, the partition function is
1
GaN PM a LHe P (CPP oe
quit 1 (ey lek
This expression is plotted in Fig. 16.4: notice that, as anticipated, q rises from 1 to
infinity as the temperature is raised,
3e.
20
o.
Fin. 162 The equally spaced infinite
energy level used inthe calculation of the
‘patition function, A harmonic oscillator
‘has the same spectrum oflevels
Comment 16.3
‘The sum ofthe infinite series $= 1 ++
x?+---is obtained by multiplying both,
sides by x, which gives 25 =x +x! +x"
+++ $= Land hence S=1/(1~2).
0 5 10
ide
Fe 164 The partition funtion forthe
system showin Fig 16.3 (ahstmonie
‘vclator) ara function of temperature
[a Deaton Plot he pation
function of: harmonic ocilator
aginst temperature for several values of
the encegy separation How does qvary
vith temperature when Tis high, nthe
sense that KD € (or ls 566 16 STATISTICAL THERMOD’
MICS 1: THE CONCEPTS,
ite
Fe.165 ‘The partition function fora two-level system asa function of ter
raphe differ in the scale of the temperature axis to show the approach to Las T-> Oand the
slow approach to2 as T+ ~,
[Gag Betrtin Consera hee evel stem wih ees and 2a Hot the parton
function against (Te,
Low High
temperature temperature
a ‘Sol-test 16.3 Find and plot an expression for the partition function of a system
Cs writh one state at zero energy and another state atthe energy €
oe ee (g=1+e%, Fig. 16.5]
oe ee
It follows from eqn 16.8 and the expression for q derived in Example 16.2 for a uni-
form ladder of states of spacing €,
ee es 1
oe ee ee - (16.12)
ee ee es ‘
TT FE = 1 hatte fraction of molecules in the state with energy 6s
a am
—— -__e_2_ Pee eT (16.13)
oe 4
1 mmm m___ Figure 16.6 shoves how p, varies with temperature. Atvery low temperatures, where q
me mm —_—_ is cose to 1, only the lowest state is significantly populated, As the temperature is
fe 30 10 «0708
105 158 1993.86
Fe 166 The populations afthe energy
levels ofthe system show in Fig. 16.3
at different temperatures,
sd the
corresponding values of the partition
function calculated in Example 16.2.
[Note that B= KT
Exploration To visualize the content
of Fig, 166in different way, plot
the functions py Pi Py and p, against RTE,
falued the population breaks out ofthe lowes fate and the upper ese become
progstsivay more highly populated. At these tne the arin fnction ves
from Land its ae giver an indistion of the eng of tates populated. The name
“parton funcon elect the seve in whch q neasues how the ttl murber of
smolecleis ditibuedparttioned—over he rable state
‘The conesponding expression fra worlevl stem derived in Sets 163 are
1 of
ee Te
(asa)
‘These functions are plotted in Fig, 16.7. Notice how the populations tend towards
equality (p,=4,p,=4) as T=. A common error is to suppose that all the molecules
in the system will be found in the upper energy state when T'= ©; however, we see 16.2 THE MOLI
'LAR PARTITION FUNCTION
5
ite
Fg. 162 ‘The fraction of populations ofthe two states ofa two-level system as function of
temperature (eqn 16.14). Note that, asthe temperature approaches infinity the populations
‘of he bwo states become equal (and the factions both approach 05),
Exploration Consider a three-level system with level 0, e, and 2. Plot the Functions py
iv and pp against kT
from eqn 16.14 that, as T'—> =, the populations of states become equal, The same
conclusion is true of multi-level systems too: as T'— o, all states become equally
populated,
Example 16.3 Using tho partion function to calculate a poputation
Calculate the proportion of I, molecules in their ground, first excited, and second,
excited vibrational states at 25°C. The vibrational wavenumber is 2146 cm"
Method Vibrational energy levels have a constant separation (in the harmonic
approximation, Section 13.9), so the partition function is given by eqn 16.12 and.
the populations by egn 16.13. To use the latter equation, we identify the index
iiwith the quantum number v, and calculate p, for v=0, 1, and 2. At 298.15 K,
KT vhe=207.226 em,
Answer First, we note that
hev__214.6em"*
1.036
“KT 207.226 cx"
‘Then it follows ftom eqa 16.13 that the populations are
P= (ee 0.6454
‘Therefore, p= 0.645, p,=0.228, p;= 0.081. TheI—Thond isnot stiffand the atoms
are heavy: as a result, the vibrational energy separations are small and at room
‘temperature several vibrational levels are significantly populated. The value of the
partition function, q= 1.55, reflects this small but significant spread of populations
Solf-test 16.4 At what temperature would the v= 1 level of, have (2) half the popu-
lation of the ground state, (b) the same population as the ground state?
((a) 445 K, (b) infinite]
567 56816 STATISTICAL THERMOD)
Entropy,
uty
Magnetic
field on
0 Temperature, T
1p 166 The vechnique of adiabatic
demagnetization is used to attain very low
temperatures. The upper curve shows that
variation ofthe entropy ofa paramagnetic
system inthe absence of an applied fed
‘The lower curve shows that variation in
entropy when a field is applied and has
sail the electron magnets more orderly
‘The isothermal magnetization step is rom
‘Ato the adiabatic demagnetization step
(at constant entropy) is from Bto C.
MICS 1: THE CONCEPTS,
It follows feom our discussion ofthe partition function that to reach low tempera
tres itis necessary to devise strategies that populate the low energy levels ofa sys-
tem at the expense of high energy levels. Common methods used to reach very low
temperatures include optical trapping and adiabatic demagnetization. In optical
trapping, atoms in the gas phase are cooled by inelastic collisions with photons from
intense laser beams, which act as walls of avery small container. Adiabatic demagne-
tization is based on the fact chat, inthe absence of a magnetic field, the unpaired elee-
ttons ofa paramagnetic material are oriented at random, but inthe presence of 3
‘magnetic field there are more f spins (m,= 4) than 0: spins (m,= +4). In thermo-
dynamic terms, the application ofa magnetic eld lowers the entropy ofa sample and,
ata given temperature, the entropy ofa sample is lower when the field ison than when
itis off. Even lower temperatures can be reached if nuclear spins (which also behave
like small magnets) are used instead of electron spins in the technique of adiabatic
nuclear demagnetization, which has been used to cool a sample of silver to about
280 pK. In certain circumstances it is posible to achieve negative temperatures, and
the equations derived ate inthis chapter can be extended to T'< 0 with interesting
consequences (see Further information 16.3).
Mlustration 16.2. Cooing a sample by adiabatic demagnetization
Consider the situation summarized by Fig. 16.8. A sample of paramagnetic
material, such asad- or f-metal complex with several unpaired electrons, is cooled
to about 1 K by using helium. The sample is then exposed to a strong magnetic
field while it is surrounded by helium, which provides thermal contact with the
cold reservoir. This magnetization step is isothermal, and energy leaves the system,
as heat while the electron spins adopt the lower energy state (AB in the illustra-
tion). Thermal contact between the sample and the surroundings is now broken
by pumping away the helium and the magnetic field is reduced to zero. This
step is adiabatic and effectively reversible, so the state ofthe sample changes from
B to C. At the end of this step the sample isthe same as it was at A except that it
nowhas alower entropy. That lower entropy in the absence of amagnetic field cor-
responds to a lower temperature. That is, adiabatic demagnetization has cooled
the sample.
(b) Approximations and factorizations
In general, exact analytical expressions for partition functions cannot be obtained.
However, closed approximate expressions can often be found and prove to be very
important in a number of chemical and biochemical applications (Impact 16.1). For
instance, the expression for the partition function for a particle of mass m free to move
in a one-dimensional container of length X can be evaluated by making use ofthe fact
that the separation of energy levels is very small and that large numbers of states are
accessible at normal temperatures. As shown in the Justification below, inthis case
on “x (16.15)
wl Tp (18.5
‘This expression shows that the partition function for translational mation increases
with the length of the box and the mass of the particle, for in each case the separation
ofthe encrgy levels becomes smaller and more levels become thermally accessible, For
a given mass and length of the box, the partition function also increases with increas-
ing temperature (decreasing f), because more states become accessible. 16.2 THE MOLI
Justifieation 16.2. The parton function fora parte na one-dimensional box:
‘The energy levels of a molecule of mass m in a container of length X ate given by
eqn 94a with
1
ox?
neL2,
‘The lowest level (n= 1) has energy i#/8mX, so the energies relative to that level are
1. However, V/N is the volume occupied by a single particle, and there-
fore the average separation of the particles is d = (VIN)!®. The condition for there
being many states available per molecule is therefore d/A?>> 1, and therefore d>> A
‘That is, for eqn 16.19 to be valid, the average separation ofthe particles must be much
_greater than their thermal wavelength. For 1, molecules at | bar and 298 K, the aver-
age separation is 3 nm, which is significantly larger than their thermal wavelength
(71.2 pm, Illustration 16.3), 116.1 IMPACT ON BIOCHEMISTRY: THE HELIX-COIL TRANSITION IN POLYPEPTIDES
WV IMPACT ON BIOCHEMISTRY
116.1 The helix-coil transition in polypeptides
Proteins are polymers that attain well defined three-dimensional structures both in
solution and in biological cells. They ate polypeptides formed from different amino
acids strung together by the peptide link, CONTI. Hydrogen bonds between amino
acids of a polypeptide give rise to stable helical or sheet structures, which may collapse
into a random coil when certain conditions are changed. The unwinding of a helix
into a random coil isa cooperative transition, in which the polymer becomes increas-
ingly more susceptible to structural changes once the process has begun. We examine
here a model grounded in the principles of statistical thermodynamics that accounts
for the cooperativity ofthe helix-coil transition in polypeptides.
To calculate the fraction of polypeptide molecules present as helix or coil we need
to set up the partition function for the various states of the molecule. To illustrate the
approach, consider a short polypeptide with four amino acid residues, each labelled
ifit contributes to a helical region and cif it contributes to a random coil region, We
suppose that conformations hhh and cece contribute terms q,and q,, respectively, to
the partition function g. Then we assume that exch of the four conformations with
one amino acid (such as hhh) contributes q,. Similarly, each of the six states with
two camino acids conteibutes aterm ,, and each of the four states with three ¢ amino
acids contributes a term q,, The partition function is then
(44% 4g
jog, + 69, +4954 9¢= 9 14 “Bt
dy do ed
64,
‘We shall now suppose that each partition function differs from q, only by the energy
of each conformation relative to hhith, and write
4
%
‘Next, we suppose that the conformational transformations are non-cooperative, in
‘the sense thatthe energy associated with changing one ht amino acid into one camino
acid has the same value regardless of how many hor camino acid residues are in the
reactant or product state and regardless of wherein the chain the conversion occurs
‘That is, we suppose that the difference in energy between ch and cH?“ has the
same value 7 forall. This assumption implies that ¢,~ €,~ iyand therefore that
nae (16.20)
sok
plltds4 6a 4s st)
a
where I= Nya sis called the stability parameter, Te term inp
form of the binomial expansion of (1 +5)4
‘ “
Sete wih ot6i=
» os
theses has the
(6.21)
G=nH
Which we interpret as the number of ways in which a state with i camino acids can be
formed,
‘The extension of this treatment to take into account a longer chain of residues is
‘now straightforward: we simply replace the upper limit of 4 in the sum by r:
4 So
Fey cin as! (16.22)
% »
A cooperative transformation is more difficult to accommodate, and depends on
building a model of how neighbours facilitate each other's conformational change. In
Comment 16.6
‘The binomial expansion of (1 +3)"is
S71 57216 STATISTICAL THERMOD’
0.5
["
o4 1
0.05|
Fe.169 The ditibution ofp, the fraction
‘of molecules that hasa number iof camino
acids for s= 0.8 (= 1.1), 1.0 (()=338),
and 15 (()= 15), with = 5.0 10°.
Fe 1610 Plots ofthe degree of conversion
6, againsl sor several values of 6: The
‘eaves show the sigmoidal shape
characteristics of cooperative behaviour
MICS 1: THE CONCEPTS,
the simple zipper model, conversion from h to cis allowed only ifa residue adjacent to
the one undergoing the conversion is already a cresidue. Thus, the zipper model allows
a transition of the type. .hhhch, ..>.. hice... But not a transition of the type
Iihhch...>.. heheh, .., The only exception to this rule is, of couse, the very
first conversion from h to ¢ in a fully helical chain. Cooperativity is included in the
zipper model by assuming that the first conversion from h to ¢ called the nucleation
sep, is less favourable than the remaining conversions and replacing sfor that step by
0s, where a 1. Each subsequent step is called a propagation step and has a stability
parameters. In Problem 16.24, you are invited to show that the partition function is
aes Denar (4.23)
where Z(r,i) is the number of ways in which a state with a number i of c amino acids
canbe formed under the strictures ofthe siper model Because Z(ni)=n—i + 1 (ee
Problem 1624),
geltone yD e- oS s2
After evaluating both geometric series by using the two relations
ket
ine"! (n+ 1)x"+1]
we find
onlst~ (nt Ds"+1]
eee
‘The fraction p,= 4\/q of molecules that has a number i of ¢ amino acids is p=
{(n 1+ 1}on'I/g and the mean value fis then () = Sip, Figure 16.9 shows the dis-
tribution ofp, for various values of s with @= 5.0 % 07, We see that most of the
polypeptide chains remain largely helical when s< 1 and that most ofthe chains exist
Jargely as random coils when s> 1. When s~ 1, there is a more widespread distibu-
tion of length of random coil segments, Because the degree of conversion, 8, of a
polypeptide with n amino acids toa random col is defined as 8=(i)/n, it is possible to
show (see Problem 16.24) that
1a
oat.
nada)
‘hiss general ssult that applies to any model ofthe helix-coil transition in which
the partition function q is expressed as a function ofthe stability parameters.
‘Amore sophisticated model fr the helix-coil transition must allow fr helical seg-
ments form in diferent regions of long polypeptide chai, with the nasces
being separated by shrinking cil segments. Calculations based on this more complete
Zimons-Bragg model give
( (= D420 )
o=4 1+
(s-1)* +4s0)'
Figure 16.10 shows plots of @ against s for several values of o. The curves show the
sigmoidal shape characteristic of cooperative behaviour, There is a sudden surge of
‘transition to a random coil as spasses through 1 and, the smaller the parameter 6, the
greater the sharpness and hence the greater the cooperativity ofthe transition. Thatis,
the harder itis to get coil formation started, the sharper the transition from helix to coi
Ing (16.25)
slices
(1826) The internal energy and the entropy
‘The importance of the molecular partition function is that it contains all the informa-
tion needed to calculate the thermodynamic properties of a system of independent
particles, In this respect, q plays a role in statistical thermodynamics very similar
to that played by the wavefunction in quantum mechanics: q is a kind of thermal
wavefunction,
163 The internal energy
‘We shall begin to unfold the importance of q by showing how to derive an expression
for the internal energy of the system,
(a) The relation between U and
“The total energy ofthe system relative to the energy of the lowest state is
B=Dng, (16.27)
Because the most probable configuration is so strongly dominating, we can use the
Boltzmann distribution for the populations and write
(16.28)
‘To manipulate this expression into a form involving only q we note that
get = ets
3 a
{Iefollows that
(16.29)
lMustration 16.4 The energy ofa two-level system
+, we can deduce that the total
From the two-level partition function
energy of N two-level systems is
‘This function is plotted in Fig. 16.11, Notice how the energy is zero at T= 0, when
only the lower state (at the zero of energy) is occupied, and rises to SNeas T=,
when the two levels become equally populated.
‘here are several points in relation to eqn 16.29 that need to be made. Because
f= 0 (remember that we measure all energies (rom the lowest available level),
E should be interpreted as the value of the internal energy relative to its value at
T=0, U(0). Therefore, to obtain the conventional internal energy U, we mustadd the
internal energy at T=0:
u
HO) VE (16.30)
163 THEINTERNALENERGY 573
04pm ’
g ~
Atte
5 70
ktie
‘Rg. 16:11 The total energy ofa tworlevel
system (expressed asa multiple of Né)
asa function of temperature, on two
temperature sales, The graph atthe top
shows the slow rise away from zero energy
as low temperatures; the lope of the graph
at T=0's 0 (thats, the heat capacity is
zero at T=0). The graph below shows the
slow rise to 054s T > = as both states
become equally populated (see Big. 167)
Exploration Draw graphs similar to
‘hose in Fig, 161 for three-level,
system with levels 0, € and 26 574
16 STATISTICAL THERMOD’
MICS 1: THE CONCEPTS,
Secondly, because the partition function may depend on variables other than the
temperature (for example, the volume), the derivative with respect to Bin eqn 16.29
is actually a partial derivative with these other variables held constant, The complete
expression relating the molecular partition function to the thermodynamic internal
energy ofa system of independent molecules is therefore
¥(3)
U=0) (16314)
a By
An equivalent form is oblained by noting that dv/x=d ln x
(Inq)
v=u@)-n| 4 (163i)
eB Jy
These two equations confirm that we need know only the partition function (as a
function of temperature) to calculate the internal energy relative to its value at T= 0.
(b) The value of 6
‘We now confirm that the parameter f, which we have anticipated is equal to VKI;,
does indeed have that value. To do so, we compare the equipartition expression for
the internal energy of a monatomic perfect gas, which from Molecular interpretation
2.2.we know to be
(16.328)
with the value calculated from the translational partition function (see the following
Justification), which is
v= v@)+— 16.326)
+35 (16.326)
It follows by comparing these two expressions that
p=——=—™s (16.33)
POoRT GNA (609)
(We have used N'= nN, where 1 isthe amount of gas molecules, N, is Avogadro's
constant, and R = Nk.) Although we have proved that B= /KT by examining a very
specific example, the translational motion ofa perfect monatomic gas, the result is
general (see Example 17.1 and Further reading)
Justification 16.3 The intemal energy of perfect gas
‘To use eqn 16.31, we introduce the translational partition function from eqn 16.19:
fa) (av) a1 ven
lap) > lane | Vana aap
‘Then we note from the formula for in eqn 16.19 that
da df hp
B al Gam
and so obtain
(22)
wv
tal apa 164 THE STATISTICAL ENTROPY
‘Then, by eqn 16313,
a) av) aN
v |laae}
V=UI0)-
asin eqn 16.32b,
164 The statistical entropy
{fits true that the partition function contains all thermodynamic information, then,
{it must be possible to use it to calculate the entropy as well asthe internal energy
Because we know (from Section 3.2) that entropy is related to the dispersal of energy
and that the partition function is a measure of the number of thermally accessible
slates, we can be confident that the (wo are indeed related,
‘We shall develop the relation between the entropy and the partition function in two
stages. In Further information 16.2, we justify one of the most celebrated equations in
statistical thermodynamics, the Boltzmann formula for the entropy:
s-klnW [16.34]
In this expression, Wis the weight of the most probable configuration of the system,
In the second stage, we express Win terms ofthe partition function.
‘The statistical entropy behaves in exactly the same way as the thermodynamic
‘entropy. Thus, a the temperature is lowered, the value of W, and hence ofS, decreases
because fewer configurations are compatible with the total energy. Tn the limit T — 0,
W=1,so ln W=0, because only one configuration (every molecule in the lowest level)
is compatible with £=0, It follows that S—> 0s T+ 0, which is compatible with the
‘Third Law of thermodynamics, that the entropies ofall perfect erystals approach the
same value as T—>0 (Section 3.4),
‘Now we relate the Boltzmana formula for the entropy to the partition function,
To do so, we substitute the expression for In W given in eqn 16.3 into eqn 16.34 and,
as shown in the Justification below, obtain
v-v0)
+NkIng (16.35)
Justifieation 16.4 The statistical entropy
‘The fs stageis to ase gn 163 Gn W= Nia NE, nian) and N=¥ tows
AZo =-AE png
= nN the faction of molecules in stat i It fellows from eqn 16,7 that
Inp.=-Be-Ing
and therefore that
SM BY 26 - Lplag) = WHlU- UO} +Nking
‘We have wsed the fact thatthe sum over the sequal to 1 and that from eqas 16.27
and 1630)
NZpe= Dvne= Dane= Dae
‘We have already established that = 1/K1, s eqn 1635 immediately follows
8=kS (mtn Nn, In
where
— U(0) 576 16 STATISTICAL THERMOD’
0 5 10
ite
Fe. 1612 The temperature variation ofthe
‘entropy ofthe system shown in Fig. 163
(exprested here asa multiple of NE). The
‘entropy approaches er0 as T-»0,and
increates without limit a8 T ©,
Exploration Plot the function 48/47,
‘the temperature coefficient of the
entropy, agains Ie. Is there a
temperature a which thie coelicient passes
through a maximum? Ifyou find
‘maximum, explain it physical origins.
Fe1612 The temperature variation ofthe
entropy of two-level system (expressed as
multiple of Ni). As T=, the two states
become equally populated and S
approaches Nkln 2
Exploration Draw graphs similar to
those in Fg. 16.13 fora three-level
system with levels 0, e, and 26
MICS 1: THE CONCEPTS,
Example 16.4 Calculating the entropy ofa collection of oscitators
Calculate the entropy of a collection of N independent harmonic oscillators, and
evaluate it using vibrational data for I, vapour at 25°C (Example 16.3).
Method To use eqn 16.35, we use the partition function for a molecule with evenly
spaced vibrational energy levels, eqn 16.12. With the partition function avaiable,
the internal energy can be found by differentiation (as in eqn 16.31a), and the two
expressions then combined to give S.
Answer The molecular partition function as given in eqn 16.12 is
1
oe
‘The internal energy is obtained by using eqn 16.31:
N(aq) _Nee* Ne
gop), 1 Pa
‘The entropy is therefore
Be
ef Pa waa
‘This function is plotted in Fig. 16.12, For kat 25°C, Be:
Sq,=8.38 JK" mol"!
u-u0o)
036 (Example 16.3), 30
Self-test 16.6 Evaluate the molar entropy of N two-level systems and plot the
resulting expression. What isthe entropy when the two states are equally thermally
accessible?
[SINK= Bet(1 + ef) +1n(1 +e); se Fig 16.13; $=Nkln2]
SING
SINK
rr ind
ite idle 16.5 THECANONICALENSEMBLE 577
The canot ion function
al pal
In this section we see how to generalize our conclusions to include systems composed
ofinteracting molecules. We shal also see how to obtain the molecular partition func-
tion from the more general form of the partition function developed here.
165 The canonical ensemble
“The crucial new concept we need when treating systems of interacting particles isthe
‘ensemble’, Like so many scientific terms, the term has basically its normal meaning of
‘collection’, but ithas been sharpened and refined into a precise significance.
{a} The concept of ensemble
To set up an ensemble, we takea closed system of specified volume, composition, and
temperature, and think ofit as replicated N times (Fig. 16.14). All the identical closed
systems are regarded asbeing in thermal contact with one another, so they can exchange
‘energy. The total energy of all the systems is E and, because they are in thermal
‘equilibrium with one another, they all have the same temperature, T. This imaginary
collection of replications of the actual system with a common temperature is called
‘the canonical ensemble
‘The word ‘canon’ means ‘according to a rule’. There are two other important
‘ensembles, In the microcanonical ensemble the condition of constant temperature is
replaced by the requirement that all the systems should have exactly the same energy:
‘ach system is individually isolated. In the grand canonical ensemble the volume and
temperature of each system is the same, but they are open, which means that matter
‘an be imagined as able to pass between the systems the composition of each one may
‘uctuate, but now the chemical potential isthe same in each system:
Microcanonical ensemble: N, V, Ecommon
Canonical ensemble: N, V; Tcommon,
Grand canonical ensemble: 1, V, Teommon
‘The important point about an ensemble s that tsa collection of imaginary replica
tions ofthe system, so we are free to let the number of members be as large as we likes
‘when appropriate, we can let NV become infinite. The number of members of the
‘ensemble in a state with energy is denoted fi, and we can speak of the configuration
‘of the ensemble (by analogy with the configuration of the system used in Section 16.1)
and its weight, W. Note that Vis unrelated to N, the number of molecules in the
actual system; Nis the number of imaginary replications ofthat system,
{b) Dominating configurations
Just asin Section 16.1, some ofthe configurations ofthe ensemble will be very much
‘more probable than others. For instance, itis very unlikely thatthe whole ofthe total
‘energy, E, will accumulate in one system. By analogy with the earlier discussion, we
‘ean anticipate that there will be a dominating configuration, and that we can evaluate
the thermodynamic properties by taking the average over the ensemble using that
single, most probable, configuration. In the thermodynamic limit of N — =, this
‘dominating configuration is overwhelmingly the most probable, and it dominates the
properties ofthe system virtually completely
“The quantitative discussion follows the argument in Section 16.1 with the modifica-
tion that N and mare replaced by N and i, The weight ofa configuration (mp,,..}i8
452+
4S2N
4x26
Ba
Vv
tr
; | |
v
Tr T
fig. 1654 A sepresentation ofthe canonical
«ensemble inthis ease for N=20, The
‘individual replications of the actual system
all have the same composition and volume.
“They are ll in mutual thermal contact, and
soall have the same temperature, Energy
raybe transferred between them as heat,
and so they do notallhave the same
energy. The total energy £ ofall 20,
replications isa constant because the
ensemble isisolated overall.
az
sz 57816 STATISTICAL THERMOD’
]Width of
range
Number of
states
Energy
1618 The energy density of states isthe
‘number of states inan energy range divided
by the width of the range.
Sasa
Eneray
Fg 1616 To construct the form of the
distribution of members of the canonical
cenemblein terms oftheir energies, we
multiply the probability that any one sin a
state of given energy, eqn 1639, by the
number of states corresponding to that
energy (a steeply rising function). The
product isa sharply peaked function at the
sean energy, which shows that almost all
the members ofthe ensemble have that
energy.
(1636)
The configuration of greatest weight, subject the constraints thatthe total energy of
the ensemble i constant at £and that the total number of members is Gxed at N, is
given by the eanonical distribution
* ade (16.37)
‘The quantity Q, which is function of the temperature is calle the eanonical parti-
tion function.
{¢) Fluctuations from the most probable distribution
The canonical distribution in eqn 16.37 is only apparently an exponentially decreas-
ing function of the energy of the system. We must appreciate that eqn 16.57 gives
the probability of occurrence of members in a single state i of the entire system of
energy E,. There may in fact be numerous states with almost identical energies. For
example, ina gas the identities of the molecules moving slowly or quickly can change
without necessarily affecting the total energy. The density of states, the number of
states in an energy range divided by the width of the range (Fig. 16.15), is a very
sharply increasing function of energy. It follows thal the probability of a member of
an ensemble having a specified energy (as distinct from being in a specified stat) is
given by eqn 16.37, a sharply decreasing function, multiplied by a sharply increasing
function (Fig. 16.16). Therefore, the overall distribution is a sharply peaked function.
We conclude that most members of the ensemble have an energy very close to the
‘mean value.
166 The thermodynamic information in the partition function
Like the molecular partition function, the canonical partition function carries all the
thermodynamic information about a system. However, Q is more general than q
because it does not assume that the molecules are independent. We ean therefore use
Qto discuss the properties of condensed phases and real gases where molecular inter~
actions are important.
{2} The internal energy
If the total energy ofthe ensemble is B, and there are N members, the average energy
‘ofamember is E= B/N. We use this quantity to calculate the internal energy ofthe sys-
tem in the limit of N (and F) approaching infinity
HO)+RIN as Noe (16.38)
‘The fraction, p, of members of the ensemble in a state i with energy Eis given by the
analogue of eqn 16.7 as
(4839)
It follows that the internal energy is given by
+ Dae word,
uv
(aso) 16.7 INDEPENDENT MOLECULES
By the same argument that led to eqn 16.31,
1 x)
v=v-2|S] =
‘ a,
(©) The entropy
‘The total weight, W, ofa configuration of the ensemble isthe produet ofthe average
‘weight W’of each member of the ensemble, W= W®. Hence, we can calculate from
(ast)
k
SokinWokla Wi =— nw (16.42)
follows, by the same argument used in Section 16.4, that
u- uo)
+kIng (16.43)
T
167 Independent molecules
‘We shall now see how to recover the molecular patttion function from the more
general canonical partition function when the molecules are independent. When
the molecules are independent and distinguishable in the sense to be described), the
relation between Q and qis
Q-a® (16.44)
Justifieation 16.5 Tho lation between Q-and
‘The total energy of collection of N independent molecules is the sum ofthe enet=
gies ofthe molecules. Therefore, we cam write the total energy ofa sate of the
systemas
Exe (i) +E(Q)+--- +6000)
In this expression, (1) isthe energy of molecule 1 when the system sin the tate
(2) the energy of molecule 2 when the eytem iin the same tate, and soon. The
canonical partition function is then
Soctensaae tuto
‘The sum over the states of the system can be reproduced by leting each molecule
center all its own individual states (although we meet aa important proviso shorty)
‘Therefore, instead of summing over the states i of the system, we can sam over all
‘the individual states § of molecule 1, all the states {of molecule 2, and so on. This
rewriting ofthe original expression leads to
aie) Bele
{a} Distinguishable and indistinguishable molecules
fall the molecules are identical and free to move through space, we cannot distin-
{guish them and the relation Q = q™ is not valid. Suppose that molecule 1 is in some
state a, molecule 2is in b, and molecule 3 isin ¢, then one member of the ensemble
hhas an energy F = €, + 6, + & This member, however, is indistinguishable from
‘one formed by putting molecule 1 in state b, molecule 2 in state ¢, and molecule 3 in
state a, or some other permutation. There are six such permutations in all, and N! in
579 580
16 STATISTICAL THERMOD’
MICS 1: THE CONCEPTS,
general. In the case of indistinguishable molecules, it follows that we have counted
‘too many states in going from the sum over system states to the sum over molecular
states, so writing Q= q* overestimates the value of Q. The detailed argument is quite
involved, but at all except very low temperatures it turns out thatthe correction factor
is LIN Therefore:
+ For distinguishable independent molecules: Q= 4 (16.45)
+ For indistinguishable independent molecules: Q=q¥/N! (16.450)
For molecules to be indistinguishable, they must be of the same kind: an Ar atom
is never indistinguishable from a Ne atom. Their identity, however, is not the only
criterion, Each identical molecule in a crystal lattice, for instance, can be ‘named! with
a set of coordinates, Identical molecules in a lattice can therefoxe be teated as dis-
tinguishable because their sites are distinguishable, and we use eqn 16.458. On the
cother hand, identical molecules in a gas are free to move to different locations, and
there is no way of keeping track of the identity of a given molecule; we therefore use
eqn 16456,
(b) The entropy of a monatomic gas.
‘An important application of the previous material is the derivation (as shown in the
Justification below) of the Sackur—Tetrode equation for the entropy of a monatomic
a8
Sank
(ev h
(16.468)
In Aa
GN (Grok
Tis equation implies that the molar entropy of a perfect gas of high molar mass is
greater than one of low molar mass under the same conditions (because the former
has more thermally accesible translational states), Because the gas is perfect, we can
use the relation V= ntTip to express the entropy in terms ofthe pressure as
(16.466)
Justification 16.6 The Sackur-Tetrode equation
For a gas of independent molecules, Q may be replaced by gM/NT, with the result
that eqn 16.43 becomes
v-ulo)
+ NkIng- kin Nt
Because the number of molecules (N'= nN) in a typical sample is large, we can use
Stirling's approximation (eqn 16.2) to write
v-u10)
+nRln g—nRln N+ nR
The only mode of motion fr a gas of atoms is translation, and the patti func-
tion isq= VIA? (eqn 16.19), where Ais the thermal wavelength. The internal energy
is given by egn 16.32 s0 the entropy is
v ) v \
Zn + nk{ ln —In aN, +1|=nk{ Ine? +In——In nN, tne]
¥ i]
which rearranges into egn 15.46, Example 16.5 Using the Sackur-Tetrode equation
CHECKLIST OF KEY IDEAS 581
Calculate the standard molar entropy of gaseous argon at 25°C.
Method ‘To caleslate the molar entropy, 5, from eqn 16.48b, divide both sides by
1n, To calculate the standard molar entropy, Sz st p= p”in the expression for Sy
Kr
ra
‘Answor ‘The mass of an Ar atom is m=39.95 u, At 25°C, its thermal wavelength is
16.0 pm (by the same kind of calculation asin stration 16.3) Therefore,
inf _ 2 (421074)
"Y GO"N a) x (60x10 >
We can anticipate, on the bass of the number of accessible states fora lighter
‘molecule, that the standard molar entropy of Neis likely to be smaller than for Ar;
its actual value is 17.60R at 298 K
cs '8.6R= 155 JK mol?
Solf-test 16.7 Calculate the translational contsibution to the standard molar
entropy of H, at 25°C. [142]
‘The Sackur~Tetrode equation implies that, when a monatomic perfect gas expands
isothermally from V, to V,, its entropy changes by
AS=nR In(aV,) —nR In(a¥) (16.47)
where aV is the collection of quantities inside the logarithm of eqn 16.46a. This is
‘exactly the expression we obtained by using classical thermodynamics (Example 3.1)
Now, though, we see that that classical expression is in fact a consequence of the
increase in the number of accessible translational states When the volume of the con-
tainer is inereased (Fig. 16.17).
Checklist of key ideas
()
(a)
Fa. 1617 Asthe width ofa container ie
increased (going from (a) to (b)), the
energy levels become closer together (as
UL?), and asa result more are thermally
accessible ata given temperat
Consequently, the entropy ofthe system
rises a the container expands
1. Theinstantancous configuration ofa system of N molecules is
8 :
the specification of the set of populations n,,... ofthe
energy levels €;,... The weight W of a configuration it
tiven by W=NUngn,!
Os
solecues)
‘The Boltzmann distribution gives the numbers of molecules
in cach sate ofa system at any temperature: N= NeP/s,
‘The entropy interme of the partition function i
S={U- UOT +Nkn g (ait
— UO)}T+Nkln NK
shale molecules) of
1) (indistinguishable
7. The canonical ensembles an imaginary collection of
a
replications ofthe actual eyetem with a common temperature.
B= Ur.
‘The partition function is defined as g= ¥,e"*andis an
indication ofthe numberof thermally accesible states
a the temperature af interest,
The internal energy is U(T) = U(0) +E, with
B= ~{Nig)(09/2f5),=-NC@ ln gy
1 5, The Boltzmann formula for the entropy is S=KIn W,
‘where Wis the number of different ways in which the
molecule of system can he arranged while keeping the
same total ener
o4
Os
Os.
Oho.
On.
‘The canonical distribution ie given by N= eV,
The canonical partition function, Q= Ze,
ropy ofan ensemble are,
V- UOT
‘The internal energy and.
respectively, U=U10) (Bn QI@B)y and
Vela @.
For distinguishable independent molecules we write Q=4%
For indistinguishable independent molecules we write
Q=qXiN
The SackurTetrode equation, eqn 16.46, isan expression for
the entropy of a monatomic gas, 582
Further reading
16 STATISTICAL THERMODYNAMICS I: THE CONCEPTS
Antics and texts
1D. Chandler, Introduction to modern statistical mechanics. Oxford
University Press (1987).
DA, McQuarrie and JD. Simon, Molecular thermodynannis,
University Science Books, Sausalito (1999).
Further information
KE. van Holde, W.C. Johnson, and PS. Ho, Principles of physical
Biochemistry. Prentice Hall, Upper Saddle River (1988)
J. Wisiak, Negative absolute temperatures, a novelty. J. Chem. Blu,
177,518 (2000)
Further information 16.1. The Botamann alstrbution
‘We remarked in Section 16. that In W is easier to handle than W.
"Therefore to find the form ofthe Boltzmann distribution, we look
forthe condition for In W being a maximum rather than dealing
izecly with W. Because In W dependis on all the , when a
configuration changes and then, change ton, + dn, the function
Jn Wehanges to ln Wt dia W, where
Allis expression state ie that a change in In Wis the sum of
Contributions arising from changes in each value fn, Ata
‘maximum, dn W=0, Hossever, when the n, change, they da so
subject to the two constraints
Lesn=0 Dan=0
“The ist constraint recognizes that the total energy must not change,
and the second recognizes that the (ol number of molecules must
not change, These two constraints prevent us from solving d In W=0
simply by setting all (ln Wian,) =O because the dn, are not all
independent.
‘The way o take constraints into account was devised bythe French
‘mathematician Lagrange, and is aed the method of undetermined
multipliers The techniques described ip Appendix Allwe need
haere the rule that a constraint should be multiplied by a constant
snd then added tothe main variation equation, The variables are
then treated as though they were all Independent, and the constants
reevaluated atthe end ofthe calculation.
‘We employ the technique as follows. "The two constraints in eqn
16.48 ae multiplied by the constants —Pland a respectively (he
‘mins sign in ~Phas been included for future convenience), and then
added tothe expression for dln W:
& RE neo an eed
aes
6.48)
dln W:
All he dr, are now treated a independent. Hence the only way of
satisfying dln W=Ois to requite that, for each,
amw
Fata fe=0 (15.49)
when then have their most probable values
Differentiation of n Was given in eqn 16.3 with respect ton, gives
2inns Ds 2 la,
amw
my,
lod (2H), [nN
“a Eee)
am, Lm,
InN on InN+1
oat
‘The In. Nin the iret term on the right in the second line arses because
1 +1 +~--and go the derivative of N with respect to-anyof the
1,61 thats, @NT@n,= 1 The second term on the right inthe second
line arises because (in N)/@n,= (1/NYANTB, The final 1s then
‘obtained inthe same way ain the preceding remat, by using
aN, 1
For the derivative ofthe second term we fist note that ‘with 6, the Kronecker delta (6, J. 6,=0 otherwise). Then
& [S)enol)]
* )on-[24}}
Atlan)
Tn
-S4loH)61)-In
and therefore
ainw
2m,
(nn, D+ (nN
Iefolows from eqn 16.49 that
win = Be,
wre ne
and therefore that
Ti gt,
N
Actin stage we note that
(650)
which is eqn 16.6
Further information 162 The Botzmann formula
Achange in the internal energy
vev@+ Dag ss)
may arse from cithera modification ofthe energy level of
system (when €, changes to €, + de) or ftom a modification ofthe
populations (when n, changes to, di). The most general change
is therefore
d= a0) + D nde Sed,
Because the energy vels do not change when a system is heated a
constant volume (Fig, 16.18) inthe absence of ll changes other than
heating
wv=Z. ean,
(16.52)
FURTHER INFORMATION 583
Heat
(a)
Work
(b)
fe 16.18 (2) When a system is heated, the energylevels are
‘unchanged but their populations are changed. (5) When work
is done ona system, the energy levels themselves are changed
‘The level this case ate the one-dimensional partclesin-a-box
energy level of
and move apart art length ie decreased
ter 9: they depend on the size of the container
We know from thermodynamics (and specifically from eqn 3.43) that
under the same conditions
(2653)
For changes in the most probable configuration (the only one we
need consider), we rearrange eqn 16.49 to
alnw:
™m 584
snd find that
(22 on sede
But because the number of molecules is constant, the sum over the
ary iszer0. Hence
(me
on=tain
‘Thierelation strongly suggests the definition $=kln W, ae in
eqn 16.34
Further information 16.3. Temperatures below zero
‘The Bollemann distsibution tells ws that the ratio of populations ins
two-level system ata temperature Tis
Ny,
N
or (654)
‘where eis the separation of the upper state N, and the lower stateN.
Ikfllowe that, if we can contrive the population of the upper state to
«exceed that ofthe lower state, then the temperature must have a
negative value. Indeed, fora general population,
fk
poh
InWIND
(16355)
sand the temperature is formally negative fr all N, > N.
All the statistical thermodynamic expressions we have derived
apply to T<0 aswell sto T> 0, the diference being that states with
T<0 are notin thermal equilibrium and therefore have to be
achieved by techniques that do not rely on the equalization of
temperatures ofthe system and ts surroundings. The Third Law of
thermodynamics prohibit the achievement of absolute zero in 2
finite number of steps. However, itis posible to circumvent this
restriction in systems that have a finite number oflevele or in
systema that ae effectively finite because they have such weak
«coupling to their surroundings. The practical realization of sucha
system isa group of spin nuclei that have very long relaxation
times, such asthe "nucle in cold solid LiF Pulse techniques in
NMR can achieve non-equilibrium populations (Section 15.8) as
«an pumping procedures in laser technologies (Section 14.8).
From now on, we shall suppose that these non-equilibrium
0, and ate shown in Fig, 16.19,
‘We see that q and U show sharp discontinuities on passing through
zet0, and T= +0 (corresponding to al population inthe lower state)
is quite distinct from I™=~0, where all he population isin the upper
sale, The entropy Sis continuous at T=0, But al these functions are
continuous ifwe use B= UATas the dependent variable (Fig, 1620),
‘which shoves that Bisa more natural, fess familar, variable than T
[Note shat U—> Oas > (thatis, as > 0, when only the lower
slate is occupied) and U—> Neas > (hati, as T—>-0);
16 STATISTICAL THERMODYNAMICS I: THE CONCEPTS
Partition function, q
Internal energy, UINE
Entropy., SINK
kNie
Fig.1619 The patition function, internal energy, and entropy of
‘s0o-level system extended to negative temperatures
wwe se that state with T= -0is‘hotte than one with T= 10. The
cniropy ofthe system is zero on citer side of T= 0, andses to
Nkin2 as T > 4,ACT=+0 ony one sat isacesible (the ower
sate), only the upper state i accesible, othe entropy it zer0in
cach eae
‘We get more insight int the dependence of thermodynamic
properties on temperature by acting the thermodynamic result
{Section 38) that T= (95/90), When Sis potted aginst U tora
two-level system (Fig, 6.21), We se that the entropy ssa energy
is supplied to the aysem at we world expect) provided thst > 0
(Che thermal equilibrium regime). However, the entropy decreases
as energy is supplied when T <0, This conclusion is consistent with,
the thermadynamic definition of envopy, dS=da,/T (where, of
course, q denotes heat and not the patton function). Physical,
the increase in entopy for I'>0 corresponds to the increasing
acest of the upper stat, and the decrease fr T<0 coresponds Partition function, @
Internal energy, U/Ne
Entropy., SINK
ekT
Fg. 1620 The partion function, internal energy, and entropy of
two-level system extended to negative temperatures but plated
against B= VAT (modified hereto the dimensionless quantity e/kT).
Discussion questions
DISCUSSION QUESTIONS 585
Internal energy, UINe
Fe 16.21 The variation ofthe entropy with internal energy fora
towo-level system extended to negative temperatures
to the shift towards population ofthe upper state alone as more
«energy is packed into the system,
“The phenomenological laws of thermodynamics survive largely
intact at negative temperatures, The First Lae (in essence, the
conservation of energy) is robust, and independent of how
populations are distributed over states, The Second Law survives
because the definition of entropy survives (as we have seen above)
‘he efficiency ofheat engines (Section 3.2), which isa direct
consequence ofthe Second Lav, iil given by 1 ~ Ta Ta
However, ifthe temperature af the cold reservoirs negative then
the efficiency of the engine may be greater than I. This condition
corresponds tothe amplification of sgnale achieved in laser.
‘Alternatively, an efficiency greater than 1 implies that heat can be
converted completely into work provided the heat is withdrave from,
reservoir at T< 0, Ifboth reservoirs ate at negative temperatures,
then the efficiency is lee than 1 ei the thermal equilibrium case
‘eated in Chapter 3. The Thied Law requires light amendment
‘on account ofthe discontinuity of the populations across T=0: itis
{impossible ina finite number of steps to cool any system down to
"80 or to heat any system above -0,
461 Describe he physical significance ofthe patton function.
162 Explain how the internal energy and entropy ofa system composed of
‘wo lees vary wi temperature
kT,
tthe way by which he parameter maybe dented with
$64 Distinguish between the ipper and Zimm-Brage modes ofthe
helical raniion.
1685 Beplinwhatis meanchyan ensebleand why iis usefulin stasis
thermodynamic,
166 Under what circumstances may dential particles be regarded as
Alstingushable? 586
Exercises
16 STATISTICAL THERMODYNAMICS 1: THE CONCEPTS
‘6:12 What ae the relative populations ofthe states of two-level stem
hen the temperntureisinfiite!
16,18 What ithe temperature of o-leve este of energy separation
equivalent. 300 em whe the population ofthe upper states onehal that
ofthe lower ate
16.20 Calclte the tasltiona parson function at (a) 00 Kand (b) 500 K
‘ofa molecule of molar mas 120 g mot ina container of volume 2.0m!
16.26 Calelate (a) the thermal wavelength, (b) the wansationl partion
function ofan At ator ina cubic box of side 109 can at 6) 300K and
(33000
416.38 Calelate the rato ofthe ransatonalparion fanctions of, and Hf,
at the sme temperature and volume,
46.3 Calculate the ratio ofthe translations prttion functions ofsenon and
Jhelum at the same temperature and volume
‘64a A certain tom has a threefold degenerate ground level. anon
degenerate lecteonialy excited level 33500 er and a theeTld degenerate
level 2 4700 cm, Calcuate the partition function ofthese electronic states at
1900 K
16.4 A-ceran tom has a doubly degenerate ground evel atrply degenerate
decroniclly exited level at 1250 can" and a doubly degenerate level at
1300 em Calculate the partion function of hese electronic states at 200K.
‘6458 Calelate the electronic conrbtion oth molar internal emery at
1900 K fora sample composed of the some specified in Exercie 1648,
4.5% Calelate the electronic contedbution tothe molar internal energy st
2000 K fora sample composed ofthe atoms specie in Execs 16.45,
1646 A certain molecule has a non-degenerate exited state lying at 540 em!
hove the non-degenerate ground state At what lemperatre wll 10 per cent
ofthe molecules bein the upper ate?
1646 A cerain molecule bas a doubly degenerate excited tating at
5360 en shove the nonegenerte ground sate At what empertare
snl 5 percent ofthe molecules bein the upper sat?
16.72 An elecuon spin can adopt ihe of two ovenations in amagnetic
Sel and it eneries are th, whet gi the Bobs magneton, Dedice
‘eapraton forthe prttion function and mean energy ofthe electeon and
sketch the variation ofthe functions with 8 Caleulat the relative poplations
ofthe spin states at (a) 40K, (b) 298 Kwhen = 10,
Problems*
16:7 A nitrogen nclesepin can adopt any ofthe onentations in a
magnetic id ands energies te 0, +7 where 7s the magnetic
so ofthe nucleus: Deuce an expression forthe partion function and
sean energy ofthe audeus and sketch the variation ofthe fanctions wait 5
Calealaethe relate populations ofthe spin tater (a) LOK, (b) 298K
when 2920007,
168 Consider asstem of distinguishable patceshaving only to non-
degenerate energy lel separated bya energy that sequal tthe value of
[Ef a 10K. Caen (a) the ati of poplationsin the two sates at(1) 10K,
(2) 10K, and (2) 100, (b) the molecule parton function a 10K, (the
solar energy a 10K, (@) she molar eat expacity at 10K, (e the molar
entropy 10K
168% Consider sytem of distinguishable parses having ony tree non-
degenerate energy level separated by an energy whichis equal othe ale of
19 250K, Calealate (a) thera of poplations inthe states at (1) 1.00,
(2)25.0K,and (3) 100K, (bth molecalr patton function at25.0K,
{) the molar energy 225K. () the molar heat epacity a 250K, (e) the
sola entropy #25.0K
{6298 Atoehat temperature would the poplaton ofthe ist excited
rational state af FIC be te est population ofthe ground sat?
16.9 Atowhattempertare would the popslaton ofthe fist exited
sotaionl level of HCI Le imei population ofthe ground state?
46:108 Calelae the standard mola entropy f eon gaa (3) 200K
()29SK
$6:100 Caleate the standard mal
()29815K
10.118 Celelate the vibrational contebution tothe entropy of Cl,
given thatthe wavenumber ofthe vibration is 580 em
entropy ofxenon gaat (a) 100K,
00K
40.110 Calelate the wbrationl contusion tothe entropy of Br 600K
ven thatthe wavenumber ofthe vibrations 321 em!
10.128 Lent the pte for which its essential to ineude factor of UN
on going fom Q toa) a sample of helium gas, (6) asample of erbon,
monoxide gs, ()asolid sample of carbon monoxide (d) water vapour
16:12 dey he yrtems for which iezential to ncade a factor of IN
on going fem Q toa: (a) a sample of carbon diode gs (b)asample of
graphite (c)asample of diamond, (@) ce
Numerical problems
10.18 Consider system A consisting ofsbsystems A, and A, for which
W,= 1-10 and W, =2 10° What isthe number of conbigratione
viable to the combined ester? Alo, compute the entropies §,§, and S,
Whatisthe significance ofthis teu?
16:28 Conde 1.00% 10" He atomsin a box of mensions Lex 1.0em,
X10 em, Caleulat the occupancy ofthe fst excited level at 0 mK, 20,
tnd 4.0K Dothe same for He What conclusions might you draw fromthe
results of your calelatione|
* Problems denoted withthe symbol 4 were suppl by Charles Trapp and Carmen Giunta, 16.3 By what factor doe the numberof eal configurations inerease
‘when 10 Jo energy is added to assem containing 1.00 mal of partclesat
‘constant volume a1 298
16.4 By what actor dos the mmber of salable confgsrationsinzease
wen 20? of ir at 1.0 tm and 300 Kis allowed to expand by 0.0010 pet
cent at constant temperature?
465 Explore the conditions under which the integral proximation forthe
‘ranslational partion fnction not vad by consdering the anaationl
Darton function ofan Ar atom ina cube bax of ide 1.0 cm. Estimate he
temperatire at which, secoing othe integral approximation, q= 103d
‘erase the exact parton funciona thet temperature
10.6 A certsn atom har a doubly degenerate ground level par and an upper
devel offour degenerate veer st 450 can above the ground level Inan atomic
‘beam ud of the tome it war observed that 30 percent othe stom were in
‘the uppe level, and the uanltinal emperaute ofthe beam was 300K Ase
the lecronic ates ofthe toma in thermal equim wit the ransatonal
167 (3) Calealte the clectronicpartson fncton of llr stom at
(01298, (i) 3000 Ky dinect summation using the folowing dat:
‘Term Degenercy_Wavenumberfem™
Ground 5 °
2 5 4751
(b) What proportion of the Te stom ae in the ground tem and in the erm
labeled a the two lermperatues(e) Calelate the electron contibution to
‘the standard molar entzopy of ascous Te atoms
168 The four lowest dectroniclevels ofa Ti stom are. "F,.Fy and,
£300,170, 87, and 6557 ca” respectively. There are many other eecwonie
‘ales at higher energies. The baling point of anium 287°C, What ae the
‘elative populations ofthe levels tthe Boling pont int. The degeneracies
ol thelevlsare 21+ 1.
169 The NO moleculehar «doubly degenerate exited electronic evel
121. ea above the doubly degenerate electronic ground tem. Calculate
snd plot the cletonic prion function of NO from =Dto 1000 K
Evaluate (4) the term populitions an (the electronic contibtion to
the molar nternal energy t 300K. Calculate the electronic eanteibuton
tothe mola enuopy of the NO melecule at 800K and 00K.
16.1 J. Sugar and A. Musgrove (Phys Chem. Ref Data22 1213 (1883)
Ihave pehliched ables of energy levels for germanium atoms and cations from
Ge" to Ge Thelowestlyng energy level in neat Ge ate as follows:
RDS,
Hex! 9 S871 M09 71253— 163673
‘Cleulate the electronic partition fonction at 298 K and 1000 Ky duet,
summation. Hix The degeneracy ofa leel 2} +1
16.1 Calculate, by explicit summation, the wibrtional partition neon
snd the vibrational contabution tothe nla internal energy of, molecules
238 (2) 100, () 298K given that brtional energy eel ie athe
Tollowing wavesusmbers above he zero-point energy level 0,213.30, 425.89,
163627, 84595 em What proportion ff, moecsles are inthe round and
‘esto excited Levels atthe two temperatures! Calculate the vibrational