NANO LETTERS
Fluctuations in Energy in Completely Open Small Systems
2002 Vol. 2, No. 6 609-613
Terrell L. Hill* Department of Chemistry and Biochemistry, UniVersity of California at Santa Cruz, Santa Cruz, California 95064
Ralph V. Chamberlin Department of Physics and Astronomy, Arizona State UniVersity, Tempe, Arizona 85287-1504 Received February 12, 2002; Revised Manuscript Received April 5, 2002
ABSTRACT General equations are derived for fluctuations in energy in small completely open µ, p, T and µ, T or µ1, µ2, T (incompressible) systems. Four examples are considered. As expected, the fluctuations are unusually large because there are no restraints on the size of the system. Also, fluctuations in N and V or in N1 and N2 contribute indirectly to the fluctuations in E.
The traditional extensive variables in statistical thermodynamics for which fluctuations are of interest are N, V, and E. Fluctuations are different in different ensembles1 (examples for one-component systems include independent or “environmental” variables N, V, T; N, p, T; µ, V, T; or µ, p, T). Excluding phase transitions, fluctuations are “normal” if there is at least one extensive environmental variable (as in N, V, T; N, p, T; or µ, V, T). That is, for an appropriate extensive variable X, the relative fluctuation(X2 - X h 2)/X h 2 is of order 1/N, whether the system is macroscopic or small (of course these fluctuations, though normal, are relatively more important in a small system, with N of the order of say 102 to 103). The completely open case2 µ, p, T is special in that no extensive environmental variable is held constant. This has two unique consequences: the system must be small in order that µ, p, and T can all be independent; and fluctuations in extensive properties are of a larger magnitude because there is no fixed extensive variable that provides some restraint on fluctuations in the size of the system (as is the case in the other ensembles). The result: the above quotient for the relative fluctuation in X is of order 1 rather than 1/N. There is a summary of fluctuation formulas for small systems, in the four environments mentioned above, at the end of section 10-3 of ref 2. Fluctuations in N and V are included. With the exception of N, V, T systems, fluctuations in E are omitted because E is not the natural “heat” variable and hence the expressions for energy fluctuations are rather complicated. However, the formulas for energy fluctuations * To whom correspondence should be addressed. Address: 433 Logan St., Santa Cruz, CA 95062. 10.1021/nl020295+ CCC: $22.00 Published on Web 05/14/2002
© 2002 American Chemical Society
in the N, p, T and µ, V, T cases, given in eqs (19.28) and (19.49) of ref 1, respectively, can be shown (using the methods below) to hold for small systems as well as macroscopic systems. Thus, what remains is the problem of fluctuations in energy in small µ, p, T systems. This is the primary subject of the present note. Also included are the special cases of the µ, T or µ1, µ2, T ensembles, where p and V do not enter (e. g., incompressible spherical particles), as well as some simple examples of µ, p, T; µ, T; and µ1, µ2, T systems. Environmental Variables µ, p, T. The completely open partition function is Υ(µ,p,T) )
∑ exp[(- pV + Nµ - Ek)/kT]
(1)
N,V,k
The subscript k refers to the kth individual energy state for given N and V. We define Z ) E + pV - µN
(2)
This is the “heat” function for this kind of system [see eq (10-138) of ref 2]. Then, on differentiating ZΥ )
∑ Z exp[(-pV + Nµ - Ek)/kT]
N,V,k
with respect to T kT2
() ∂Z ∂T
µ,p
) Z2 - Z2
) (E2 - E2) + p2(V2 - V2) + µ2(N2 - N2) + 2p(EV - E V) - 2µ(EN - E N) - 2pµ(NV - NV)
[( ) ( ) ( ) ] ∂E ∂T
) kT2
µ,p
+p
∂V ∂T
µ,p
-µ
∂N ∂T
- kT(∂N/∂p)µ,T ) NV - NV ) kT(∂V/∂µ)p,T
(5)
kT2(∂V/∂T)µ,p ) p(V2 - V2) - µ(NV - NV) + (EV - E V) (6) If we also use
() ()() ()() () () () () ∂E
∂N
∂E
∂N
µ,p
) -T
T,V
∂µ ∂T
T,V
+ µ,
∂N ∂T
∂V
µ,p
∂E
∂V
N,V
∂E
+
)T
T,N
T,N
∂p ∂T
2
+
µ,T
∂V ∂µ
p,T
T2
∂V ∂T
µ,p
(∂T∂µ)
N,V
+ µp
()] ∂V ∂p
µ,T
(13)
(3)
µ,p
kT2(∂N/∂T)µ,p ) p(NV - NV) - µ(N2 - N2) + (EN - E N) (4)
) CV +
[( ) ( ) ] ( ) [ ( ) - T(∂V/∂p)µ,T
Next, we differentiate N h Υ and V h Υ, and again encounter the above fluctuation ingredients
∂E ∂T
V terms ) ∂V ∂V T +p ∂T µ,p ∂p
∂V ∂T
These general results, eqs 10-13, will now be illustrated using two very simple theoretical models. Example: Ideal Monatomic Gas. This is the model on pp 90-91 of ref 2 (Part 2). The completely open partition function is Υ(µ,p,T) ) 1/(1 - f), f ) kTλ/pΛ3
(14)
λ ≡ eµ/kT, Λ ≡ h/(2πmkT)1/2 A small system exists if f < 1. The macroscopic relation between µ, p, and T is given by f ) 1. We then find2 N ) f/(1 - f), V ) NkT/p
(15)
E ) 3NkT/2, CV ) 3Nk/2
(16)
(7)
µ,p
The various quantities in eqs 10-13 can now be calculated, leading to the final result
-p
(8)
N,V
where CV ≡ (∂E h /∂T)Nh ,Vh , we can deduce (E2 - E2)/kT2 ) CV + N terms + V terms
(9)
N terms ) (15/4)Nk/(1 - f)
(17)
V terms ) - (3/2)Nk/(1 - f)
(18)
(E2 - E2)/kT2 ) (3/2)Nk + [(9/4)Nk/(1 - f)]
(19)
) (3/2)Nk + (9/4)N(1 + N)k
(20)
where N terms )
( )[ ()
V terms )
∂N ∂T
-T
µ,p
] ( )
µ2 ∂N ∂µ + 2µ + + ∂T N,V T ∂µ p,T µp ∂N T ∂p
( ) [( ) ∂V ∂T
µ,p
T
] ()
∂p p2 ∂V - 2p ∂T N,V T ∂p
( )
µ,T
The first term on the right is of conventional order N h k but the second term is larger: of order N h 2k. Alternatively, we can also write
µ,T
(10)
()
µp ∂V T ∂µ p,T (11)
Thus, fluctuations in N and V have an indirect effect on fluctuations in E, as should be expected. Note the close relationship between N terms and V terms: NTV, µT-p [as pointed out in eq (6-5) of ref 2]. Alternatives to eqs 10 and 11 can be derived (after further thermodynamic manipulations) that are related to eqs (19.28) and (19.49) of ref 1, already mentioned N terms ) ∂N ∂N T +µ ∂T µ,p ∂µ
[( ) ( ) ] ( ) [ ( ) 2
p,T
∂N + ∂p
∂N T ∂T 2
µ,T
µ,p
∂p ∂T
( )]
∂N + µp N,V ∂µ
( )
p,T
T(∂N/∂µ)p,T
(12) 610
(E2 - E2)/E2 ) O(1/N) + O(1) ) O(1)
(21)
This large relative fluctuation is to be expected for a completely open small system. From (∂N h /∂µ)p,T and eq (10-136) of ref 2, we find for this µ, p, T system (N2 - N2)/N2 ) (1 + N)/N ) O(1)
(22)
which is again a large fluctuation. On the other hand, from (∂N h /∂µ)V,T and the grand partition function Ξ ) exp(Vλ/Λ3), we deduce for a µ, V, T system (not completely open) (N2 - N2)/N2 ) 1/N
(23)
which has the usual order of magnitude. Incidentally, from Ξ(µ,V,T) above and from ∆(N,p,T) in eq (10-84) of ref 2, it is easy to show (a) that the extra Nano Lett., Vol. 2, No. 6, 2002
term beyond CV in eq (19.49) of ref 1 for (E2 - E h 2)/kT2, in a µ, V, T system, is (9/4)N h k and (b) that the corresponding extra term in eq (19.28) in an N, p, T system is zero. These results should be compared with eq 20 for the µ, p, T system. Thus, for an ideal monatomic gas, fluctuations in N in a µ, V, T system add to the energy fluctuations, but fluctuations in V in an N, p, T system have no effect (as might be expected). Example: Ideal Lattice Gas. In this model, there are N molecules bound on B identical sites, at most one molecule per site. There are no interactions between bound molecules. A bound molecule has a partition function j(T) (e.g., binding energy plus vibration). This model is discussed in detail in section 7-1 of ref 3. The thermodynamic term pdV is replaced here by pdB (i.e., B replaces V and p has dimensions of energy). Again using λ ≡ eµ/kT, the grand partition function3 is Ξ (µ,B,T) ) (1 + jλ)B
(24)
and the completely open partition function is ∞
Υ(µ,p,T) )
∑ [e-p/kT(1 + jλ)]B B)0
) 1/(1-K) (K < 1)
(25)
Then (E2 - E2)/kT2 ) CV + [jλP(1 - P)D2k/k2T2(1 - K)2] (34) ) CV + (D2/k2T2)N(1 + N)k ) O(Nk) + O(N2k) (35) The large relative fluctuation in E is again as given in eq 21. The relative fluctuation in N for this µ, p, T system is found to be (1 + N h )/N h , as in eq 22: a large fluctuation. Using the Ξ in eq 24, we find for the µ, B, T system, on the other hand (N2 - N2)/N2 ) (B - N)/NB ) O(1/N)
(36)
which is the conventional magnitude. As in the previous example (following eq 23), we mention h 2)/kT2 in µ, B, T and N, p, here the expressions for (E2 - E T systems. The results are similar to those for an ideal monatomic gas. Here, using Ξ(µ,B,T) in eq 24 and ∆(N,p,T) in eq (A4.10) of ref 1, we find (a) that the extra term beyond CV in eq (19.49) of ref 1, for a µ, B, T system, is (D2/k2T2)N(B - N)k/B ) O(Nk)
K ≡ P(1 + jλ), P ≡ e-p/kT
and (b) that the corresponding extra term in eq (19.28) for an N, p, T system is zero.
A small system exists if K < 1; the macroscopic relation between µ, p, and T is given by K ) 1. As on pp 87-90 of ref 2 (Part 2), where this model with j ) 1 is considered, we find the properties
Environmental Variables µ, T. This is a completely open system but the small systems are assumed incompressible so that p and V drop out. Corresponding to eq 1, we have here
B ) K/(1 - K)
(26)
N ) Pjλ/(1 - K) ) Bjλ/(1 + jλ)
(27)
µ ) kTln[N/(B - N)j]
(28)
p ) - kTln[(B - N)/(1 + B)]
(29)
Υ(µ,T) )
exp[(Nµ - Ek)/kT] ∑ N,k
(37)
The general procedure used in eqs 2-8 can be repeated, though much simplified, to yield [T(∂N/∂T)µ + µ(∂N/∂µ)T]2 E2 - E2 )C+ kT2 T(∂N/∂µ)T
(38)
E ) ND, D ≡ kT dlnj/dT
(30)
CV ) (∂E/∂T)N,B ) NdD/dT ) O(Nk)
(31)
where C ) (∂E h /∂T)Nh . Note the resemblance to part of eq 12 and to eq (19.49) in ref 1.
The second form of eq 27 is the Langmuir adsorption isotherm. We can now introduce eqs 26-29 into eqs 10-13 to find, in eq 9
Example: Linear Aggregate. Consider a linear chain of N molecules, each of which has an intrinsic partition function j(T). There is an interaction energy between pairs (there are N-1 pairs). This model is discussed at length in section 10-2 of ref 2. We add here a consideration of the fluctuation in E. The canonical partition function is clearly
2
N terms ) jλPDk[D(1 - P) + p]/k2T2(1 - K)2
(32)
V terms ) - jλpPDk/k2T2(1 - K)2
(33)
Nano Lett., Vol. 2, No. 6, 2002
Q(N,T) ) j(T)N exp[-(N - 1)/kT]
(39) 611
and the completely open partition function is2
and the completely open partition function is ∞
∞
Υ(µ,T) ) )
∑ N)0
Υ(µ1,µ2,T) )
Q(N,T)λN
1 - x + cx (x
