162 Canonical ensemble
0 2 4 6 8 10
q (Å
-1
)-1
0
1
2
S
(q)100 K
200 K
300 K
400 K
0 2 4 6 8 10
q (Å
-1
)-1
0
1
2
S
(q)213 K
273 K
(a) (b)Fig. 4.5(a) Structure factors corresponding to the radial distribution functions in Fig. 4.2.
(b) N–N partial structure factors for liquid ammonia at 213 Kand 273 K from Ricciet al.
(1995).
4.6.3 Thermodynamic quantities from the radial distribution function
The spatial distribution functions discussed previously can be usedto express a number
of important thermodynamic properties of a system. Consider first the total internal
energy. In the canonical ensemble, this is given by the thermodynamic derivative
E=−
∂
∂βlnQ(N,V,T). (4.6.33)SinceQ(N,V,T) =Z(N,V,T)/(N!λ^3 N), it follows that
E=−
∂
∂β
[lnZ(N,V,T)−lnN!− 3 Nlnλ]. (4.6.34)Recall thatλis temperature dependent, so that∂λ/∂β=λ/(2β). Thus, the energy is
given by
E=
3 N
λ∂λ
∂β−
∂lnZ
∂β=
3 N
2
kT−1
Z
∂Z
∂β. (4.6.35)
From eqn. (4.6.4), we obtain
−
1
Z
∂Z
∂β=
1
Z
∫
dr 1 ···drNU(r 1 ,...,rN)e−βU(r^1 ,....,rN)=〈U〉, (4.6.36)and the total energy becomes