## Liquid-State Physical Chemistry: Fundamentals, Modeling, and Applications (2013)

### 6. Describing Liquids: Structure and Energetics

### 6.5. Energetics

If we assume that the correlation function *g*(*r*) is known and the pairwise additivity (of spherical potentials) holds for the potential energy, that is, *Φ*(**r**) = ½Σ* _{ij}ϕ_{ij}*(

*r*), we can write down for the internal energy almost by inspection the

*energy equation*

__(6.32)__

where the first term is the kinetic contribution. More formally, the canonical average, as given by __Eq. (6.9)__, results in

(6.33)

and using *Φ* = ½Σ* _{ij}ϕ_{ij}* leads to

*N*(

*N*− 1)/2 identical terms so that

(6.34)

to which the kinetic term has to be added. In the thermodynamic limit (*N* → ∞, *V* → ∞ but *N*/*V* → *ρ*), the factor (*N* − 1)/*V* becomes *ρ*, so that the result is __Eq. (6.32)__.

Similarly, though in a somewhat more complex fashion, an expression for the pressure *P* can be obtained. We start with the previously derived expression (see __Chapter 5__)

(6.35)

with the thermal wavelength *Λ* ≡ (*h*^{2}/2π*mkT*)^{1/2}. Before differentiating, we consider that the pressure is independent of the shape of the container, so that we can take a cube and employ the coordinate transformation *r** _{i}* =

*V*

^{1/3}

**r**

*′. The expression for*

_{i}*Q*then becomes

_{N}(6.36)

while for the intermolecular distance *r _{ij}* = [(

*x*−

_{i}*x*)

_{j}^{2}+ (

*y*−

_{i}*y*)

_{j}^{2}+ (

*z*−

_{i}*z*)

_{j}^{2}]

^{1/2}we obtain the expression . Therefore ∂

*Q*/∂

_{N}*V*becomes

(6.37)

(6.38)

Transforming back, noting that we have *N*(*N* − 1)/2 identical terms and taking the thermodynamic limit, one obtains the *pressure* (or *virial*) *equation* (see Section 3.8)

__(6.39)__

In __Eqs (6.32)__ and __(6.39)__, the first terms are due to the momenta (i.e., the kinetics), while the second term is due to the coordinates (i.e., the configuration). Thus, both equations in principle relate a thermodynamic quantity to the structure, as represented by *g*(*r*).

For future reference we note that the compressibility *κ _{T}* can be calculated using the grand canonical ensemble (see

__Justification 6.1__). The final result

__(6.40)__

is known as the *compressibility equation*, and this expression is derived without assuming any pairwise additivity for the potential energy of the system, and is thus generally valid.

Justification 6.1: The compressibility equation*

The derivation of the compressibility equation requires the grand canonical ensemble. We recall first the definition of the isothermal compressibility

From *PV* = *kT* lnΞ we have, using ,

We further need the absolute activity given by *λ* = exp(*βμ*), and note that ∂ln*Ξ*/∂*ρ* = (∂ln*Ξ*/∂*λ*)/(∂*ρ*/∂*λ*). For the numerator ∂ln*Ξ*/∂*λ* we obtain, defining *I _{N}* ≡ ∫ … ∫exp(−

*βΦ*)d

**r**

_{1 }… d

**r**

*,*

_{N}where the distribution as defined for the grand canonical ensemble

is used. The denominator ∂*ρ*/∂*λ* is given by

so that

where for the last step the definition *g*(**r**_{1},**r**_{2}) = *ρ*^{(2)}(**r**_{1},**r**_{2})/*ρ*^{(1)}(**r**_{1})*ρ*^{(2)}(**r**_{2}) = *ρ*^{(2)}(**r**_{1},**r**_{2})/*ρ*^{2} is used. Combining the expressions for ∂ln*Ξ*/∂*λ* and ∂*ρ*/∂*λ* yields

The isothermal compressibility is then given by

Note that the structure factor *S*(*s*) for *s* = 0 reduces to

and therefore *S*(0) = *ρkTκ _{T}*, a fact that can be used for reconstructing

*g*(

*r*).

One might wonder whether the compressibility equation and the pressure equation yield the same result for the pressure. On integrating __Eq. (6.40)__, one obtains

(6.41)

to be compared with *P _{P}* as obtained from

__Eq. (6.39)__. For the ideal gas

*g*(

*r*) = 1, leading for both expressions to the ideal gas EoS

*P*=

*ρkT*. This is no longer true for a nonideal fluid for which

*ϕ*(

*r*) ≠ 0, and thus

*g*(

*r*) is unknown. Using approximations for

*g*(

*r*) usually leads to different values for

*P*and

_{P}*P*, indicating a loss of

_{κ}*thermodynamic consistency*(see Section 6.3).

In order to obtain a complete description of the thermodynamic state, some extra information is required. This is probably most clear from the Gibbs equation

(6.42)

which shows that information is required about the energy *U*, the pressure *P*, and the chemical potential *μ* to obtain a complete description of the thermodynamic state of the system.

If the correlation function *g*(*r*) were to be known over a wide temperature range, we could use the energy __Eq. (6.32)__ and integrate the Gibbs–Helmholtz equation and so obtain the required information, since the chemical potential *μ* can be calculated from the Helmholtz energy *F*. Similarly, if *g*(*r*) were to be known over a wide density range, we could use the pressure __Eq. (6.39)__ and integrate ∂(*F*/*N*)/∂(1/*ρ*) = *ρ*^{2}*P* (as obtained from d*F* = −*P*d*V* − *S*d*T* + *μ*d*N*). Alas, neither the temperature nor density dependence of *g*(*r*) is usually available.

As an alternative we introduce the coupling parameter approach, also known as thermodynamic integration. A *coupling parameter* *ξ* is defined, varying from 0 to 1, which controls the interaction between the reference molecule 1 and another, say *j*, by replacing *ϕ*_{1j} by *ξϕ*_{1j}. The potential energy then becomes

__(6.43)__

and a molecule is added to the system when *ξ* varies from 0 to 1. The partition function *Z _{N}*(

*ρ*,

*T*) reads now

*Z*(

_{N}*ρ*,

*T*,

*ξ*) while the correlation function

*g*(

*r*;

*ρ*,

*T*) becomes

*g*(

*r*;

*ρ*,

*T*,

*ξ*). Because the number of particles

*N*is very large, we can write

(6.44)

From *F* = −*kT*ln*Z*, we obtain −*βF* = ln *Q _{N}* − ln

*N*! − 3

*N*

_{ }ln

*Λ*, and hence we get

__(6.45)__

Further, we have *Q _{N}*(

*ξ*

*=*1) =

*Q*and

_{N}*Q*(

_{N}*ξ*

*=*0) =

*VQ*

_{N}_{−1}. Therefore,

(6.46)

(6.47)

Calculating ∂*Q _{N}*(

*ξ*)/∂

*ξ*using

__Eq. (6.43)__results in

(6.48)

Further, employing *ρ*^{(N)}(*ξ*) = *N*!*n*^{(N)}(*ξ*) with *n*^{(N)}(*ξ*) = *Q _{N}*(

*ξ*)

^{−1}exp[−

*βΦ*(

*ξ*)], and recognizing that we have

*N*− 1 identical integrals, we obtain

(6.49)

where in the last step *ρ*^{(2)}(*ξ*) = *ρ*^{2}*g*(*ξ*) is employed. Substitution in __Eq. (6.45)__ yields the required result

__(6.50)__

From , *P* and *μ*, we can calculate all of the thermodynamic properties if *g*(*r*;*ξ*) is known. The theoretical calculation of *g*(*r*;*ξ*) is the topic of __Chapter 7__.

Problem 6.7*

Check the steps necessary to obtain __Eqs (6.39)__, __(6.40)__, and __(6.50)__.