Marisa B. Rydzy
Statistical Thermodynamics
“Model for Gas Hydrate Equilibria using a Variable Reference Chemical Potential”
by Lee & Holder (2002)
Gas Hydrates
Clathrate hydrates are nonstoichiometric ice-like crystalline solids comprised of a threedimensional network of hydrogen-bonded water molecules, which confines guest
molecules in well-defined cavities. Gas Hydrates are stable only at conditions of low
temperatures and elevated pressures. Being able to predict gas hydrate stability pressure
and temperatures is of vital importance to the oil and gas industry as hydrate plug
formation in pipelines can cause severe damage. The paper published by Lee & Holder
(2002) is mainly concerned with the prediction of gas hydrate equilibrium pressures
applying a statistical thermodynamics model. Correlation of experimental and theoretical
data is then successfully used to demonstrate the validity of the model.
The Model
The model presented by Lee & Holder is based on the statistical thermodynamic model
for describing gas hydrate phase equilibrium, which was developed, generalized, and
further modified by Van der Waals & Platteeuw (1959), Parrish & Prausnitz (1972) and
Holder et al. (1981), respectively. The model describes the hydrate phase, when liquid
water is present and assumes that a) classical statistical thermodynamics is valid, b) one
cavity can only accommodate one molecule, and c) the mutual interaction between
enclathrated molecules is negligible. Unlike Van der Waals and Platteeuw who assumed
a rigid cage in which the guest rotates freely; Lee & Holder assumed that the size of guest
molecules does indeed impact the host-host interactions by distorting the lattice, thus
changing the theoretical chemical potential of the theoretical (empty) lattice. To account
for this, the authors introduce a guest depended reference chemical potential difference,
which is derived from experimental data and can be used to predict equilibrium pressure
and temperatures for simple gas hydrates. An empirical correlation relating cage radius
and the reference chemical potential difference as well as smooth cell potentials are
integrated into the equations determining the Langmuir constants to account for the
lattice distortion as well as as for the contribution of more distant water molecules and
the distortion of the lattice, respectively.
Application
At equilibrium the chemical potential differences (the chemical potential of the
theoretical empty lattice serves as a reference state) of the pure liquid water and the
hydrate are equal. Following the recipe given below, the chemical potential differences of
the pure water and the hydrate phase can be calculated. Thereby an initial value for the
reference chemical potential difference is assumed for 273.15 K and the respective
(experimentally derived) dissociation pressure. If the chemical potential at the end of the
calculation are not equal, a new reference chemical potential difference, using an
appropriate convergence method, i.e. the calculations are repeated until the two chemical
potential differences become identical.
I. Calculating the guest-dependent reference chemical potential of the hydrate phase
a. A first approximation for the guest-dependent reference chemical potential is
assumed based on experimentally obtained dissociation pressure data at 273.15 K
b. The guest-dependent cavity radii are calculated for each cell using the guestdependent reference chemical potential.
c. The Langmuir Constants for the respective cages are calculated from the radii for
the small and large cages. Here, the smooth cell potentials are incorporated.
d. The Langmuir Constants are then used to determine the fractional cage
occupancies of i-type cavities with j-type guests using fugacity coefficients fj
which are using Peng-Robinson Equation.
e. Knowing the fractional occupancy of the hydrate, the chemical potential
difference of the hydrate phase can be calculated using the VdWP Model.
2


 Hw (T, P)  RT   i ln 1    j  (1)
i 1
j


II. Calculating the guest-dependent reference chemical potential of the liquid phase
a. To obtain the guest-dependent reference enthalpy difference Equation (1) and (3)
are set equal, rearranged and the integrals solved with appropriate limits. The
result is a linear equation of the form Y=αX, where the term α includes the
reference enthalpy difference between theoretical empty hydrate cage and the
pure water phase at 273.15 K. The slope of the graph X vs. Y thus yields the
reference enthalpy difference for the liquid water phase. The method to calculate
the heat capacity difference and b is given elsewhere.
1
h 0w    C 0pw T0  bT02
2
(2)
b. The reference enthalpy difference for the liquid water phase is then used to
calculate the temperature-dependent enthalpy, which is thereafter applied to
determine the guest-dependent reference chemical potential of the liquid phase
given by Equation (8)
P
  0w T h w

Vw
  RT 

dT

dP

ln



w
w
2
 RT
P0
 RT0 T0 RT

L
w
(3)
[Most of the equations presented by Lee and Holder have been omitted from this review
to not exceed the two page limit.]