Hydration numbers by the Miller and Strong method

After de Forcrand s Clapeyron, and Handa s methods, a third method for the determination of hydrate number, proposed by Miller and Strong (1946), was determined to be applicable when simple hydrates were formed from a solution with an inhibitor, such as a salt. They proposed that a thermodynamic equilibrium constant K be written for the physical reaction of Equation 4.14 to produce 1 mol of guest M, and n mol of water from 1 mol of hydrate. Writing the equilibrium constant K as multiple of the activity of each product over the activity of the reactant, each raised to its stoichiometric coefficient, one obtains  

The thermodynamic reaction equilibrium constant K, is only a function of temperature. In Equation 4.18, m, the activity of the guest in the vapor phase, is equal to the fugacity of the pure component divided by that at the standard state, normally 1 atm. The fugacity of the pure vapor is a function of temperature and pressure, and may be determined through the use of a fugacity coefficient. The method also assumes that an, the activity of the hydrate, is essentially constant at a given temperature regardless of the other phases present. 

The activity of water tfw in Equation 4.18 is normally taken as unity disregarding the solubility of the gas. At a given temperature, if an inhibitor such as a salt is present, the activity of the water decreases and the activity of the gas must increase in order to maintain a constant product K aH at that temperature. Thus writing a second equation for the formation of a hydrate from an inhibited liquid we get 

Subtracting Equation 4.19 from Equation 4.18a, with the right sides constant, one may replace the activity of the guest M with its fugacity (if the same standard 

Recalling that the ( ) denotes the presence of an inhibitor, Equation 4.20 may be rearranged to obtain the hydrate number n as 

---