Semigrand partition function

G[T P] = G = —RTIn A, where A is the isothermal-isobaric partition function and U[T,n ] = —RTInE, where S is the grand canonical ensemble partition function. When a system involves several species, but only one can pass through a membrane to a reservoir, L/(7 jux] = — PTlnT, where T is the semigrand ensemble partition function. The last chapter of the book is on semigrand partition functions. 

Introduction to Semigrand Partition Functions Transformed Gibbs Energy for a System Containing a Weak Acid and Its Basic form at a Specified pH... 

Semigrand Partition Function for a Biochemical Reaction System at Specified Concentrations of Coenzymes Discussion... 

If a system contains two types of species, but the membrane is permeable only to species number 1, the natural variables for the system are T, K //, and N2, where N2 is the number of molecules of type 2 in the system. The thermodynamic potential for this system containing two species is represented by U[T, //,]. The corresponding ensemble is referred to as a semigrand ensemble, and the semigrand partition function can be represented by P(71 K /q, N2). The thermodynamic potential of the system is related to the partition function by... 

When a system at specified T and P contains two species and one of them is in equilibrium with that species in a reservoir through a semipermeable membrane, the transformed Gibbs energy of the system is calculated from the semigrand partition function F(7( P, /q, N2) by use of... 

The semigrand partition function F corresponds with a system of enzyme-catalyzed reactions in contact with a reservoir of hydrogen ions at a specified pH. The semigrand partition function can be written for an aqueous solution of a biochemical reactant at specified pH or a system involving many biochemical reations. The other thermodynamic properties of the system can be calculated from F. 

The forms of semigrand partition functions for biochemical rection systems can be illustrated, starting with an aqueous solution of a weak acid and its basic form at a specified pH (Alberty, 2001c). The semigrand partition function F for this... 

Note that G = H — TS. This shows that the semigrand partition function T contains all the thermodynamic information about the system. 

SEMIGRAND PARTITION FUNCTION FOR A SYSTEM CONTAINING TWO PSEUDOISOMER GROUPS AT A SPECIFIED pH... 

For a system containing two pseudoisomer groups A and B at a specified pH, the semigrand partition function is given by... 

SEMIGRAND PARTITION FUNCTION FOR A BIOCHEMICAL REACTION SYSTEM AT SPECIFIED CONCENTRATIONS OF COENZYMES... 

At level 3 the semigrand partition function T" for a sum of reactants that are pseudoisomers at specified concentrations of coenzymes is given by a sum of exponential terms raised to a power equal to the number of molecules in the pseudoisomer group (Alberty, 2001g) ... 

The semigrand partition function can be written for a system of biochemical reactions at specified concentrations of coenzymes, and G" = —kT n T". The fundamental equation for G" can be used to calculate the other thermodynamic... 

The thermodynamics of biochemical rections at specified pH and specified concentrations of coenzymes can be represented by semigrand partition functions. Partition functions are always sums of exponential terms. For a single reactant at specified pH, T has a term for each species that is weighted by a factor that is exponential in AnjijpH. For a mixture of reactants, the partition function T for the system is a product of partition functions of reactants each raised to the power of the number of molecules of the reactant. Thus the partition function for a mixture of reactants is also a summation of exponential terms. For a sum of reactants at specified concentrations of coenzymes, for example, C( the partition function T" is a sum of exponential terms, with a term for each reactant (e.g.,... 

This is the semigrand partition function for the system. It was calculated from Af G °, but it could have been written down directly. 

Thus the semigrand partition function contains all the information on the thermodynamics of the system containing acetate at a specified pH. 

In 1957 Hill introduced a binary solution theory based on an analysis of the semigrand partition function in which the pressure P, temperature T, and number of solvent particles, Nj, are held fixed (10, 11). In this section, we extend his derivation to a four-component system containing solvent (component 1), two polymers (components 2 and 3) and protein (component 4). The objective of the calculation is to derive expressions for the chemical potentials of all components. Later, by equating the chemical potentials of each component in each phase, we will determine the composition of each phase and hence the protein partition coefficient which is defined to be the ratio of protein compositions in the top and bottom phases. 

The theory begins with a derivation of the semigrand partition function, T, which is defined for a system at constant P and T that is open with respect to components 2, 3 and 4 but not with respect to 1. The semigrand partition function is given in terms of the canonical partition function, Q, by... 

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