General expressions for the partition function

At any temperature in the presence of a catalyst, the equilibrium ratio of ortho- to / ara-hydrogen is given by 

The slight difference in the average rotational energy of the two forms enhances the heat capacity due to the LeChatelier shift in the equilibrium position as the temperature is changed. This effect is exhibited in the curve for ot labeled e-H2. Equilibrium-hydrogen, e-H2, is hydrogen that is kept in the presence of a catalyst to ensure that the equilibrium between 0-H2 and P-H2 is established at all temperatures. The curve for e-H2 is typical of the heat capacity of a reactive mixture maintained in equilibrium as the temperature is changed. 

At 0 K, J = 1 for ortho-hydrogen, and 2J + 1 = 3. These three rotational states combine with the three nuclear spin states to yield nine quantum states available to the 0-H2 molecule at 0 K. In normal hydrogen, the f mole fraction of 0-H2 is evenly distributed over all nine of these states. On the other hand, for P-H2 at 0 K, J = 0 and 2J + 1 = 1. This one rotational state combines with the one nuclear spin state to yield a single quantum state that is available to the P-H2 molecule. The entire mole fraction of P-H2 in n-H2 occupies this single quantum state at 0 K. 

At 0 K normal hydrogen has a residual entropy due to the entropy of mixing the mole fraction of P-H2 with the nine different states occupied by the J mole fraction of 0-H2. Each of the nine states has (J) = mole fraction in it. The AS of mixing is 

The nuclear spin can be oriented in two ways this leads to an entropy of In 2 per nucleus or a total of 2R In 2. Since this contribution to the entropy persists through all changes, it is not taken as part of the residual entropy and so must be subtracted from the entropy of mixing above. This yields for the residual entropy of hydrogen at 0 K, 18.38 — 2i In 2 = 6.8 J/K mol, which is in good agreement with the observed value of 6.2 J/K mol. 

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The theory begins with a very general expression for the partition function. A given configuration of the entire system is specified by sets of solvent molecule coordinates [rsi] with i = I,.Ns for each S, homopolymer space curves (rH/Cr) for j = I,..., Nu for each H, and copolymer space curves for each block of each copolymer. In the first instance, we consider only diblock copolymers, so we need rcA/t(r) and rcBjt(r) for A = 1,..., iVc for each C. The partition function can be... 

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