Molecules indistinguishable

For N indistinguishable molecules that do not interact with each other, the total partition function QN is given by (1/N ) times the Mh power of (Q,r nSQ, ,), where we take the Qs without the subscript Was the partition functions for individual molecules. The translational portion of the partition function is relatively easy to evaluate for ideal gases (see, for example, Atkins 1994). Substituting its value into Equation (15) gives... 

For a gas containing N molecules of the same chemical species, the molecules would all be indistinguishable from one another. The factor W has to be divided by Nl in this case. The proper explanation can only be understood through a detailed discussion of quantum mechanics and Bose-Einstein statistics. This explanation is beyond the realm of interest here, and we simply state the proper weighting for a collection of N indistinguishable molecules as... 

We note that the molecular partition function q that we have derived is for the special case of a collection of N indistinguishable molecules of one particular species. Mixtures of gases will have a different partition function, and the general case is not treated (and will not be needed) here. 

Recall that Eq. 8.28 gave the general expression for IV (for indistinguishable molecules)... 

C must have the units of (xpx) in order to make the probability a dimensionless quantity. The absolute magnitude of the dimensions are of little importance, since, as we shall see, it is only the relative probability of two states that is important. Quantum mechanics enables us to set this multiplicative constant equal to where h is Planck s constant. It should also be divided by iV for a system of N indistinguishable molecules, since we are not able to distinguish configurations in which molecules have been interchanged. 

Figure 1.55. The original lattice containing indistinguishable molecules in which a proper unit cell is shaded (left) and a new lattice derived by switching 1/2 of atoms from happy to sad faces in a regular fashion.

The hrst multiplier in the right-hand side of Equation 5.10 expresses the number of ways indistinguishable molecules can be distributed among M labeled sites the partition function for a single adsorbed molecule is g = where q qy, and q are one-dimensional harmonic-oscillator... 

If the temperature of the system is held constant, the entropy of the system decreases because 3 moles of two differing reactants have been combined to form 2 moles of a single product. The gas now consists of a uniform set of indistinguishable molecules. However, the reaction releases a significant amount of heat into the surroundings, and this heat will increase the entropy of the surrounding molecules by increasing their random movement. 

Suppose that three indistinguishable molecules are distributed among three energy levels. The... 

Functional Form and Limiting Behavior. The fnnctional form of the Langmuir isotherm can be rationalized by calcnlating the partition function of N indistinguishable molecules adsorbed on a solid surface, where the internal degrees of... 

An ideal gas consists of N indistinguishable molecules that do not interact with each other. The total energy of the ensemble is NE, where E represents the energy of a single molecule. The partition function can be expressed as a product of N single-molecule partition functions because the sum of states includes the same set of stationary states for each molecule. Z for a non-ideal gas cannot be factored as conveniently as Z for an ideal gas. Finally, statistics suggests that it is necessary to divide by N, due to the indistinguishability of N gas molecules. The partition function for the ideal gas ensemble is... 

The partition function 2 of a system of m non-interacting, identical and thus indistinguishable molecules is given by... 

In the demixed state (i.e., for the pure components), there is only one way to distribute indistinguishable molecules over sites, which means that In = 0. Similarly, In = 0, for pure 2. For the mixture of 1 and 2,... 

Figure 3.1 Example of the number of ways of representing a system of three indistinguishable molecules. The six indistinguishable systems are physically identical.

Example 3.2 Reduced Liouviiie Equation for Interacting Dipole It is desired to write the lowest-order version of the contracted Liouviiie equation for a system of interacting dipoles. An important physical example of this is a polar gas composed of iV indistinguishable molecules, such as those listed in Table 1.2. 

Gas molecules are considered as indistinguishable molecules the canonical partition function is therefore given by relation [5.25], By applying this relation and by using Stirling s first approximation [4.1], relation [6.105] becomes ... 

A molecule in solution can be seen as a system complying with constant F, V, and N. Let Z9) denote its partition function. Having the partition function a probabilistic interpretation, the partition function of a composite system equals the product of the components partition functions, unless they are identical and indistinguishable. Consider for instance a system composed of N identical, indistinguishable molecules in solution. In this case, the system partition function is ... 

If it is possible to identify all the molecules, such as in case of a solid, where the positions in the crystal lattice can be defined in terms of the coordinate positions, eq. (11.17) can be used. However, for the case where the molecules cannot be differentiated, e.g. with an ideal gas, it is necessary to count over all the sums of the molecular states of indistinguishable molecules, and weigh the result by the number of possible permutations, A . For this case... 

---