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Residual Entropy

The standard entropies measured experimentally (by using third law of thermodynamics)*are in good agreement with those calculated from statistical mechanics. However in some cases, the experimental value is found to be less than the calculated value. The explanation, as give above, is that at T=0, The crystals of these substances are not perfect i.e., there is some disorder present in the solid even at T=0. The entropy which the crystal of a substance has at T=0 is called the residual entropy. It can be calculated by applying Boltzmann formula viz. S = K In W as follows.

Suppose a sample consists of N molecules and each molecule can have two orientations that are equally probable (e.g. in case of CO we can have CO CO OC CO OC). Then the same energy can be achieved in 2Ndifferent ways i.e.e, W = 2N. Hence
                                S = k In 2N = k N In 2
If 1 mol of the sample is taken Kn= R . Then
                                S = R In 2 = 2.303 R log 2 = 5.85 JK-1mol-1
Thus residual molar entropy = 5.85 JK-1 mol-1

In general if a molecule can have p  possible orientations with about the same energy then residual molar entropy will be
                                 SR = R In p
For example, FCIO3 ( ) molecule can have four possible orientations with approximately same energy, hence its residual entropy will be =11.5 JK-1 mol-1.

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