Temperature Derivatives of Pressure and Volume

It follows that in the limit of absolute zero, the entropy of a perfect crystalline substance must be independent of changes in pressure or volume (or any other variable of state except T). Thus, 

In other words, the temperature gradients of the pressure and volume vanish as absolute zero is approached. 

In the statement that we have adopted for the third law, it is assumed (arbitrarily) that the entropy of each element in some crystalline state is zero at 0 K. Then for every perfect crystalline substance, the entropy is also zero at 0 K. Consequently we can set S(0 K) in Equation (11.14) equal to zero. Thus, we may write 

To calculate the entropy of a substance at a temperamre at which it is no longer a sohd, it is necessary to add the entropy of transformation to a hquid or gas and the subsequent entropies of warming. The same procedure would apply to a solid that exists in different crystalline forms as the temperature is increased. The procedure can be illustrated by some sample calculations. 

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