Summary:  As T → 0 K ,  S → 0.

For the General Chemistry student, the important point about the third law is that entropy is an absolute quantity which depends upon temperature.  This is in contrast to ΔH for reactions which have as a reference the elemental state. 

Thus, when one looks up the  ΔH^o_f of an elements, the answer is 0.  In contrast, S^o for an element (note difference in symbols as well) has a value for temperature above 0 K.  Careful when doing calculations for  ΔS^o of reactions that you do not use 0 for the S^o of the elements.

The entropy change with respect to temperature can be thought of a continuous summation of all the increments of heat added to the system divided by the temperature at the time of the addition. Or symbolically:

        ΔS =   (dq/T) dT     which is approximately SUM of the ( Δq /T) s

Thus, to calculate a change in S one simply adds up the little increments of heat added divided by temperature.

The question then is, what if the addition of these increments start with the temperature at 0 K?  The answer is, that at 0K the q added is also 0.  0 divided by 0 presents a dilemma and the third law answers this by the following:

For a pure component in the most stable condition,  S =   0 at T = 0 K.

This leads to the assumption needed above, that the S^o s for pure components are absolute values and are not referenced against some arbitrary initial condition like the ΔH ^o s are.  As an illustration, see the example thermodynamic table and notice that the elements do have S^o s listed.  

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