Does Gibbs free energy of formation always equal zero for elements in standard state?

I was looking through the appendix in my chemistry textbook when I noticed that every compound that had a $\Delta H\_f$ of zero also had a $\Delta G\_f$ of zero. So of course the compounds that have a $\Delta H\_f$ of zero are the elements in their standard states but why does that make their $\Delta G\_f$ zero? Does every compound that has a $\Delta H\_f$ of zero also have a $\Delta G\_f$ of zero? What exactly is $\Delta G\_f$?





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> Does Gibbs free energy of formation always equal zero for elements in standard state?
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No. Elements occur in different allotropes. 


$\Delta H\_f^o$ and $\Delta G\_f^o$ are defined to be zero at 298K, 1 bar for the lowest energy allotrope, with the exception that the values for white phosphorous are defined to be zero even though it is not the lowest energy allotrope. 


The other allotropes do not have zero $\Delta H\_f^o$ and $\Delta G\_f^o$ even though they are still elements. 



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> So of course the compounds that have a $\Delta H\_f$ of zero are the elements in their standard states but why does that make their $\Delta G\_f$ zero?
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$\Delta H\_f = 0$ does not imply $\Delta G\_f = 0$


The two values are independently defined to be zero for a particular allotrope as explained above. 



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> Does every compound that has a $\Delta H\_f$ of zero also have a $\Delta G\_f$ of zero?
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If a compound (other than by definition) happens to have $\Delta H\_f = 0$, then $\Delta G\_f$ could be any value, it is not necessarily zero. 



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> What exactly is $\Delta G\_f$ 
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It is the change in Gibbs energy upon forming a compound in its standard state from its constituent elements in their standard states at 298K. 


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If you look more closely through your textbook, you will find it pointed out that the heats of formation and free energies of formation of all elements in their natural states at 298 and 1 bar are taken to be zero by convention. This in no way limits your ability to use these tables to determine the changes in enthalpy and free energy for arbitrary chemical reactions involving compounds comprised of these elements. 


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