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can we say that $$\Delta G =\Delta H-T\Delta S$$ is spontaneity criteria only if $$\tag{$t,p =$ const}T_\text{system} = T_\text{surrounding}$$

please see the image attached. In gibs function what exactly is $t$?

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    $\begingroup$ If $T_{sys}\neq T_{surr}$, I don't think either $T$ would be constant, which is an assumption when stating spontaneous processes have $\Delta G < 0$ $\endgroup$ – Quantumness May 28 '18 at 18:31
  • $\begingroup$ Even if the $T_{sys}$ changes, it's generally assumed that $T_{surr}$ doesnt change. Other than that, I agree with you $\endgroup$ – 22134484 May 31 '18 at 5:49
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To generalize a little, if we assume $T$ is affected only by its surroundings then It is valid to require (by product rule and $q=mC_P\Delta T$) that $\Delta G < -\dfrac{Sq}{mC_P}$, where $q$ is the heat transferred in the time it takes for the system to convert from state A to state B, and $\Delta G$ is the "normal" definition:$\Delta H-T\Delta S$. It is then valid to say $\Delta G < 0$ if $T_{sys}=T_{surr}$ but not only if, as $S=0$ is also sufficient.

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