Gibbs Free Energy Correction Term

[laser1]

TL;DR

$dG$ has correction term $T-T_0$, can't find information on it.

By definition, we have that $G=H-TS$, which means that $dG=dU+PdV+VdP-TdS-SdT$, and at constant temperature and pressure, $dG=dU+PdV-TdS$. As $dU=TdS-PdV$, I asked my lecturer why $dG=0$ isn't true for all processes at constant temperature and pressure.

He then tells me that there is actually a correction term $(T-T_0)dS$. So in reality, $dG=dU+P_0dV-TdS+(T-T_0)dS=dU+P_0dV-T_0dS$, where I also substituted in $P=P_0$ (which is always true for constant pressure), where the $0$ quantities denote the properties of the surroundings.

However, I cannot find any information about this mysterious $(T-T_0)dS$ term. Would anyone be able to provide any insight on this? Thanks!

 

[anuttarasammyak]

$$dG=Vdp-SdT=0$$
for dp=dT=0 with dN=0 as you derived.

TL;DR Summary: $dG$ has correction term $T-T_0$, can't find information on it.

why dG=0 isn't true for all processes at constant temperature and pressure.

Please let me share counter examples you find.

 

[laser1]

$$dG=Vdp-SdT=0$$
for dp=dT=0 with dN=0 as you derived.

Please let me share counter examples you find.

I don't understand what you mean in relation to my question.

 

[Chestermiller]

TL;DR Summary: $dG$ has correction term $T-T_0$, can't find information on it.

By definition, we have that $G=H-TS$, which means that $dG=dU+PdV+VdP-TdS-SdT$, and at constant temperature and pressure, $dG=dU+PdV-TdS$. As $dU=TdS-PdV$, I asked my lecturer why $dG=0$ isn't true for all processes at constant temperature and pressure.

He then tells me that there is actually a correction term $(T-T_0)dS$. So in reality, $dG=dU+P_0dV-TdS+(T-T_0)dS=dU+P_0dV-T_0dS$, where I also substituted in $P=P_0$ (which is always true for constant pressure), where the $0$ quantities denote the properties of the surroundings.

However, I cannot find any information about this mysterious $(T-T_0)dS$ term. Would anyone be able to provide any insight on this? Thanks!

I've never seen anything like that "correction term" before. Please ask you professor where it came from.

G is a state function, and it's changes are independent of the specific process that took the material from equilibrium state 1 to equilibrium state 2.

More generally, if there are changes in composition of the system (due to adding or removing or reacting chemical species to the system), the equation for dG is: $$dG=-SdT+VdP+\mu_1dN_1 +\mu_2dN_2...$$where the $\mu's$ are the chemical potentials of the various species.

 

[laser1]

I've never seen anything like that "correction term" before. Please ask you professor where it came from.

I asked him a few weeks ago and he said it's due to "irreversible work", but I will ask him again sometime next week when I see him.

 

[Chestermiller]

I asked him a few weeks ago and he said it's due to "irreversible work", but I will ask him again sometime next week when I see him.

This makes no sense to me. The change in G between two end states is independent of the path whether reversible or reversible.