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The heat q we provide is change in enthalpy. Also we measure entropy by the heat we provide with respect to Temperature(S=Q/T).

In gibbs equation, dG= dH-TdS dH is the energy we supply. Also TdS is also the energy we provide. This makes me think that dG=0. But why it is not?

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If the system is in equilibrium you have $dG = 0$. In this event the pressure is constant and you have $dq_P = dH$, if you replace in the definition of $dG$ you have

dG = dq - TdS

so it is correct that dG = 0 since TdS = $\delta q$

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Your equation for dG is incorrect. It should read $$dG=dH-TdS-SdT$$. When combined with the equation $dH=TdS+VdP$, you then get$$dG=VdP-SdT$$At constant temperature, this reduces to $dG=VdP$

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