Gibbs free energy and chemical potential

If you add a particle to a system, then the internal energy will change by an amount which we call the chemical potential $mu$. Thus we add a term $mu dN$ to the expression $$dU=TdS-pdV+mu dN$$

I would like to know when is the summation term zero.

The summation term would be zero if $$sum_i mu_idN_i=0Rightarrow dN_i=0 text{closed system}$$

Can it be non-zero in a closed system?

Yes! In chemical reactions such as $$Arightarrow B$$ $$dG=mu_AdN_A+mu_BdN_B$$ However, since an increase in B is always accompanied by a corresponding decrease in $A$, we have that $$dG=(mu_B-mu_A)dN_Bnot=0$$ In equilibrium at constant temperature and pressure, we have that the Gibbs function is minimized $$dG=0Rightarrow mu_A=mu_B$$

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In a similar way, the closed system might be composed of different phases of different substances. In general $dGnot=0$ unless there is an equilibrium between different phases.