 Solution

Away from singular regimes, assume enthalpy and heat capacity at constant pressure are extensive: $⟨H⟩=Nh$ and $C_P=N{c}_P$, with nonzero intensive $h$. The enthalpy fluctuation formula then givesAbsolute fluctuations grow as $\sqrt{N}$, while the mean grows as $N$. In the thermodynamic limit, enthalpy per particle therefore concentrates around its mean; it is relative, rather than absolute, fluctuations that vanish. This is the usual relative fluctuations of an extensive observable scaling. It assumes ordinary extensive heat capacity and a nonzero mean with a stated energy reference; a critical divergence or an artificially zero mean requires separate analysis.