= Solution
For a bulk liquid and vapor at common fixed <temperature> and <pressure>, transfer $dN_l$ molecules from vapor to liquid while holding their sum fixed. The <Gibbs free energy> change is $dG=(\mu_l-\mu_v)dN_l$. <Thermodynamic equilibrium> with both phases present requires stationarity under either sign of transfer, hence $\boxed{\mu_l=\mu_v}$. This is the bulk coexistence condition; an interface's curvature adds a surface-energy cost.
In the capillarity approximation, take an incompressible drop with constant liquid <number density> $n_l$, bulk <chemical potential> $\mu_l$ at the specified reservoir conditions, and size-independent <surface tension> $\Gamma$. The liquid bulk is extensive, so its Gibbs <thermodynamic free energy> is $N_l\mu_l$. Creating interface area reversibly at fixed <temperature> adds $\Gamma A$. Equivalently, along growth at fixed bulk parameters the stated differential integrates as $dG_d=\mu_l\,dN_l+\Gamma\,dA$, with the no-drop reference set to zero. For a sphere,
$$
\boxed{G_d=\frac43\pi R^3n_l\mu_l+4\pi R^2\Gamma.}
$$
This approximation uses bulk properties for the interior and neglects the molecular interface thickness, curvature dependence of <surface tension> and higher compressibility corrections. It does not apply quantitatively to drops only a few molecules across.
The vapor contains $N-N_l$ molecules. Treat its <chemical potential> as approximately unchanged by forming the drop, as for a large surrounding reservoir or negligible vapor depletion. Then
$$
\boxed{G_{\rm tot}(R)=N\mu_v+\frac43\pi n_l(\mu_l-\mu_v)R^3+4\pi\Gamma R^2.}
$$
The <radius> is limited by the available molecules in a finite sample; the constant-reservoir expression is a local nucleation model, not a description of arbitrarily complete vapor depletion. The excess over the vapor reference has derivative
$$
\Delta G'(R)=4\pi R[2\Gamma+n_l(\mu_l-\mu_v)R].
$$
If $\mu_l>\mu_v$, it increases for every $R>0$, leaving only the stable boundary minimum $R=0$. If $\mu_l<\mu_v$, $R=0$ is a metastable local minimum and the additional stationary <radius> is
$$
\boxed{R_c=\frac{2\Gamma}{n_l(\mu_v-\mu_l)}.}
$$
At that <radius> $\Delta G''=-8\pi\Gamma<0$, so it is an unstable maximum. The <classical nucleation theory> barrier is $16\pi\Gamma^3/[3n_l^2(\mu_v-\mu_l)^2]$.
\Image[/past-exam-of-the-quantum-condensed-matter-course-of-the-university-of-cambridge/2004-droplet-nucleation-barrier.png]
{title=Surface cost and bulk gain produce a critical droplet in supersaturated vapor}
{description=Original dimensionless capillarity curves. The supersaturated case has a barrier at the critical radius; the other curve rises monotonically. These are reservoir-model free energies, before finite-sample depletion changes the chemical potentials.}
A fluctuation below $R_c$ tends to shrink because reducing its surface cost lowers <thermodynamic free energy>. \b[A supercritical drop tends to grow], because its bulk free-energy gain outweighs the added surface cost. Growth continues while material supply and supersaturation allow it, eventually changing the vapor state or leading to macroscopic liquid. Curved-interface <pressure> and chemical <thermodynamic equilibrium> can also be described through the <Young–Laplace equation> and the <Kelvin equation>.
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