= Solution
Let $P_{\rm sat}$ denote the flat-interface <saturation pressure> at the same <temperature>, $P_V$ the surrounding equilibrium <vapor pressure>, and $\overline V_l$ the incompressible <liquid> <molar volume>. Equating <liquid> and ideal-vapor <chemical potentials> to their common reference at $P_{\rm sat}$ gives
$$
\overline V_l(p_l-P_{\rm sat})=RT\ln\frac{P_V}{P_{\rm sat}}.
$$
Thus the <pressure> contribution to the attractive normal <force> on a wetted circular area is
$$
F_p=\pi a^2(P_V-p_l)=\pi a^2\left[\frac{RT}{\overline V_l}\ln\frac{P_{\rm sat}}{P_V}+P_V-P_{\rm sat}\right].
$$
For <capillary pressures> large compared with the vapor-pressure difference, and a bridge <wetting> essentially the plate area,
$$
\boxed{F\simeq\frac{\pi a^2RT}{\overline V_l}\ln\frac{P_{\rm sat}}{P_V}}.
$$
Here $F$ is the magnitude of the external separating <force> balancing attraction, and $P_V<P_{\rm sat}$. A finite <contact angle> adds a contact-line <force> $2\pi a\gamma\sin\theta$ at each plate's rim, counted once in the <force> on that plate, while the exact <pressure> also retains both curvatures. The rim term is normally smaller than the narrow-gap <pressure> term by order $d/a$ for appreciable <wetting>; it cannot be neglected uniformly near $\theta=90^\circ$. For complete <wetting> the leading equivalent expression is $F\simeq2\pi a^2\gamma/d$. The corresponding equilibrium humidity is $P_V/P_{\rm sat}\simeq\exp[-2\gamma\overline V_l/(RTd)]$.
These are quasistatic, equilibrium expressions for an existing bridge, neglecting viscous suction, contact-angle hysteresis, disjoining <forces> and deformation of the plates. The wetted radius must replace $a$ if the <liquid> does not reach the plate edge. During actual separation a bridge can shrink, depin or break, so this local <force> balance is not a promise of a constant <force> for arbitrary travel.
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