= Solution
Let the closed atmospheric parcel exchange <heat> and volume work with reservoirs at fixed <temperature> $T$ and <pressure> $P$, while reactions rearrange its conserved elemental inventory. For a reversible differential between equilibrium states, the <first law of thermodynamics>, with chemical work included, is
$$
dU=T\,dS-P\,dV+\sum_i\mu_i\,dN_i.
$$
The <Gibbs free energy> $G=U+PV-TS$ therefore obeys
$$
dG=-S\,dT+V\,dP+\sum_i\mu_i\,dN_i.
$$
The <first law of thermodynamics> supplies this differential but does not alone choose the direction of a reaction. The <Second law of thermodynamics> does: for the parcel plus its reservoirs, at fixed $T,P$ and with no non-volume work,
$$
dS_{\rm total}=dS-\frac{dU+P\,dV}{T}=-\frac{dG}{T}\ge0.
$$
Thus spontaneous reactions decrease <Gibbs free energy>, and stable <thermochemical equilibrium> is its constrained minimum. For an allowed reaction of extent $d\xi$, let $dN_i=\nu_i\,d\xi$, with positive <stoichiometric vector> entries for products and negative entries for reactants. At an interior minimum,
$$
\boxed{\left(\frac{\partial G}{\partial\xi}\right)_{T,P}
=\sum_i\nu_i\mu_i=0,\qquad
\delta^2G\ge0\text{ on allowed variations}.}
$$
The condition must hold for every independent allowed reaction, with fixed total atoms of each element and any relevant charge constraint. At a boundary with absent species, only feasible one-sided variations are permitted. These are the hypotheses of <constrained Gibbs minimization for chemical equilibrium>.
For an <ideal gas> mixture, $\mu_i=\mu_i^\circ(T)+\mathcal RT\log(x_iP/P^\circ)$, where $x_i$ is the <volume mixing ratio> and $\mathcal R$ the molar gas constant. Therefore
$$
\prod_i\left(\frac{x_iP}{P^\circ}\right)^{\nu_i}
=\exp\left(-\frac{\Delta_rG^\circ}{\mathcal RT}\right)\equiv K_p(T).
$$
This is a dimensionless activity-based equilibrium constant, not a bare product of dimensional pressures.
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