Unit of equilibrium constant in formula for Gibbs free energy change at arbitrary temperature

We know the relation for Gibbs free energy change of a reaction at any stage and any temperature
$$\Delta G=\Delta G^{\circ}+RT\ln Q$$
where Q is reaction quotient. Subsequently, we get the rate constant as
$$\Delta G^{\circ}=-RT\ln k\_{eq}$$
My question is how do we deal with the units of $k\_{eq}$, for example, if we have the reaction
$$N\_{2}+3H\_{2}\rightleftharpoons2NH\_{3}$$
Rate constant at equilibrium can be written in terms of partial pressure as
$$k\_{p}=\frac{P\_{NH\_{3}}^{2}}{P\_{N\_{2}}\cdot P\_{H\_{2}}^{3}}$$
I could calculate partial pressure in Pascal, torr, bar and so on which will change $\Delta G$. Please give clarification on the convention as to how we write $k\_{p}$ when partial pressures in different units change $k\_{p}$ according to equation above


