Let $X$ be a topological space. Then its chain complex $C_{*}(X)$ is naturally a coalgebra (as explained here Does homology have a coproduct?). In particular if $X$ is simply connected we have that the homology of the cobar construction $\Omega C_{*}(X)$ of $C_{*}(X)$ is isomorphic to the homology of the pointed path space $\Omega X$ (Adam's theorem). I'm looking for reference about similar statements using the bar construction:

a) Consider the cochain complex $C^{*}(X)$ as a dg algebra equipped with the cup product. Assume that $X$ is simply connected. Then under which conditions the cohomology of the bar construction $BC^{*}(X)$ is isomorphic to the cohomology of the path space?

b) $C_{*}(X)$ is a coalgebra where the coproduct is the composition of the Alexander-Whitney map with the diagonal map. By taking the dual we get a dg algebra $C^{*}(X)$ with a product $\mu$. Let $B'C^{*}(X)$ be the bar construction of ($C^{*}(X)$, $\mu$). What is the relation between $BC^{*}(X)$ and $B'C^{*}(X)$?