A cell in which the following reaction occurs:
\[
\text{Zn}_{(s)} + \text{Cu}^{2+}_{(aq)} \rightarrow \text{Zn}^{2+}_{(aq)} + \text{Cu}_{(s)}
\]
has standard electrode potential \(E^\circ_{\text{cell}} = 1.1~\text{V}\) at \(298~\text{K}\). Calculate the standard Gibbs energy and the equilibrium constant of the cell reaction.