  Let $\mathcal{L}$ be a finite distributive lattice and $S=K[x_\alpha: \alpha
\in \mathcal{L}]$ be a polynomial ring over a field $K$ and $I=\langle x_\alpha
x_\beta - x_{\alpha\vee \beta} x_{\alpha\wedge\beta} : \alpha \nsim
\beta,\alpha,\beta \in {\mathcal{L}} \rangle$ an ideal of $S$. In this article
we describe the first syzygy of the Hibi ring $R[\mathcal{L}]=S/I$, for a
planar distributive lattice $\mathcal{L}$. We also derive an exact formula for
the first Betti number of a planar distributive lattice. We give a
characterization of planar distributive lattices for which the first syzygy is
linear.
