  We investigate the asymptotic behavior of the eigenvalues of the Laplacian
with homogeneous Robin boundary conditions, when the (positive) Robin parameter
is diverging. In this framework, since the convergence of the Robin eigenvalues
to the Dirichlet ones is known, we address the question of quantifying the rate
of such convergence. More precisely, in this work we identify the proper
geometric quantity representing (asymptotically) the first term in the
expansion of the eigenvalue variation: it is a novel notion of torsional
rigidity. Then, by performing a suitable asymptotic analysis of both such
quantity and its minimizer, we prove the first-order expansion of any Robin
eigenvalue, in the Dirichlet limit. Moreover, the convergence rate of the
corresponding eigenfunctions is obtained as well. We remark that all our
spectral estimates are explicit and sharp, and cover both the cases of
convergence to simple and multiple Dirichlet eigenvalues.
