We give a proof of the semi- ical expansion of the thermodynamic limit for a model introduced in statistical mechanics by M.Kac. For this family (parametrized by m) of Schrödinger operators $P^{(m)}(h) =-\sum _{k=1}^m h^2\partial ^2/\partial x^2_k + V^{(m)}(x)$ defined on $\mathbb {R}^m$, this corresponds to the study of the expansion in power of $h$ of $\lim _{m\to \infty }\lambda (m, h)/m$ where $\lambda (m, h)$ is the first eigenvalue of $P^{(m)}(h)$.