  We analyze the specific heat for the $O(N)$ vector model on a $d$-dimensional
film geometry of thickness $L$ using ``environmentally friendly''
renormalization. We consider periodic, Dirichlet and antiperiodic boundary
conditions, deriving expressions for the specific heat and an effective
specific heat exponent, $\alpha\ef$. In the case of $d=3$, for $N=1$, by
matching to the exact exponent of the two dimensional Ising model we capture
the crossover for $\xi_L\ra\infty$ between power law behaviour in the limit
${L\over\xi_L}\ra\infty$ and logarithmic behaviour in the limit
${L\over\xi_L}\ra0$ for fixed $L$, where $\xi_L$ is the correlation length in
the transverse dimensions.
