We examine the one-dimensional Hubbard model and derive expressions for the chemical potential and momentum distribution in the limit of strong electron-electron interactions to first order in perturbation theory. By showing that the perturbation matrix is stochastic, the chemical potential is found to depend only on the model parameters and not on the system size, consistent with the well-known result in the thermodynamic limit. The momentum distribution for electrons of a given spin is found to depend only on the fraction of electrons with that spin.