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Recent developments in X-ray detectors have created the potential to perform energy-sensitive X-ray computed tomography (CT); that is, to reconstruct a series of CT images associated with different X-ray energies from a single scan. In this paper we propose a penalized weighted least squares (PWLS) algorithm for reconstruction of polychromatic energy-differentiated X-ray CT data and a unique experimental setup to take energy-differentiated X-ray CT data. The experimental setup is designed to acquire a complete X-ray spectrum for every projection ray. We use these data to estimate the linear attenuation coefficient as a function of energy for every pixel in the reconstructed attenuation map. We use prior knowledge of the properties of attenuation spectra to smooth the reconstructions, significantly reducing the noise and improving the contrast-to-noise ratio (CNR) in the reconstructed images without significantly biasing the data. We conclude that this algorithm is an effective method for reconstructing energy-sensitive CT data and provides justification for further research in energy sensitive CT systems.