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In this paper, we model breathing at the level of a human alveolus by developing a novel mathematical framework that links pulmonary mechanics and gas exchange. Unlike previous models, the developed alveolar model comprises the alveolar gas mixture, capillaries perfused in the alveolar wall, pulmonary surfactant lining the alveolus and the pleural cavity. The inclusion of the multiple compartments affords interested researchers the chance to study the effect of several respiratory factors on breathing. Here, we investigate the effect of holding breath, breathing at different altitudes and frequencies, and varying cardiac output. The important role of pulmonary surfactant in breathing is also highlighted. In addition, we propose a new method to model the transport of oxygen and carbon dioxide between the alveolus and perfused capillaries, and provide a Lagrangian description of surfactant dynamics. Finally, the mechanical work of breathing at the level of the alveolus is derived and implemented to compare the required breathing effort in different scenarios. Numerical results are consistent with published experimental and computational work. [ABSTRACT FROM AUTHOR]