Final answer:
The normal alveolar-to-arterial difference for a 56-year-old female in the emergency department is approximately 14 mm Hg, calculated from the expected A-a gradient formula.
Step-by-step explanation:
The normal alveolar-to-arterial (A-a) gradient increases with age. The partial pressure of oxygen in alveolar air is about 104 mm Hg, and the partial pressure of oxygen in the blood of the pulmonary capillaries is about 40 mm Hg. The expected A-a gradient can be roughly estimated as (age in years/4) + 4, which for a 56-year-old would be approximately (56/4) + 4 = 14 mm Hg. Therefore, the normal alveolar-to-arterial difference for a 56-year-old female in the emergency department would be 14 mm Hg.