Question

A 32-year-old woman is evaluated in the emergency department for sudden-onset severe shortness of breath and right-sided chest pain that worsens with deep inspiration. She recently returned from a 14-hour international flight. Her medical history is unremarkable, and her only medication is an oral contraceptive pill. On physical examination, her lungs are clear to auscultation bilaterally. Her right calf is mildly swollen and tender.

Vital signs:
Temperature: $37.1^\circ\text{C}$ ($98.8^\circ\text{F}$)
Blood pressure: $118/74\text{ mmHg}$
Pulse: $115/\text{min}$
Respiratory rate: $26/\text{min}$

An arterial blood gas is obtained while the patient is breathing room air at sea level:
pH: $7.52$
$PaCO_2$: $28\text{ mmHg}$
$PaO_2$: $56\text{ mmHg}$
$HCO_3^-$: $22\text{ mEq/L}$

To further evaluate the pathophysiology of her hypoxemia, the physician decides to calculate the alveolar-arterial (A-a) oxygen gradient. Assume the partial pressure of inspired oxygen ($PiO_2$) is $150\text{ mmHg}$ and the respiratory quotient (R) is $0.8$.

Based on the laboratory results, what is the patient's calculated A-a gradient?