Rib Fracture: Given the history of an acute fall, a rib fracture is a highly plausible cause for left chest pain radiating to the back. The mechanism of injury (fall) directly correlates with the potential for rib fractures, which can cause sharp, localized pain that may radiate to the back due to the involvement of the posterior aspects of the ribs or the associated musculature.

Musculoskeletal Strain: The fall could have resulted in a strain to the muscles of the chest wall or back, leading to pain. This is a common and relatively benign condition but can be quite painful.
Pneumothorax: Although less common than rib fractures, a pneumothorax (collapsed lung) could occur after a fall, especially if there was significant trauma to the chest. The pain from a pneumothorax can be sharp and may radiate to the back.
Hemothorax: Similar to pneumothorax, a hemothorax (blood in the pleural space) could result from trauma to the chest and cause chest pain radiating to the back.

Aortic Dissection: Although less likely, an aortic dissection is a life-threatening condition that could present with sudden, severe chest pain radiating to the back after a traumatic event. The force of the fall could potentially cause a tear in the aorta.
Pulmonary Embolism: While not directly related to the fall, if the patient has been immobilized after the fall, there's an increased risk of deep vein thrombosis and subsequent pulmonary embolism, which can cause chest pain.
Myocardial Infarction: Chest pain radiating to the back can also be a symptom of a heart attack, especially in women or individuals with atypical presentations. The stress of a fall could potentially trigger a cardiac event.

Oesophageal Rupture: A rare but serious condition that could occur after a fall, especially if there was significant abdominal or chest trauma. It would present with severe chest pain.
Spinal Fracture: While more commonly associated with back pain, certain types of spinal fractures (e.g., thoracic spine fractures) could radiate pain to the chest area, especially if there's associated musculature or nerve root involvement.
Diaphragmatic Rupture: This is a rare injury that could occur after significant trauma to the chest or abdomen, leading to chest pain and potentially breathing difficulties.