**Core Concept**
The visualization of air under both sides of the diaphragm is indicative of a condition where gas enters the peritoneal cavity, leading to free intraperitoneal air. This can occur due to various gastrointestinal or abdominal pathologies.

**Why the Correct Answer is Right**
The correct answer is related to a condition where there is a rupture or perforation in the gastrointestinal tract, allowing gas to escape into the peritoneal cavity. In this case, the duodenal ulcer perforation leads to free intraperitoneal air, which can be visualized under both sides of the diaphragm on an erect chest X-ray. This is due to the communication between the duodenum and the peritoneal cavity, allowing gas to spread throughout the abdominal cavity.

**Why Each Wrong Option is Incorrect**
* **Option A:** Perforated Meckel's diverticulum may lead to free intraperitoneal air, but it is less likely to be visualized under both sides of the diaphragm as it is a smaller structure and the air may be contained within the pelvic cavity.
* **Option B:** Uterine rupture following illegal abortion may lead to free intraperitoneal air, but it is more likely to be visualized in the lower abdomen rather than under both sides of the diaphragm.
* **Option D:** Liver abscess may lead to free intraperitoneal air, but it is typically a contained process and not likely to be visualized under both sides of the diaphragm.
* **Option E:** Ruptured appendix may lead to free intraperitoneal air, but it is typically located in the right lower quadrant and the air may be contained within the pelvic cavity, making it less likely to be visualized under both sides of the diaphragm.

**Clinical Pearl / High-Yield Fact**
In cases of suspected gastrointestinal perforation, an erect chest X-ray can be a valuable diagnostic tool, especially in the early stages of the condition. The presence of free intraperitoneal air under both sides of the diaphragm is a critical finding that suggests a perforation in the upper gastrointestinal tract.

**Correct Answer: C. Perforation of duodenal ulcer.**