The scaling coefficient for metabolic rate being 2/3 is based on which relationship?

The scaling coefficient of 2/3 for metabolic rate is indeed based on the relationship between surface area and volume. This relationship arises from the principles of allometric scaling, which studies how biological measurements change with size.

In animals, metabolic rate is closely linked to how heat is produced and lost, which correlates with surface area. As an animal's size increases, its volume (which determines the mass and thus the total metabolic 'work') increases at a faster rate than surface area (which affects heat loss). The surface area to volume ratio decreases as animals get larger, leading to a 2/3 exponent in the metabolic rate equation when considering these dimensions.

This relationship is crucial because it helps explain why larger animals tend to have lower metabolic rates per unit mass compared to smaller animals. Smaller animals have a higher surface area to volume ratio, allowing them to lose heat more rapidly and thus requiring a higher metabolic rate to maintain body temperature. As a result, this relationship is foundational for understanding metabolic rates across different body sizes in the animal kingdom.