Question 17.3: Determine the basal metabolic rate (BMR) per unit mass of an...

The basal metabolic rate per unit mass of a warm-blooded animal is given by Eq. (17.17) as

\text{BMR}/m = 293 ( \text{m}^{−0.25})

where \text{BMR} is in \text{kJ/d} and m is in \text{kg}. Then, the \text{BMR} per unit mass of an 80.0  \text{kg} human is

(\text{BMR}/m)_\text{human} = 293(80.0^{−0.25}) = 98.0 \frac{\text{kJ}}{\text{kg.d}}

and the \text{BMR} per unit mass of an 8.00  \text{gram} mouse is

(\text{BMR}/m)_\text{mouse} = 293(0.00800^{−0.25}) = 980. \frac{\text{kJ}}{\text{kg.d}}

This calculation shows that the basal metabolic rate per unit mass of a mouse is ten times that of a human. This large difference is primarily due to the difference in surface area to volume ratio of these mammals. Since heat loss from the body is primarily by convection heat transfer, which is proportional to surface area, and internal heat generation inside the body is proportional to its volume, then as the ratio of surface area to volume increases the internal heat generation rate must also increase if a mammal is to maintain its body temperature. To produce higher internal heat generation rates, small animals must feed very often if they are not to starve.

It is a fact that the smaller any object becomes, the larger its surface area to volume ratio becomes. This is easiest to understand with spherical objects. The surface area of a sphere is 4πR^2 whereas its volume is (4/3)πR^3 . Therefore, its surface area to volume ratio is

(\frac{\text{Surface area}}{\text{Volume}})_\text{sphere} = \frac{4π\text{R}^2}{\frac{4}{3} π\text{R}^3} = \frac{3}{\text{R}}

and this ratio decreases inversely with increasing R. Thus, there is a lower limit to the size of warm-blooded animals. The shrew and the hummingbird are the smallest known animals of this kind.

The body temperature of insects and cold-blooded animals is approximately equal to the temperature of their surroundings. Consequently, there is no thermodynamic lower limit to their size.