Problems involving dinosaurs are sure to spark introductory student interest. In this paper, we describe an intriguing example in which the embryonic metabolism of non-avian dinosaurs, crocodiles, and birds are determined via conservation of energy from the incubation time of the egg and the mass of the hatchling. All students, particularly those interested in the life sciences, understand the relevance of metabolism to living processes. Given the importance of temperature in metabolism, a simple derivation of the Boltzmann factor is presented and used by the students to calculate the activation energy *E*_{A} of the reactions of metabolism in *Crocodylus johnstoni* (the crocodile of the *Crocodile Dundee*^{®} movies). Intriguingly, it is found that the activation energy is lower at the highest incubation temperatures.

## Introduction and background

The total metabolism of an animal during its growth phase provides the energy per unit time to maintain the currently existing cells plus the energy required to grow new cells, as given in Eq. (1). This equation is due to conservation of energy for the animal.

*B*is the average metabolic rate of the animal,

*B*

_{c}is the metabolic rate of the average cell,

*N*

_{c}is the number of cells in the animal,

*E*

_{c}is the energy required to produce a new cell, and

*t*is time. The mass of the animal

*m*is related to the number of cells

*N*

_{c}it has via the mass of each cell,

*m*

_{c}, which is assumed to be fixed:

*m*=

*N*

_{c}

*m*

_{c}⇒

*N*

_{c}=

*m*/

*m*

_{c}. Studies

^{1,2}on mammals, birds, reptiles, and fish have shown that the metabolic rate of an animal

*B*obeys a power law:

*B*

_{0}is the mass-independent metabolic prefactor. Lee

^{3}and Lee et al.

^{4}have shown that Eqs. (1) and (2) can be solved to yield the mass of the animal as a function of its age:

*M*is the mass of the adult animal, and the growth fitting parameter

*p*is equal to

*B*

_{0}

*m*

_{c}/

*E*

_{c}. For this work on embryonic growth,

*m*

_{0}is the mass of the fertilized ovum, assumed to be 6.4 mg.

^{4}The energy necessary to grow an embryonic cell per unit mass

*E*

_{c}/

*m*

_{c}has been determined to be 3107 ± 219 kJ/kg.

^{5}

*B*

_{0}:

Equation (4) is used to determine the embryonic metabolic prefactor from the incubation time *t*_{h}, hatching mass *m*_{h}, the energy *E*_{c} necessary to create a cell of mass *m*_{c}, the mass of the fertilized ovum *m*_{0}, and the mass of the adult animal *M*.

Birds incubate their eggs by constructing a nest and brooding the eggs via body heat. We assume that those eggs incubate at a single temperature. Since crocodiles are ectothermic animals, their body temperature changes significantly depending on environmental conditions. They build nests but do not brood the eggs, though some heat can be provided to the eggs by the decay of organic material added to the nest by the mother. Therefore, crocodile eggs incubate at variable temperatures. The table in the Appendix^{6} contains the data for *C. johnstoni* incubated at a variety of temperatures. For simplicity, we refer to non-avian dinosaurs simply as dinosaurs in this activity, while acknowledging that birds are living members of the clade Dinosauria. The incubation times of the dinosaurs *Troodon formosus*, *Protoceratops andrewsi*, and *Hypacrosaurus stebingeri* have been determined by counting the number of daily growth lines in the teeth of nearly hatched embryos and assuming that these teeth first form at 42% of the incubation time (the same time in incubation as in modern crocodilians).^{7,8} Of the three dinosaurs, *Troodon* is believed to have brooded its eggs. Since it is likely that *Troodon* was endothermic, it presumably maintained an elevated body temperature. Nevertheless, its eggs are believed to have been partially buried in the ground itself. Consequently, the eggs were not as well insulated as for modern birds. There is no evidence that *Protoceratops* brooded its eggs. The mass of the adult members of the species was 180 kg (about 400 lb), making it difficult for the adult to brood its eggs. The situation for *Hypacrosaurus* is even worse. With its adult mass of about 4000 kg (8800 lb), it would have been impossible for the adults to brood the eggs.

The incubation time, hatchling mass, and adult mass are measured directly for the extant birds and *C. johnstoni*, with the values from the literature given in the Appendix.^{6}

The Boltzmann factor can be naturally included in the curriculum of introductory physics courses just after the derivation of the ideal gas law via the kinetic theory of gases, as we now show.

*f*(

*v*), the classical probability that an atom of mass

*m*at temperature

*T*(measured in kelvins) has speed

*v*, is

*k*

_{B}is Boltzmann’s constant, and

*b*≡

*m*/2

*k*

_{B}

*T*. Each atom is assumed to have an insignificant volume compared to the volume of the gas. The gas atoms do not interact with each other except via collisions. This distribution is shown in Fig. 1, and a collection of

*N*atoms is described by multiplying this distribution by

*N*. In the derivation of the ideal gas law, this distribution is used to connect the root-mean-square kinetic energy of the atoms to the temperature of the gas.

Before proceeding with the evaluation of the root-mean-square speed *v*_{rms}, we discuss how one calculates an average value with a distribution function by considering the case of flipping a group of 10 coins. This is necessary since the students usually have not calculated averages via this method previously.

*v*

_{rms}is given by

Equation (8) is used in the derivation of the ideal gas law from the kinetic theory of gases.

Only a few more steps are required to give the students an understanding of the meaning of the Boltzmann factor (the probability that a system in thermal equilibrium will be in a particular energy state) and its mathematical form. The students are alerted to the fact that this is not a rigorous derivation, but rather a simplified physical argument for a special case that uses an approximation. Nonetheless, it yields the correct expression with a helpful physical picture. The general derivation of the Boltzmann factor is presented to the students in the junior-level thermal physics course.

Figure 2 shows the number of atoms as a function of energy for our system of *N* gas atoms. Note that, for energies larger than (3/2)*k*_{B} *T* (the root-mean-square kinetic energy of the gas), the function looks very similar to an exponential. We test this by taking the natural logarithm of the number of atoms with energy greater than (3/2)*k*_{B} *T* and plotting that as a function of their energy, shown in Fig. 3. The fit to a straight line for this data is very good, though not perfect. Increasing the energy of the low energy cutoff improves the linearity of this graph. By repeating this analysis for gases at different temperatures, we find that the slope of the fit is inversely proportional to the absolute temperature *T*.

*k*

_{B}

*T*yields the following for the number of atoms at the two energies shown by the vertical red lines:

*B*is a constant. Dividing one equation by the other yields

Since the probability that an atom will be in a particular energy state is proportional to the number of atoms in that state in our classical model, Eq. (11) is the Boltzmann factor.^{9}

In our classroom discussions, the case of evaporation in which a molecule inside a liquid escapes the liquid at temperatures well below the liquid’s boiling point is examined. The long exponential tail to higher energies in Fig. 2 guarantees that, at any temperature, there are some atoms with enough energy to escape the liquid. Evaporation happens at a faster rate at a higher temperature since there are more atoms at higher energies.

Humans perspire in order to remove excess heat from their body since the higher-energy water molecules in a droplet of sweat are more likely to escape than the lower-energy water molecules. Since the higher-energy water molecules preferentially leave, the average kinetic energy of the remaining water molecules will be lower. That is, the water droplet will have cooled. This cooling will facilitate the movement of thermal energy from the skin into the sweat droplet, cooling the person.

*B*

_{0}at two temperatures (

*T*

_{0}and

*T*) are connected by the Boltzmann factor as follows:

Though the temperatures in these equations are in kelvins, the incubation temperatures are given in degrees Celsius merely for convenience.

## Activity

Replicas and pictures of dinosaur, crocodile, and bird eggs are brought to the classroom to initiate a discussion about the embryonic development of these animals. The different sizes of these eggs is the most notable difference, and we discuss what inferences about incubation time, hatching mass, and embryonic metabolism can be drawn from the size of the egg.

We discuss how conservation of energy is incorporated into Eq. (1). Since differential equations are beyond the scope of this first-year course, we discuss the derivation of Eq. (4) in detail.

As described in junior-level physics and chemistry courses on thermodynamics, the Boltzmann factor describes the temperature dependence of chemical reactions. Having derived Eq. (13), we note that the activation energy of the metabolic chemical reaction is given by the slope of the data plotted via this equation.

The Appendix^{6} provides the necessary data for three dinosaurs, a crocodile, and 13 species of birds. The students calculate *B*_{0} by using Eq. (4). The importance of accounting for experimental uncertainties is emphasized in class, and the students produce results with the associated uncertainties. The students are given the data and calculate the embryonic metabolic prefactor *B*_{0} for all entries, as shown in Fig. 4.

From Fig. 4, we see that modern birds had the highest metabolism in the egg. Of the three dinosaurs, we see that *Troodon* had the highest metabolism, presumably due to the fact that *Troodon* brooded its eggs, which would result in a higher average incubation temperature than observed in *Protoceratops* and *Hypacrosaurus*.

The students also plot *B*_{0} as a function of incubation temperature for *C*. *johnstoni*, as is shown in Fig. 5. This figure shows that *B*_{0} almost doubles in going from 28 to 34 °C. To explore the nature of this effect, the students make a graph of their data via Eq. (13). As shown in Fig. 6, the results show two straight lines with different slopes.

From the measured slopes, we find that the activation energy *E*_{A} for the metabolic processes is 0.978 ± 0.061 eV for the 28–32 °C temperature range and 0.288 ± 0.017 eV for temperatures between 32.5 and 34 °C. Context for this observation is provided by discussing the fact that sex in all crocodilians and many other reptiles is determined not by chromosomes, but by the temperature of incubation.^{10} Like most other crocodilians, *C. johnstoni* exhibits a female–male–female pattern of temperature-dependent sex determination in which more females are produced at the lower and higher incubation temperatures within the viable range for the species. The proportion of females decreases with increasing incubation temperature until about 32 °C, at which temperature the highest proportion of males is produced. This pivotal temperature marks a shift toward increasing proportion of females with increasing incubation temperature.^{10} Our results imply that something is fundamentally different about the metabolism of embryos incubating on either side of this pivotal temperature of 32 °C. Whether this shift is mechanistically linked to the pivotal temperature or coincidental could be determined by examining the relationship in other crocodilian species when the requisite data become available. It is noted that increased incubation temperature might cause unknown changes in the embryo, which could affect the metabolic rate in a manner unrelated to the Boltzmann factor. However, the Boltzmann factor is the most parsimonious explanation.

## Acknowledgments

Useful conversations with Lauren Grundy of the St. Augustine Alligator Farm and Colin Stevenson of Crocodiles of the World are gratefully acknowledged.

## References

*Bioenergetics and Growth*(

*Nature*

*Phys. Rev. E*

*Phys. Teach*.

*Crocodylus johnstoni*and

*Crocodylus porosus*

*Physiol. Zool.*

*TPT Online*at https://doi.org/10.1119/5.0141595, in the “Supplementary Material” section.

*Proc. Natl. Acad. Sci. U.S.A*.

*Sci. Rep.*

*Phys. Rev. Spec. Top. Phys. Educ. Res.*

*J. Anim. Ecol.*

**Scott Lee** *loves exciting students with new approaches to interesting problems. He enjoys bird watching, tries to avoid crocodiles, and would love to find a dinosaur fossil.* **Scott.Lee@utoledo.edu**

**Joanna Larson** *is the assistant curator of the Museum of Biodiversity and an assistant professor of the practice at the University of Notre Dame. She enjoys teaching her students about birds, the true living dinosaurs, and showing them American crocodiles on field trips.*

**Joshua Thomas** *is now the director of Stull Observatory and an assistant professor of physics and astronomy at Alfred University. He enjoys teaching and finding new ways to engage students with materials. He has also found that interdisciplinary research helps to broaden his ability to get students interested in learning physics and astronomy.*