Problem 11 - Animal Physiology

Question

Using the set of data that follows, calculate how many of the molecules of \(\mathrm{O}_{2}\) that were used in aerobic catabolism by Julius Caesar are in each liter of atmospheric air today. All values given are expressed at Standard Conditions of Temperature and Pressure (see Appendix C) and therefore can be legitimately compared. Average rate of \(\mathrm{O}_{2}\) consumption of a human male during ordinary daily activities: \(25 \mathrm{~L} / \mathrm{h}\). Number of years after his birth when Caesar was mortally stabbed near the Roman Forum: 56 years. Number of liters of \(\mathrm{O}_{2}\) per mole: \(22.4 \mathrm{~L} / \mathrm{mol}\). Number of moles of \(\mathrm{O}_{2}\) in Earth's atmosphere: \(3.7 \times 10^{19} \mathrm{~mol}\). Number of molecules per mole: \(6 \times 10^{23} \mathrm{molecules} / \mathrm{mol}\). Amount of \(\mathrm{O}_{2}\) per liter of air at sea level \(\left(20^{\circ} \mathrm{C}\right): 195 \mathrm{~mL} / \mathrm{L}\). Be prepared to be surprised! Of course, criticize the calculations if you feel they deserve criticism.

Step-by-Step Solution

Answer

Given the oxygen consumption rate and lifespan of Julius Caesar, there would be approximately one molecule from his breathed oxygen in every liter of today's atmospheric air. This is obtained after completing the series of conversions and computations in Steps 1-5.

Step 1: Calculate total oxygen consumption

First, calculate how much oxygen Julius Caesar consumed over his lifetime. This is done by multiplying his average oxygen consumption rate by the hours in a year and by the number of years he lived, using the following equation: \((25 \, \text{L/h} \times 24 \, \text{h/day} \times 365 \, \text{days/year} \times 56 \, \text{years})\).

Step 2: Convert liters to moles

To be able to compare this value to the moles of oxygen in the atmosphere, convert liters of oxygen consumed to moles, using the conversion 22.4 \, \text{L/mol}. Thus, use the following equation: \(\text{Moles of O}_2 \text{ consumed} = \text{(Total oxygen consumption in liters)} / 22.4 \, \text{L/mol}\).

Step 3: Calculate molecules of oxygen

Convert moles of oxygen to molecules, using Avogadro's number, and through the following equation: \(\text{Molecules of O}_2 \text{ consumed} = \text{(Moles of O}_2 \text{ consumed)} \times 6 \times 10^{23} \, \text{molecules/mol}\).

Step 4: Calculate the mole fraction of oxygen in the atmosphere

Calculate the mole fraction of oxygen in the atmosphere given that there are 195 mL of O2 per liter of air (or 0.195 \, \text{mol/L}). So, \(\text{Mole fraction of O}_2 = 0.195 \)

Step 5: Calculate the number of molecules of oxygen per liter

Multiply the number of molecules of oxygen consumed by Julius Caesar by the mole fraction of oxygen in the atmosphere to find the number of molecules of Julius Caesar's oxygen in each liter of air today. \(\text{Molecules of O}_2 \text{ per liter} = \text{(molecules of O}_2 \text{ consumed)} \times \text{(mole fraction of O}_2)\).

Key Concepts

Oxygen Consumption Rate

The oxygen consumption rate is a vital measurement in understanding how the human body uses oxygen during aerobic catabolism—the process by which cells convert food into energy in the presence of oxygen. For example, an average adult male has an oxygen consumption rate of about 25 liters per hour during ordinary daily activities. This rate could vary based on activity level, health state, and environmental factors.

To provide improved context, it's interesting to consider a historical figure like Julius Caesar. By figuring out his lifetime oxygen consumption—that is, 25 L/h multiplied by the number of hours in a year, then by the number of years he lived—we gain a unique perspective on the quantity of oxygen a single person might use in a lifetime.

Avogadro's Number

Enshrined in the halls of chemistry is Avogadro's number, a constant that defines the number of particles—typically atoms or molecules—in one mole of a substance. This number is approximately 6.022 x 10²³ particles per mole and plays a crucial role in converting between microscopic and macroscopic amounts of a substance. In our example regarding Caesar's oxygen usage, Avogadro's number is used to transform moles of oxygen into a comprehensible count of oxygen molecules, thereby linking how much oxygen he consumed over his lifetime to the number of individual oxygen molecules he used.

Mole Fraction of Oxygen

The mole fraction of oxygen is an expression of the concentration of oxygen molecules in a given mixture of gases, in this case, Earth's atmosphere. It is the ratio of the number of moles of oxygen to the total number of moles of all gases present. At sea level and at a temperature of 20°C, 195 mL of oxygen is present in every liter of air, which we can convert to a mole fraction to better understand the portion of air that is oxygen. Knowing the mole fraction of oxygen is essential when calculating the spread of oxygen molecules attributable to a person across the atmosphere, as it allows us to estimate the number of these molecules in a specified volume of air today.

Standard Conditions of Temperature and Pressure

Standard Conditions of Temperature and Pressure (STP) are a set of conditions used in chemistry to enable comparisons between different sets of data. STP is defined as a temperature of 0°C (273.15 K) and a pressure of 1 atmosphere (1 atm or 101.325 kPa). These conditions are referenced in our problem to establish a common baseline, ensuring that volumetric measurements, like liters of oxygen consumed or in the atmosphere, are comparable. When we discuss the volume of gases, such as the 22.4 liters per mole of oxygen, it's crucial to note that this is under STP, as gas volumes can vary significantly with changes in temperature and pressure.