The volume of air taken in during a typical breath is 0.5 L. The inhaled air is heated to 37^∘

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The volume of air taken in during a typical breath is 0.5...

The volume of air taken in during a typical breath is $0.5 \mathrm{~L}$. The inhaled air is heated to $37^{\circ} \mathrm{C}$ (the internal body temperature) as it enters the lungs. Because air is about $80 \%$ nitrogen $\mathrm{N}_{2}$, we can model it as an ideal gas. Suppose that the outside air is at room temperature $\left(20^{\circ} \mathrm{C}\right)$ and that you take two breaths every $3.0 \mathrm{~s}$. Assume that the pressure does not change during the process. (a) How many joules of heat does it take to warm the air in a single breath? (b) How many food calories (kcal) are used up per day in heating the air you breathe? Is this a significant amount of typical daily caloric intake?

Key Concepts

Ideal Gas Law

The ideal gas law relates pressure, volume, temperature, and the number of moles of a gas through the equation PV = nRT. It provides a useful model to approximate the behavior of gases under many conditions, allowing calculations of changes in state when one of the variables (such as temperature) is altered while holding others constant, as in the case of breathing where pressure is assumed constant.

Heat Capacity at Constant Pressure

Heat capacity at constant pressure (Cp) is a measure of the amount of heat required to raise the temperature of a given amount of a substance by one degree at constant pressure. In thermodynamics, for ideal gases, this property is used to calculate the energy needed to heat the gas when its temperature increases, which is essential for determining the heat absorbed by the air during warming in the lungs.

Molar Quantities and Conversion of Volume to Moles

In thermodynamic calculations involving gases, it is often necessary to convert the given volume of a gas to the number of moles using the ideal gas law. This conversion enables further computations involving heat capacity and energy, since most thermal properties of gases are given in a per-mole basis, linking macroscopic measurements like volume to molecular-scale quantities.

Energy Unit Conversions

Energy unit conversions are crucial when comparing energy values expressed in different units, such as converting joules to food calories (kilocalories). This step is important in applied thermodynamics to relate physical energy calculations to dietary energy, thereby providing context about the significance of the energy expenditures in biological processes like breathing.

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