The excess internal energy of metabolism is exhausted...

The excess internal energy of metabolism is exhausted through a variety of channels, such as through radiation and evaporation of perspiration. Consider another pathway for energy loss: moisture in exhaled breath. Suppose you breathe out $22.0$ breaths per minute, each with a volume of $0.600$ L. Suppose also that you inhale dry air and exhale air at $37^{\circ} \mathrm{C}$ containing water vapor with a vapor pressure of $3.20 \mathrm{kPa}$. The vapor comes from the evaporation of liquid water in your body. Model the water vapor as an ideal gas. Assume its latent heat of evaporation at $37^{\circ} \mathrm{C}$ is the same as its heat of vaporization at $100^{\circ} \mathrm{C}$. Calculate the rate at which you lose energy by exhaling humid air.

Key Concepts

Evaporative Cooling

Evaporative cooling is the process of heat loss from a system due to the phase change of a liquid (typically water) into vapor. This process consumes energy (latent heat), which is taken from the system, thereby cooling it. In biological contexts, evaporation (through mechanisms like perspiration or exhalation) plays a crucial role in thermal regulation.

Ideal Gas Law

The ideal gas law establishes a relationship among pressure, volume, temperature, and the number of moles of a gas. It is widely used to calculate the quantity of a particular gas under specified conditions and is essential for determining the number of moles of water vapor in a given volume of exhaled air using measured pressures and temperatures.

Latent Heat of Vaporization

The latent heat of vaporization is the amount of energy required to change a unit mass of a liquid into a vapor without a change in temperature. It is a fundamental property when calculating energy loss due to phase changes, such as the evaporation of water from the body, since the process consumes this energy from internal metabolic heat.

Partial Pressure

Partial pressure refers to the pressure that a single gas in a mixture would exert if it alone occupied the entire volume. In calculations involving humid air, the water vapor exerts its own partial pressure, which is used alongside the ideal gas law to determine the moles of water vapor present, and subsequently the energy associated with its evaporation.

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