The excess internal energy of metabolism is exhausted through a variety of chann

Question

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 20.0 breaths per minute, each with a volume of 0.540 L. Suppose also that you inhale dry air and exhale air at 31.1°C containing water vapor with a vapor pressure of 3.20 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 31.1°C is the same as its heat of vaporization at 100°C. Calculate the rate at which you lose energy by exhaling humid air.
__________ W

Explanation / Answer

Using the ideal gas law:
PV = nRT
T needs to be in kelvin, hence the +273.
(3.20kPa)(0.540L) = n(8.314 L*kPa/mol*K)(31.1 + 273)
n = 6.835*10^-4 mol H2O per breath

Water vapor has a molar mass of 18.02g/mol, so really you are exhaling 0.01232g H2O per breath.

Water has a latent heat of vaporization of 2260J/g (at 100 degrees C, but we're assuming it is the same for 31.1 per the problem), so the heat energy to vaporize one breath would be:

Q = mL = (0.01232g)(2260J/g) = 27.83J

If you are exhaling 20 breaths per minute, then that would be a power of:
27.83J/breath * 20breaths/60s = 9.278 W

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