Jill takes in 0.021 mol of air in a single breath. The air...

Jill takes in $0.021 \mathrm{mol}$ of air in a single breath. The air is taken in at $20^{\circ} \mathrm{C}$ and exhaled at $35^{\circ} \mathrm{C} .$ (a) How much heat leaves her body in a single breath due to the temperature increase of the air? Ignore the humidification of the air in the lungs and treat air as an ideal diatomic gas. (b) Her respiration rate is 14 breaths per minute. At what average rate does heat leave her body due to the temperature increase of the air? Compare this with $72 \mathrm{W}$, the total rate of heat loss from her body.

Key Concepts

Molar Heat Capacity for Diatomic Gases

For diatomic gases, the molar heat capacity at constant pressure (C?) is typically (7/2)R, where R is the universal gas constant. This concept reflects the degrees of freedom available in diatomic molecules, which include translational and rotational motions (with vibrational modes typically ignored at moderate temperatures). This molar heat capacity is used to calculate the amount of heat needed to raise the temperature of a mole of gas by one degree under constant pressure conditions.

Heat Transfer Calculation

Heat transfer in a substance undergoing a temperature change can be calculated using the formula Q = n · C? · ?T, where Q is the heat exchanged, n is the number of moles, C? is the molar heat capacity at constant pressure, and ?T is the temperature difference. This fundamental relation in thermodynamics allows us to determine the thermal energy absorbed or released when a gas is heated or cooled.

Temperature Difference and Unit Consistency

In thermodynamics, temperature differences are identical whether measured in degrees Celsius or Kelvin because the size of one degree is the same in both scales. This consistency allows for direct use of Celsius differences in calculations involving heat transfer, simplifying the process when converting between temperature scales is unnecessary.

Rate of Heat Transfer Calculation

The rate of heat transfer, often expressed in watts (joules per second), is computed by dividing the total amount of heat transferred by the time interval over which the process occurs. This concept is essential for comparing energy flows or losses over time—for example, determining average power outputs from intermittent processes such as breathing.

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