In very cold weather, a significant mechanism for heat loss...

In very cold weather, a significant mechanism for heat loss by the human body is energy expended in warming the air taken into the lungs with each breath. (a) On a cold winter day when the temperature is $-20^{\circ} \mathrm{C},$ what is the amount of heat needed to warm to internal body temperature $\left(37^{\circ} \mathrm{C}\right)$ the $0.50 \mathrm{~L}$ of air exchanged with each breath? Assume that the specific heat of air is $1020 \mathrm{~J} /(\mathrm{kg} \cdot \mathrm{K})$ and that $1.0 \mathrm{~L}$ of air has a mass of $1.3 \mathrm{~g}$. (b) How much heat is lost per hour if the respiration rate is 20 breaths per minute?

Key Concepts

Specific Heat Capacity

Specific heat capacity is a property that indicates the amount of energy required to raise the temperature of a unit mass of a substance by one degree Celsius (or Kelvin). This concept is fundamental in energy calculations, as it allows us to quantify how much heat input is necessary to achieve a desired temperature change in a material.

Heat Transfer Calculations

Heat transfer calculations involve determining the quantity of energy exchanged when a substance's temperature changes. Typically, the equation Q = m * c * ?T is used, where Q is the heat energy, m is the mass, c is the specific heat capacity, and ?T is the temperature difference. This forms a basic method for evaluating energy requirements in thermal processes.

Unit Conversion

Unit conversion is the process of converting measurement units from one system to another to maintain consistency in calculations. In the context of thermal energy problems, converting between units such as liters and kilograms (using density) is crucial for ensuring that all quantities are expressed in compatible units for accurate computations.

Rate Calculations in Energy Loss

Rate calculations assess how a quantity, such as energy, accumulates or is expended over a given period. In scenarios involving periodic processes like breathing, it is important to extrapolate the energy expenditure per event (per breath) to a longer duration (such as per minute or per hour) to determine the overall heat loss rate.

Respiratory Heat Loss

Respiratory heat loss refers to the energy expenditure incurred by the body when inhaling cold air and warming it to internal body temperature. This process is a significant pathway of heat loss in cold environments, as each breath requires energy to elevate the temperature of the inhaled air, contributing cumulatively to the body's overall heat loss.

Transcript

00:01 So in this problem first we will calculate mass of the air which we exchange in one breath so mass of air that is exchanged in one breath is equal to as volume of the air which we exchange in one breath is 0 .50 liter and we have the mass of one liter air which is equal to 1 .3 gram now we can also write this is 1 kilogram that is equal to thousand gram from here litter and a liter will be cancelled gram and gram will be cancelled so this gives us mess of air which we exchange in one breath is equal to 0 .3 65 times 10 to the power minus 3 kilogram.

01:29 Now the incoming air from outside has a temperature of minus 20 degrees celsius and inside the body its temperature becomes 37 degrees celsius.

01:49 Now we want to find the amount of heat that is needed to change the temperature of air in one breath from minus 20 degrees celsius to 37 degrees celsius so here we will use a relation that is delta q equal to m c delta t so here m is mass of the air which is exchanged in one breath which is 0 .65 times 10 to the power minus 3 kilogram and specific heat for air is 1 to 1020 june per kilogram per degree celsius and change in temperature is final temperature is 37 degrees celsius minus initial temperature is minus 20 degrees celsius this gives us the amount of heat that is needed to change the temperature of air in one breath equal to that is 7 .7 .71 jule so this is the answer of part a of this problem now in the second part of this problem we want to find the total heat lost per hour from the body in order to change the temperature of the incoming air from minus 20 degrees celsius to 37 degrees celsius now we can easily calculate this as total heat lost per hour that is equal to first we will find here the total number of breaths per hour best per hour into the heat lost lost per breath here we can put different values as heat lost per breath which we have already calculated in part a of this problem which is 37 .7 .7 .91 jule per breath and the best per hour that is the rate of breath which is given in the problem which is 20 breast per minute.

05:24 Now we want to convert the bread per minute into breaths per hour as one hour we have 60 minutes...

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