Problem 26 - University Physics with Modern Physics

Question

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\(^\circ\)C, what amount of heat is needed to warm to body temperature (37\(^\circ\)C) the 0.50 L of air exchanged with each breath? Assume that the specific heat of air is 1020 J / kg \(\cdot\) K and that 1.0 L of air has mass \(1.3 \times 10{^-}{^3} kg\). (b) How much heat is lost per hour if the respiration rate is 20 breaths per minute?

Step-by-Step Solution

Answer

Each breath requires 37.9 J and 45471.6 J is lost per hour.

Step 1: Understand the problem

To solve this problem, we must calculate the heat required to warm a specific volume of air from a cold temperature to body temperature for each breath. Then, multiply this by the number of breaths taken in an hour.

Step 2: Calculate temperature change

Determine the temperature change (in Kelvin) from the cold air temperature to body temperature. We know the temperatures in Celsius, so we convert these to Kelvin. \[ T_{cold} = -20 + 273.15 = 253.15 \text{ K} \] \[ T_{body} = 37 + 273.15 = 310.15 \text{ K} \]The temperature change is: \[ \Delta T = T_{body} - T_{cold} = 310.15 - 253.15 = 57 \text{ K} \]

Step 3: Calculate heat per breath

Use the formula for heat transfer: \[ Q = mc\Delta T \]- The mass \( m \) of 0.50 L of air is calculated by \[ m = \left( \frac{1.3 \times 10^{-3} \text{ kg}}{1 \text{ L}} \right) \times 0.50 \text{ L} = 0.65 \times 10^{-3} \text{ kg} \]- Specific heat \( c \) of air is 1020 J/kg・K.The heat \( Q \) is:\[ Q = 0.65 \times 10^{-3} \times 1020 \times 57 \approx 37.893 \text{ J} \]

Step 4: Calculate heat loss per hour

Now calculate how much heat is lost per hour, knowing the respiration rate is 20 breaths per minute.First, find the total number of breaths in an hour:\[ 20 \text{ breaths/min} \times 60 \text{ minutes/hour} = 1200 \text{ breaths/hour} \]Total heat lost per hour is:\[ 1200 \text{ breaths} \times 37.893 \text{ J/breath} \approx 45471.6 \text{ J} \]

Step 5: Conclusion

On a cold winter day, approximately 37.9 J of heat is needed to warm the air in each breath to body temperature, and about 45471.6 J of heat is lost per hour due to respiration.

Key Concepts

Heat transfer

Heat transfer is the movement of thermal energy from a warmer object to a cooler one. In the context of respiration, when you inhale cold air, your body has to heat it up to match your body's temperature. This is an example of heat transfer.

In physics, heat transfer can occur in three ways: conduction, convection, and radiation. In respiration, it primarily happens through convection, where the air comes into direct contact with the warmer body tissues, causing the heat to flow from your body to the air.

This process is crucial for maintaining the body's internal temperature. It plays a significant role in thermodynamics—a field that studies the movement of heat and energy. It ensures that your body's systems function correctly despite changes in the external environment.

Respiration and heat loss

Respiration is not only vital for oxygen exchange but also influences heat loss. When you breathe in cold air, your body uses energy to warm it to body temperature, leading to heat loss. Each breath contributes a small amount to the overall heat lost, which can add up quickly in cold conditions.

In the given exercise, we explore how much heat your body loses through respiration in cold weather. By knowing the temperature difference and specific heat capacity, we can calculate the energy expended per breath. This is essential not only in understanding human biology but also in designing effective clothing and environments that minimize unnecessary heat loss.

Specific heat capacity

Specific heat capacity is a property of a material that indicates how much heat energy is required to change its temperature by a certain amount. For air, the specific heat capacity is important because it determines how much energy is needed to raise the temperature of the air when breathing.

In our exercise, the specific heat capacity of air is given as 1020 J/kg・K. This means that it takes 1020 Joules to increase the temperature of 1 kilogram of air by 1 Kelvin.

Knowing the specific heat capacity allows us to derive how much energy is needed for the process of warming up the inhaled air, which is a crucial part of respiratory physiology and thermoregulation within the human body.

Temperature conversion

Temperature conversion is the process of changing a temperature reading from one scale to another. In the exercise, we converted temperatures from Celsius to Kelvin, which is necessary for consistent calculations in thermodynamics.

Celsius and Kelvin scales are related directly: 0 degrees Celsius is equivalent to 273.15 Kelvin. Thus, to convert any Celsius temperature to Kelvin, you add 273.15. For instance, cold air at -20°C becomes 253.15 K and the body's temperature 37°C is 310.15 K.

This conversion is essential because many physical equations, including those related to heat transfer, require temperature values in Kelvin for accuracy and consistency. Understanding how to convert temperatures ensures that you can effectively perform calculations in thermal physics and related fields.