Problem 36 - University Physics with Modern Physics

Question

Heat Loss During Breathing. 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 amount of heat is needed to warm to 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 mass \(1.3 \times 10^{-3} \mathrm{kg}\) . (b) How much heat is lost per hour if the respiration rate is 20 breaths per minute?

Step-by-Step Solution

Answer

37.719 J per breath; 45,262.8 J per hour.

Step 1: Convert Temperatures to Kelvin

First, convert the given temperatures from Celsius to Kelvin, since the formula for heat transfer uses temperatures in Kelvin. \(-20^{\circ}\, C\) becomes \(-20 + 273.15 = 253.15\, K\) and \(37^{\circ}\, C\) becomes \(37 + 273.15 = 310.15\, K\).

Step 2: Calculate Temperature Change

Now, find the change in temperature \(\Delta T\). It is the difference between the final and initial temperatures in Kelvin: \(\Delta T = 310.15\, K - 253.15\, K = 57\, K\).

Step 3: Determine Mass of Air per Breath

With the given density of air, calculate the mass of 0.50 L of air. This is done using: \(\text{mass} = \text{volume} \times \text{density} = 0.50\, \text{L} \times 1.3 \times 10^{-3}\, \text{kg/L} = 6.5 \times 10^{-4}\, \text{kg}\).

Step 4: Calculate Heat Required per Breath

Use the formula for heat transfer: \(Q = mc\Delta T\), where \(m\) is the mass, \(c\) is the specific heat, and \(\Delta T\) is the temperature change. \(Q = 6.5 \times 10^{-4}\, \text{kg} \times 1020\, \text{J/kg}\, K \times 57\, K = 37.719\, J \).

Step 5: Calculate Heat Lost per Minute

Multiply the heat per breath by the respiration rate to find the heat lost per minute. \(Q_{per\, minute} = 37.719\, J \times 20 = 754.38\, J/min\).

Step 6: Calculate Heat Lost per Hour

Multiply the heat lost per minute by 60 to convert to heat lost per hour. \(Q_{per\, hour} = 754.38\, J/min \times 60\, min = 45262.8\, J\).

Key Concepts

Specific Heat Capacity

Specific heat capacity is an intrinsic property of substances that measures the amount of heat per unit mass required to raise the temperature by one degree Celsius (or one Kelvin). In our scenario, air has a specific heat capacity of 1020 J/kg·K, meaning for every kilogram of air, 1020 Joules are needed to raise its temperature by 1 Kelvin. This property is crucial for calculating heat transfer as it directly influences how much energy is needed to change the temperature of the air we inhale from a cold temperature to body temperature.

• Helps understand energy changes in thermal scenarios.
• Essential for calculating energy requirements in processes such as breathing in cold environments.

Heat Transfer

Heat transfer is the process by which thermal energy moves from a warmer object or region to a cooler one. In the context of breathing, this involves the body transferring heat to the cold air we inhale during winter to warm it up to body temperature. The calculation of heat transfer involves using the formula:\[Q = mc\Delta T\]where:

• \( Q \) is the heat transferred (in Joules),
• \( m \) is the mass of the air breathed in (in kg),
• \( c \) is the specific heat capacity (J/kg·K),
• \( \Delta T \) is the change in temperature (in K).

By understanding how heat is transferred, one can appreciate the body's energy expenditure in maintaining an optimal temperature for processes like respiration.

Respiration Rate

The respiration rate, or breathing rate, refers to the number of breaths a person takes per minute. It plays a significant role in calculating the total heat lost by the body due to breathing. This rate affects the total volume of air inhaled and exhaled, thereby influencing the overall heat that needs to be transferred. For this exercise, a respiration rate of 20 breaths per minute is considered. This means that over the course of an hour, the number of breaths will increase the total energy expenditure:

• Multiply the energy required per breath with the respiration rate to find the total heat expenditure over a given time period, such as one minute or one hour.
• This contributes to understanding how physical activity and temperature impact energy loss during respiration in cold weather.

Temperature Conversion

In thermodynamics, temperature conversion is often needed to facilitate calculations that require temperature differences. While humans generally use the Celsius scale, scientific calculations with heat transfer prefer Kelvin. The conversion between Celsius and Kelvin is straightforward: add 273.15 to the Celsius temperature to convert it to Kelvin.

• This ensures that temperature differences, which are the same for Celsius and Kelvin, are accurately represented in calculations.
• Using Kelvin helps avoid negative values when working with temperature changes, which is particularly useful when calculating work and energy transfer.

For example, in this case, converting the temperatures to Kelvin before finding the temperature difference ensures that all subsequent calculations for heat transfer use the correct values.