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On a cold winter day the temperature is \(2^{\circ} \mathrm{C}\) and the relative humidity is \(15 \% .\) You inhale air at an average rate of \(5500 \mathrm{mL} / \mathrm{min}\) and exhale a gas saturated with water at body temperature, roughly \(37^{\circ} \mathrm{C} .\) If the mass flow rates of the inhaled and exhaled air (excluding water) are the same, the heat capacities \(\left(C_{p}\right)\) of the water-free gases are each \(1.05 \mathrm{J} /\left(\mathrm{g} \cdot^{\circ} \mathrm{C}\right),\) and water is ingested into the body as a liquid at \(22^{\circ} \mathrm{C},\) at what rate in \(\mathrm{J} /\) day do you lose energy by breathing? Treat breathing as a continuous process (inhaled air and liquid water enter, exhaled breath exits) and neglect work done by the lungs.

Step by step solution

01

Calculate Mass Flow Rate of Inhaled Air

Recognize that the volume flow rate of 5500 mL/min for air can be converted to a mass flow rate using the density of air. Using an approximate density value for air at room temperature (roughly 1.18 g/L), mass flow rate becomes: \((5500 \, \text{mL/min}) \times (1.18 \, \text{g/L}) \times \left(\frac{1 \, \text{L}}{1000 \, \text{mL}}\right) \times \left(\frac{60 \, \text{min}}{1 \, \text{hr}}\right) \times \left(\frac{24 \, \text{hr}}{1 \, \text{day}}\right)\)

02

Calculate Heat Required to Warm Inhaled Air

The energy needed to warm this inhaled air from 2°C to 37°C is calculated using the specific heat formula: \(q = mC\Delta T\), where m is mass, C is specific heat capacity, and \(\Delta T\) is change in temperature. So, \(q_{\text{air}} = (\text{mass flow rate of air}) \times (C_{p_{\text{air}}}) \times (\text{temperature difference})\)

03

Calculate Mass Flow Rate of Exhaled Water Vapor

Assuming the exhaled breath is saturated with water vapor, the mass of the water exhaled can be determined using the ideal gas law (\(PV=nRT\)) and the density of water vapor at body temperature (37°C). Convert the volume flow rate of exhaled breath (assumed to be the same as the volume flow rate of the inhaled air) into a mass flow rate using the molar mass and density of water vapor. Remembering that at body temperature, the saturated vapor pressure of water is about 47 mmHg, or 0.0621 atm.

04

Calculate Heat Required to Vaporize Exhaled Water

The energy needed to produce this water vapor is calculated by first raising the temperature of the liquid water ingested into the body to body temperature, and then vaporizing the water. This can be computed using the heat of vaporization (\(q_{vap} = m \Delta H_{vap}\)) for water at 37°C, which is approximately 2.43 kJ/g. So, \(q_{\text{water}} = (m \times C_{p_{\text{water}}}\times \Delta T) + (m \times \Delta H_{vap})\)

05

Compute the Total Energy Loss

Combine the energies calculated in Step 2 and Step 4 to find the total energy loss due to breathing. Energy lost (\(E_{lost}\)) is the sum of the heat used to warm the inhaled air (\(q_{air}\)) and the energy used to produce the exhaled water vapor (\(q_{water}\)), that is \(E_{lost} = q_{air} + q_{water}\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Heat Capacity

When discussing thermodynamics and biological systems, heat capacity is a key concept. It refers to the amount of heat required to change the temperature of a substance by a specific amount. In our scenario, it is crucial to calculate how much energy your body uses to warm inhaled air. The specific heat capacity formula is given by:

• \( q = mC\Delta T \)
• \( m \) represents the mass flow rate of air
• \( C \) is the heat capacity, here \( 1.05 \text{ J/g⋅°C} \)
• \( \Delta T \) is the change in temperature from \( 2^{\circ} \text{C} \) to \( 37^{\circ} \text{C} \)

Understanding this helps us see how the body uses energy just to bring air to body temperature. This is vital, especially when considering overall energy expenditure without accounting directly for work done.

Phase Change

In the context of breathing, phase change becomes pertinent when considering the water vapor exhaled. As we breathe, air at lower humidity is inhaled and exhaled at body temperature as nearly saturated with water. This involves a key phase change:

• Raising the temperature of ingested liquid water to body temperature \( (37^{\circ} \text{C}) \)
• Vaporizing it into steam

The formula for the energy required during these changes is:

• The heating part: \( q = mCD \Delta T \)
• Then, vaporization energy : \( q_{vap} = m \Delta H_{vap} \)
• Where \( \Delta H_{vap} \) is the latent heat of vaporization, \( 2.43 \text{kJ/g} \) for water at body temperature

The phase change process here efficiently models how energy is needed to transition liquid water to vapor, emphasizing the body's invisible but substantial energy use in this vital function.

Ideal Gas Law

The Ideal Gas Law, expressed as \( PV = nRT \), is fundamental in examining the behavior of gases. In this scenario, we primarily utilize it to estimate the mass of exhaled water vapor:

• \( P \) is the pressure, specifically the saturated vapor pressure at \( 37^{\circ} \text{C} \), which is approximately \( 0.0621 \text{atm} \)
• \( V \) represents the volume of exhaled air, equal to the inhaled volume
• \( n \) is the number of moles of water vapor
• \( R \) is the ideal gas constant
• \( T \) is the absolute temperature in Kelvin

This law allows us to assess the behavior of the gaseous mixture upon exhalation, which is crucial for ascertaining how much water vapor is released. Consequently, it's a vital step in calculating the total energy loss due to breathing, giving insight into how variables like temperature and volume influence gas behavior.

Energy Balance in Processes

Balancing energy within biological systems encompasses various processes, releasing or absorbing energy. With breathing, the energy balance equation considers multiple {

• Heat required to warm the inhaled air
• Energy expended in the phase change of water

By combining these calculated energies:

• Total energy loss due to breathing: \( E_{lost} = q_{air} + q_{water} \)

This highlights how thermodynamic principles apply to biological functions, indicating the efficiency and energy requirements of natural processes like breathing. Understanding the energy balance provides valuable insights, especially in optimizing conditions for health or developing related biomedical technologies.