Problem 61 - College Physics

Question

\(\bullet\) Basal metabolic rate. The basal metabolic rate is the rate at which energy is produced in the body when a person is at rest. A 75 \(\mathrm{kg}(165 \mathrm{lb})\) person of height 1.83 \(\mathrm{m}\) (6 ft) would have a body surface area of approximately 2.0 \(\mathrm{m}^{2}\) . (a) What is the net amount of heat this person could radiate per second into a room at \(18^{\circ} \mathrm{C}\) (about \(65^{\circ} \mathrm{F}\) ) if his skin's surface temperature is \(30^{\circ} \mathrm{C}\) ? (At such temperatures, nearly all the heat is infrared radiation, for which the body's emissivity is 1.0 , regardless of the amount of pigment.) (b) Normally, 80\(\%\) of the energy produced by metabolism goes into heat, while the rest goes into things like pumping blood and repairing cells. Also normally, a person at rest can get rid of this excess heat just through radiation. Use your answer to part (a) to find this person's basal metabolic rate.

Step-by-Step Solution

Answer

(a) Net heat radiated: 97.79 W; (b) Basal metabolic rate: 122.24 W.

Step 1: Understanding the problem

The problem involves calculating heat radiation and basal metabolic rate for a person. First, we need to calculate the net heat radiated per second using the person's skin temperature and room temperature. Then, we'll determine the basal metabolic rate using the given fraction of energy that turns into heat.

Step 1: Calculate the net radiant heat loss (Q_net)

We use the Stefan-Boltzmann law to calculate the heat radiated by a body: \[ Q = \epsilon \sigma A (T_{skin}^4 - T_{room}^4) \] where \( \epsilon = 1 \) (emissivity), \( \sigma = 5.67 \times 10^{-8} \text{ W/m}^2\text{K}^4 \) (Stefan-Boltzmann constant), \( A = 2.0 \text{ m}^2 \), \( T_{skin} = 303 \text{ K} \), and \( T_{room} = 291 \text{ K} \). Plug in these values to calculate \( Q_{net} \).

Step 2: Perform calculations

Substitute the given values into the Stefan-Boltzmann law. \[ Q_{net} = 1.0 \times 5.67 \times 10^{-8} \times 2.0 \times (303^4 - 291^4) \] Calculate each term step-by-step: Firstly, find \(303^4\) and \(291^4\), then subtract the two results and complete the multiplication to find \(Q_{net}\). The calculation results in approximately 97.79 W.

Step 3: Calculate the basal metabolic rate (BMR)

Given that 80% of the energy goes into heat, we use the equation: \[ BMR = \frac{Q_{net}}{0.8} \] Substitute \(Q_{net} = 97.79 \text{ W}\) into the equation to get: \[ BMR = \frac{97.79}{0.8} \] This calculation results in a BMR of approximately 122.24 W.

Key Concepts

Heat Radiation

Heat radiation is the process through which heat energy is transferred from one object to another without the need for a physical medium. This means it can occur even in a vacuum, unlike conduction and convection which require matter to facilitate the transfer of heat energy.
In the context of the human body, heat radiation primarily involves the emission of infrared radiation. When our skin surface is warmer than the surrounding environment, our body loses heat through radiation to the cooler surroundings. This is an essential process for maintaining a stable internal temperature.
The amount of heat radiated is determined by the temperature difference between the body and its surroundings, as well as the surface area through which the heat is radiated. Greater temperature differences and larger surface areas increase the rate of heat radiation.

Stefan-Boltzmann Law

The Stefan-Boltzmann law is a fundamental principle that describes how radiant heat energy is emitted from a black body in terms of its temperature. According to this law, the total energy radiated per unit surface area of a black body is directly proportional to the fourth power of its absolute temperature.
Mathematically, it is expressed as:
\[Q = \epsilon \sigma A (T_{skin}^4 - T_{room}^4) \]where:

• \(Q\) is the net radiant heat energy emitted,
• \(\epsilon\) is the emissivity of the surface (1 for the human body),
• \(\sigma\) is the Stefan-Boltzmann constant \(5.67 \times 10^{-8} \text{ W/m}^2\text{K}^4\),
• \(A\) is the surface area through which heat is radiated,
• \(T_{skin}\) and \(T_{room}\) are the absolute temperatures in Kelvin of the skin and the room, respectively.

This equation helps us predict the amount of energy emitted by bodies of different temperatures and surface areas, making it crucial for calculating the balance between heat production and loss in humans.

Infrared Radiation

Infrared radiation is a type of electromagnetic radiation with wavelengths longer than visible light but shorter than microwaves. It is often associated with heat because many objects (including humans) emit infrared radiation when they are warm.
Infrared radiation is invisible to the human eye but can be felt as heat. This radiation is crucial for maintaining thermal equilibrium since it allows warm bodies to radiate energy into cooler environments, ensuring that excess body heat is lost when necessary.
Interestingly, most human body heat loss at room temperature occurs through infrared radiation. This is why devices like thermal cameras can detect humans in low-light conditions; they "see" the infrared radiation emitted by our bodies rather than visible light.

Metabolism

Metabolism refers to the chemical processes that occur within a living organism to maintain life. These processes include converting the food we eat into energy required for cellular functions, growth, and repair.
An essential aspect of metabolism is the production of heat. When our bodies process food and burn calories, around 80% of this energy is converted into heat, while the rest is used for activities like circulating blood and repairing tissues. This heat production is what we refer to as the basal metabolic rate (BMR), which represents the energy expended while at rest to maintain vital body functions.
Understanding metabolism is crucial for insight into how energy balance works in our body and influences body temperature regulation. A stable BMR ensures that our body's internal environment remains constant, allowing us to function efficiently even when external conditions change.