Problem 60 - College Physics

Question

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 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) 147.64 W; (b) 184.55 W.

Step 1: Calculate the Skin and Room Temperature in Kelvin

To use the Stefan-Boltzmann law for calculating the radiated heat, both temperatures need to be converted to Kelvin. Use the conversion formula: \[ T(K) = T(C) + 273.15 \] Thus, for the skin temperature:\[ T_\text{skin} = 30 + 273.15 = 303.15 \, K \]and for the room temperature:\[ T_\text{room} = 18 + 273.15 = 291.15 \, K \]

Step 2: Apply the Stefan-Boltzmann Law

The Stefan-Boltzmann law states that the power radiated by a black body per unit area is given by:\[ P = \varepsilon \sigma A (T_\text{skin}^4 - T_\text{room}^4) \] where \( \varepsilon = 1.0 \) (emissivity), \( \sigma = 5.67 \times 10^{-8} \, \mathrm{W/m^2K^4} \) (Stefan-Boltzmann constant), and \( A = 2.0 \, \mathrm{m^2} \) (surface area). Plugging in the values, we find:\[ P = 1.0 \times 5.67 \times 10^{-8} \times 2.0 \times ((303.15)^4 - (291.15)^4) \, \mathrm{W} \]

Step 3: Calculate the Radiated Power

Calculate the net heat power radiated using the Stefan-Boltzmann Law from Step 2. First, compute \[ (303.15)^4 = 847,792,924.51 \] and \[ (291.15)^4 = 717,831,744.10 \].Then the power becomes:\[ P = 1.0 \times 5.67 \times 10^{-8} \times 2.0 \times (847,792,924.51 - 717,831,744.10) \].Simplifying:\[ P = 1.0 \times 5.67 \times 10^{-8} \times 2.0 \times 129,961,180.41 \].Thus, \[ P \approx 147.64 \, \mathrm{W} \].

Step 4: Determine the Basal Metabolic Rate (BMR)

Given that 80% of the metabolically produced energy dissipation is in the form of heat, the BMR can be found knowing that this is equal to the radiation found in Step 3:Let \( BMR \approx 1.25 \times P \). Since\[ P = 147.64 \, \mathrm{W} \],the BMR satisfies \[ 0.8 \times BMR = 147.64 \, \mathrm{W} \]. Solving for BMR gives:\[ BMR = \frac{147.64}{0.8} \approx 184.55 \, \mathrm{W} \].

Key Concepts

Basal Metabolic Rate

The Basal Metabolic Rate, or BMR, is a measure of the energy expenditure of a person at complete rest. It's the amount of calories required to keep your body functioning while not doing any physical activities. This includes fundamental processes like breathing, circulating blood, and maintaining body temperature. For instance, when factoring in the Stefan-Boltzmann law, BMR can be calculated based on the body's ability to radiate heat at rest.

• The calculation considers the net heat radiation into the surrounding environment; thus, it relies on both body and room temperature comparisons.
• When a person is at rest, roughly 80% of their energy is released as heat, which is crucial in evaluating BMR.

BMR is a vital concept in thermodynamics, offering insights into how the body balances energy input and output in metabolic processes.

Stefan-Boltzmann Law

The Stefan-Boltzmann Law is fundamental in thermodynamics. It describes how the radiation power of a black body is related to its temperature. This principle is key when calculating the amount of heat energy a body, like the human body, radiates into its surroundings. The law is expressed mathematically as:\[ P = \varepsilon \sigma A (T_\text{body}^4 - T_\text{ambient}^4) \]Where:

• \( \varepsilon \) is the emissivity, which is 1 for a perfect black body.
• \( \sigma \) is the Stefan-Boltzmann constant, \( 5.67 \times 10^{-8} \mathrm{W/m^2K^4} \).
• \( A \) is the surface area of the body.

The Stefan-Boltzmann Law allows us to predict how much energy is emitted by a body, based on its temperature difference with the environment.

Black Body Radiation

Black body radiation refers to the theoretical concept of an idealized body that absorbs all incoming radiation and emits thermal radiation based on its temperature, without any reflection, which makes it an ideal emitter. In thermodynamics, this concept helps us understand how objects radiate energy.

• Any real object can approximate a black body when its emissivity is close to 1, meaning it behaves similarly to this ideal model in terms of heat emission.
• Understanding black body radiation allows us to calculate temperature differences between a body and its surroundings, which is essential in thermodynamic calculations, such as determining body heat loss.

Applying this concept to real bodies, like human skin, aids in understanding how heat is managed by our metabolic processes.

Emissivity in Physics

Emissivity is a measure of an object's ability to emit energy as thermal radiation. It's a dimensionless value ranging from 0 to 1, where 1 indicates a perfect black body, which radiates energy most efficiently. In physics, understanding emissivity helps us determine how much heat an object radiates compared to this perfect black body.

• Human skin has an emissivity close to 1, which is why it can efficiently radiate the heat generated by metabolic processes.
• Emissivity impacts how we calculate the radiated power from real objects, informing us about their energy conservation and transfer dynamics.

In practical applications, knowing an object's emissivity allows for more accurate calculations in energy budgets and thermodynamic predictions, such as determining how well the body maintains homeostasis.