Question

Your skin temperature is about \(308 \mathrm{~K}\), and the walls in a typical room are about \(295 \mathrm{~K}\). If you have about \(1 \mathrm{~m}^{2}\) of outward-facing surface area, how much power do you radiate as infrared radiation, in Watts? Compare this to the typical metabolic rate of \(100 \mathrm{~W}\).

Short answer

How does this compare to a typical metabolic rate of 100 W? Answer: The power emitted as infrared radiation by the person is 526.11 W, which is more than five times the typical metabolic rate of 100 W. This demonstrates the importance of regulating body temperature, as a significant portion of energy is emitted as infrared radiation.

Step-by-step solution

Understand the Stefan-Boltzmann Law

The Stefan-Boltzmann Law states that the power emitted per unit area of an object due to its temperature is given by the equation: P/A = σ * T^4 where, P is the power emitted, A is the surface area, T is the temperature in Kelvin, σ is the constant called Stefan-Boltzmann constant (\(5.67 \times 10^{-8}\, W/m^2\mathrm{K}^4\)).

Calculate the emitted power

We are given the person's surface area, A = \(1\, m^2\) and temperature, T = \(308\, K\). Using this information, we first compute the power per surface area (\(P/A\)): P/A = σ * T^4 \(P/A = 5.67 \times 10^{-8} * (308)^4\) Now, we compute the total emitted power (P) by multiplying by the surface area (A): P = A * (P/A)

Compare the emitted power with the metabolic rate

We now compare the calculated emitted power (P) with the given typical metabolic rate (\(100\, W\)) to analyze the difference.

Calculate the power emitted by the person in Watts

Calculating the power emitted, P: \(P = 1 * 5.67 \times 10^{-8} * (308)^4 = 526.11\, W\)

Compare the emitted power with the metabolic rate

The emitted power as infrared radiation is \(526.11\, W\), and the typical metabolic rate is \(100\, W\). The radiated power is more than five times the metabolic rate. This means that a significant portion of energy is emitted as infrared radiation, and it shows the importance of regulating body temperature.

Key concepts

Infrared Radiation

Infrared radiation is a type of electromagnetic radiation, which lies beyond the visible spectrum at its red end. Objects emit this form of energy naturally as heat. When we think about a person emitting infrared radiation, what we're really talking about is the body's heat being radiated away as invisible light.

The skin temperature, which in the textbook exercise is about 308 K (Kelvin), directly affects the amount of infrared radiation released. Since people are warmer than their surroundings, they constantly emit infrared radiation. This heat transfer can be explained through the Stefan-Boltzmann Law, which relates the power radiated by an object to its temperature.

The law suggests that as the object's temperature increases, the amount of infrared radiation emitted will rise to the fourth power of temperature in Kelvin. This relationship is key to understanding how we can measure the power output of a human body in terms of infrared radiation.

Human Metabolic Rate

The human metabolic rate is the rate at which the body expends energy or burns calories. It is a crucial factor for maintaining life, as it includes the energy used for all basic bodily functions while at rest, known as the basal metabolic rate (BMR), as well as any additional energy expended during physical activities.

In the exercise, a typical metabolic rate of 100 W (watts) represents the energy conversion within the body under resting conditions. It is fascinating to see how this rate compares with the power we radiate as infrared radiation. The comparison shows that our bodies emit more energy as heat than what is used for basal metabolic functions. This excess heat needs to be dissipated effectively through radiation and other means to prevent overheating and maintain our core temperature within a safe range.

Temperature in Kelvin

Temperature in Kelvin is a fundamental unit of measurement in the physical sciences. It is the base unit of temperature in the International System of Units (SI) and is used widely to express both scientific and everyday temperatures.

The Kelvin scale is an absolute temperature scale starting at absolute zero, the point where all molecular motion ceases. In the context of the exercise, human skin temperature is represented as 308 K, which is about 35°C (Celsius) or 95°F (Fahrenheit). The conversion between Kelvin and Celsius is straightforward: 0 K is -273.15°C.

The importance of using Kelvin in calculations like the Stefan-Boltzmann equation cannot be overstated because the law requires temperature in an absolute scale for accurate computations. Incorrect use of temperature scale can result in erroneous outcomes, thus the emphasis on temperature in Kelvin in scientific calculations involving heat and thermodynamics.