Question

If a human body having an outward surface area of \(1 \mathrm{~m}^{2}\) continued to put out \(100 \mathrm{~W}\) of metabolic power in the form of infrared radiation in the cold of space (naked; no sun), what would the equilibrium temperature be? Would this be comfortable (put in context)?

Short answer

Answer: In space, the equilibrium temperature of a human body with a metabolic power output of 100 W and a surface area of 1 m² would be around -39.60°C. This temperature would be extremely uncomfortable and potentially life-threatening without proper thermal protection.

Step-by-step solution

Define the Stefan-Boltzmann Law

Stefan-Boltzmann Law states that the power radiated by a black body is proportional to its surface area and the fourth power of its temperature. Mathematically, it is expressed as follows: \(P = σAT^4\) where \(P\) is the power radiated, \(σ\) is the Stefan-Boltzmann constant (\(5.67 × 10^{-8} \mathrm{~W~m}^{-2} \mathrm{~K}^{-4}\)), \(A\) is the surface area, and \(T\) is the temperature in Kelvin.

Plug in the given values

In our problem, we have \(P = 100 \mathrm{~W}\) and \(A = 1 \mathrm{~m}^{2}\). We need to find \(T\). Using the Stefan-Boltzmann law, we have: \(100 \mathrm{~W} = (5.67 × 10^{-8} \mathrm{~W~m}^{-2} \mathrm{~K}^{-4})(1 \mathrm{~m}^{2})T^4\)

Solve for the equilibrium temperature \(T\)

Now, we will isolate the temperature \(T\) and solve the equation: \(T^4 = \frac{100 \mathrm{~W}}{5.67 × 10^{-8} \mathrm{~W~m}^{-2} \mathrm{~K}^{-4}}\) \(T^4 = \frac{100}{5.67 × 10^{-8}}\) Solve for \(T^4\): \(T^4 = 1.763 × 10^9\) Now find \(T\): \(T = (\sqrt[4]{1.763 × 10^9}) \mathrm{K}\) \(T \approx 233.55 \mathrm{K}\) To convert this temperature to Celsius, we use the formula \(T_{C} = T - 273.15\): \(T_{C} \approx 233.55 - 273.15\) \(T_{C} \approx -39.60^{\circ}\mathrm{C}\)

Determine if the equilibrium temperature is comfortable

The equilibrium temperature of the human body in the cold of space would be around \(-39.60^{\circ}\mathrm{C}\), which is far below human body's normal temperature (around \(37^{\circ}\mathrm{C}\)). This would be extremely uncomfortable and potentially life-threatening without proper thermal protection.

Key concepts

Equilibrium Temperature

The concept of equilibrium temperature is vital in understanding how objects balance energy in various environments. When we refer to an object's equilibrium temperature, we're talking about the point at which the energy it emits is equal to the energy it absorbs. In thermal physics, this means the energy lost through radiation is identical to the energy gained, often from an external source.

In the case of the human body in space, it continues to radiate heat energy in the form of infrared radiation without gaining any heat from its surroundings. Hence, the equilibrium temperature would be the temperature at which the energy emitted as infrared radiation matches the metabolic power of the body. The problem statement calculates this to be around -39.60°C, which would be unbearable and unsafe for a human being, as our natural metabolic processes cannot compensate for such a severe loss of heat in the vacuum of space.

Black Body Radiation

Black body radiation is a cornerstone concept in thermal physics. A black body is an idealized physical entity that perfectly absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. It does not reflect or transmit any light; instead, it is a perfect emitter of radiation.

According to the Stefan-Boltzmann Law, the power or intensity of radiation emitted by a black body is directly proportional to the fourth power of the black body's temperature and can be calculated using the law's formula. Although human bodies are not perfect black bodies, we assume them to be such for simplification in many physics problems. This assumption allows us to estimate the equilibrium temperature using the Stefan-Boltzmann Law, as seen in the provided exercise.

Infrared Radiation

Infrared radiation is a type of electromagnetic radiation with wavelengths longer than those of visible light, and it is commonly associated with heat. The human body naturally emits infrared radiation as a form of thermal energy. This energy is indeed invisible to the naked eye but can be felt as heat and seen through infrared imaging.

In the context of the exercise, when a person is in the cold vacuum of space, the body loses heat only through infrared radiation, as there are no particles around to conduct or convect heat away. Understanding the role of infrared radiation in heat transfer is crucial, especially when studying the equilibrium temperature of objects in different conditions.

Thermal Physics

Thermal physics is the branch of physics that deals with the relationships and conversions between heat and other forms of energy. It encompasses the study of thermodynamics, kinetic theory, and statistical mechanics, providing a foundation for understanding how temperature affects matter and energy flow.

In our scenario, thermal physics principles guide us in determining how thermal energy emitted by a human body in space would lead to a calculated equilibrium temperature. The principles involve considering the methods of heat transfer (conduction, convection, and radiation), the role of the Stefan-Boltzmann constant, and the calculation of temperature in degrees Kelvin. Thermal physics also helps predict the consequences of such temperatures on biological organisms, emphasizing the need for protective gear in extremely cold environments like space.