A person with skin temperature of 37.00°C is in a room when the temperature of the room is (i) 30.00°C and then (ii) 15.00°C. What is the ratio (ii):(i) of heat loss from the person's body, assuming the emissivity remains constant?

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Quizplus · Accepted Answer

The heat loss from a person's body can be approximated by the Stefan-Boltzmann law, which states that the power radiated per unit area of a black body is proportional to the fourth power of the temperature of the body in Kelvin. The formula for heat loss is $P = \epsilon \sigma A (T_{body}^4 - T_{environment}^4)$, where $\epsilon$ is the emissivity, $\sigma$ is the Stefan-Boltzmann constant, $A$ is the surface area, $T_{body}$ is the temperature of the body, and $T_{environment}$ is the temperature of the environment. Both temperatures must be in Kelvin.For (i) 30.00°C = 303.15 K and (ii) 15.00°C = 288.15 K, and the body temperature is 37.00°C = 310.15 K.The ratio of heat loss (ii):(i) is then given by:$$\frac{\epsilon \sigma A (310.15^4 - 288.15^4)}{\epsilon \sigma A (310.15^4 - 303.15^4)}$$Since $\epsilon$, $\sigma$, and $A$ are constants and do not change between the two scenarios, they can be canceled out, leaving:$$\frac{310.15^4 - 288.15^4}{310.15^4 - 303.15^4}$$Calculating the values gives a ratio of approximately 2.92, which corresponds to choice B.

A Person with Skin Temperature of 37