A person's body is producing energy internally due to metabolic processes. If the body loses more energy than metabolic processes are generating, its temperature will drop. If the drop is severe, it can be life-threatening. Suppose a person is unclothed and energy is being lost via radiation from a body surface area of 1.51 m2, which has a temperature of 35.6 °C and an emissivity of 0.720. Suppose that metabolic processes are producing energy at a rate of 130 J/s. What is the temperature (in °C) of the coldest room in which this person could stand and not experience a drop in body temperature? Do not enter unit.
I know, I know. I am posting another question. we have hw on the stuff before we learn it so it's confusing.
To answer this question, we need to consider the energy balance between the energy being generated by the body's metabolic processes and the energy being lost via radiation. The body will not experience a drop in temperature if the rate of energy loss via radiation is equal to or less than the rate of energy generation by metabolic processes.
First, we need to calculate the rate of energy loss via radiation using the Stefan-Boltzmann Law:
E = σ * A * ε * T^4
where E is the rate of energy loss via radiation, σ is the Stefan-Boltzmann constant (5.67 x 10^-8 W/m^2K^4), A is the body surface area (1.51 m^2), ε is the emissivity (0.720), and T is the body temperature in Kelvin.
Converting the body temperature from Celsius to Kelvin:
T = 35.6 + 273.15 = 308.75 K
Calculating the rate of energy loss via radiation:
E = (5.67 x 10^-8) * (1.51) * (0.720) * (308.75^4)
Next, we compare the rate of energy loss via radiation to the rate of energy generation by metabolic processes (130 J/s). The body will not experience a drop in temperature if the rate of energy loss is less than or equal to the rate of energy generation:
E ≤ 130 J/s
We can rearrange the equation to solve for the minimum temperature T that the room needs to be in order to satisfy this condition:
(5.67 x 10^-8) * (1.51) * (0.720) * (T^4) ≤ 130
Now we can solve for T by dividing both sides of the inequality and taking the fourth root:
T ≤ (130 / [(5.67 x 10^-8) * (1.51) * (0.720)])^(1/4)
Plug in the values and calculate:
T ≤ (130 / (6.82506984 x 10^-8))^(1/4)
T ≤ (1.90400938 x 10^9)^(1/4)
T ≤ 660.14
Finally, subtract 273.15 to convert from Kelvin to Celsius:
T - 273.15 = 660.14 - 273.15
T - 273.15 = 386.99
So, the temperature of the coldest room in which this person could stand and not experience a drop in body temperature is approximately 387 °C.