I'd love thorough solutions to these problems, but simply providing the specific formulas that I should solve each part of the problem with would be really good too. :-)
Thanks!
Human energetics
You are sitting down to study in the library. Let’s find out what limits the time you are able to spend studying.
a) Calculate how much power you are radiating, in watts. While your skin temperature is about 33 degrees Celsius, the outside of your clothing is at about 28 degrees Celsius. In the absence of perspiration, black body radiation at this temperature is your main radiation mechanism. You can take your surface area to be 2 m^2.
b) What is the wavelength of the peak of your spectrum? Comment why we cannot see each other in a dark room.
c) Your body is constantly emitting thermal radiation and is also absorbing radiation from the surroundings. The library is unusually chilly today, and the ambient temperature is 15 degrees Celsius. How much power are you absorbing from the room, in watts? You can use the black body formula at the temperature of the room.
d) What’s your net energy loss rate, in watts? Compare this to a typical 60W light bulb. How much energy will you lose per day, in mega-Joules?
e) On a normal dorm diet, your daily energy intake from food should be close to 2000 kilo-calories (kcal), where one calorie is defined as 4.2 Joules. How long will you be able to study in this library before your daily energy intake is exhausted?
a) You radiate away heat at your T = 301 K clothing temperature. The blaclbody emission rate is
(sigma)*T^4*area
sigma is the Stefan-Boltzmann contstant. Its value is
0.567*10^-7 W/m^2*K^4
(b) Learn about and apply the Wien displacement law.
(peak wavelength)*(Temperature)= 0.290 cm K
(c) Use the blackbody law again, but this time with T = 288K, and the same body aarea
(d) Subtract answer (c) from answer (a)
(e) Energy loss rate * Time = 2*10^6 cal*4.18 J/cal.
Solve for Time
a) To calculate the power you are radiating, we can use the Stefan-Boltzmann Law, which states that power radiated by a black body is proportional to its surface area and the fourth power of its temperature. The formula is given as:
P = εσAT⁴
Where:
P = power radiated (in watts)
ε = emissivity of the object (assumed to be 1 for a black body)
σ = Stefan-Boltzmann constant (5.67 x 10^-8 W/m²K⁴)
A = surface area (in square meters)
T = temperature (in Kelvin)
First, we need to convert the temperatures to Kelvin:
Temperature in Kelvin = Temperature in Celsius + 273.15
Substituting the values into the formula, we have:
P = (1)(5.67 x 10^-8)(2)(33+273.15)⁴
Simplifying the expression, we can find the power radiated.
b) To calculate the wavelength of the peak of your spectrum, we can use Wien's Law, which states that the peak wavelength of black body radiation is inversely proportional to the temperature. The formula is given as:
λ_max = b / T
Where:
λ_max = peak wavelength (in meters)
b = Wien's constant (2.9 x 10^-3 m·K)
T = temperature (in Kelvin)
Using the temperature from part a), substitute the values into the formula to find the peak wavelength.
c) To calculate the power absorbed from the room, we can use the Stefan-Boltzmann Law again. The formula is the same as in part a), but we will use the temperature of the room and the surface area.
P = (1)(5.67 x 10^-8)(2)(15+273.15)⁴
Simplifying the expression will give us the power absorbed from the room.
d) The net energy loss rate is the difference between the power radiated and the power absorbed. Calculate the difference and convert it to watts.
To compare it to a typical 60W light bulb, calculate the difference and determine if the net energy loss rate is greater or smaller.
To find the energy loss per day, multiply the net energy loss rate by the number of seconds in a day and convert it to megajoules.
e) To determine how long you can study before your daily energy intake is exhausted, divide your daily energy intake by the net energy loss rate calculated in part d). This will give you the time in seconds. You can convert it to hours or minutes if desired.
Remember to convert kilocalories to joules before performing the calculation