In an old house, the heating system uses radiators, which are hollow metal devices through which hot water or steam circulates. In one room the radiator has a dark color (emissitivity = 0.810). It has a temperature of 65.0 oC. The new owner of the house paints the radiator a lighter color (emissitivity = 0.414). Assuming that it emits the same radiant power as it did before being painted, what is the temperature (in degrees Celsius) of the newly painted radiator?
To solve this problem, we can use the Stefan-Boltzmann law, which states that the radiant power emitted by an object is proportional to its emissivity (e), the Stefan-Boltzmann constant (σ), and the fourth power of its temperature (T).
The formula for the radiant power is:
P = e * σ * T^4
Given that the initial radiator has a temperature of 65.0°C, an emissivity of 0.810, and its radiant power remains the same after being painted, we can set up the equation as follows:
P_initial = e_initial * σ * T_initial^4
For the newly painted radiator, we need to find its temperature (T_new). The emissivity of the new radiator is 0.414. The equation for the new radiator becomes:
P_new = e_new * σ * T_new^4
Since the radiant power remains the same, P_initial = P_new. Therefore, we have:
e_initial * σ * T_initial^4 = e_new * σ * T_new^4
We can simplify by canceling out the σ term and isolating T_new:
(e_initial * T_initial^4) / e_new = T_new^4
Taking the fourth root of both sides, we get:
T_new = (e_initial * T_initial^4 / e_new)^(1/4)
Plugging in the given values:
T_new = (0.810 * 65.0^4 / 0.414)^(1/4)
Calculating this expression gives us:
T_new ≈ (336730.92)^(1/4) ≈ 66.6°C
Therefore, the temperature of the newly painted radiator is approximately 66.6°C.