At high noon, the Sun delivers 1 000 W to each square meter of a blacktop road. If the hot
asphalt transfers energy only by radiation, what is its steady-state temperature?
Stefan’s Law: P = σAeT4
σ = 5.669 x 10-8 W/m2K4
(e = 1 for blacktop)
Pin = Pout
σAineT4 = σ Aout e T4
(1000) = 5.669 x 10-8 1*1* T4
T = 364 K
To determine the steady-state temperature of the hot asphalt, we need to understand how the energy delivered by the Sun is balanced by the energy radiated by the asphalt.
First, we need to recognize that the energy received from the Sun is equal to the energy radiated by the asphalt. This is known as the Stefan-Boltzmann Law, which states that the power radiated by an object is proportional to the fourth power of its temperature.
So, the power radiated by the asphalt can be given by the equation:
Power radiated = σ * A * T^4
Where:
- Power radiated is the energy radiated by the asphalt (in watts)
- σ is the Stefan-Boltzmann constant (approximately 5.67 x 10^-8 W / (m^2 K^4))
- A is the surface area of the asphalt (in square meters)
- T is the temperature of the asphalt (in Kelvin)
In this case, the power delivered by the Sun is 1000 W/m^2. So, we can equate the power delivered by the Sun to the power radiated by the asphalt:
Power delivered by the Sun = Power radiated by the asphalt
1000 W/m^2 = σ * A * T^4
Now, we solve for the temperature T:
T^4 = (1000 W/m^2) / (σ * A)
T^4 = 1000 W/m^2 / (5.67 x 10^-8 W / (m^2 K^4) * A)
T^4 = (1000 W/m^2) / (5.67 x 10^-8 W/(m^2 K^4) * A)
Finally, we take the fourth root of both sides to find the temperature T:
T = (1000 W/m^2) / (5.67 x 10^-8 W / (m^2 K^4) * A)^(1/4)
By plugging in the value of A (the surface area of the asphalt road), you can calculate the steady-state temperature of the asphalt. Keep in mind that A will depend on the dimensions of the road, so you will need more information to determine an exact value.
To determine the steady-state temperature of the hot asphalt, we can use the Stefan-Boltzmann law and the fact that energy received by the asphalt equals energy radiated by the asphalt.
The Stefan-Boltzmann law states that the power radiated by an object is proportional to the fourth power of its temperature, expressed as:
P = σ * A * T^4
Where:
P is the power radiated
σ is the Stefan-Boltzmann constant (approximately 5.67 x 10^-8 W/m^2K^4)
A is the area of the asphalt (m^2)
T is the temperature of the asphalt in Kelvin
Given that the Sun delivers 1,000 W to each square meter of the blacktop road, the power received by the asphalt is 1,000 W/m^2.
Therefore, we can equate the power received to the power radiated:
1,000 W/m^2 = σ * A * T^4
We need to solve for T, so we rearrange the equation:
T^4 = (1,000 W/m^2) / (σ * A)
T^4 = (1,000 W/m^2) / (5.67 x 10^-8 W/m^2K^4 * A)
Now, let's assume that the area of the asphalt is 1 m^2.
T^4 = (1,000 W/m^2) / (5.67 x 10^-8 W/m^2K^4 * 1 m^2)
T^4 = 1.76 x 10^16 K^4
Taking the fourth root of both sides:
T = (1.76 x 10^16 K^4)^(1/4)
T ≈ 876.26 K
Therefore, the steady-state temperature of the hot asphalt is approximately 876.26 Kelvin.