A sphere of radius 0.500 m, temperature 25.7°C, and emissivity 0.915 is isolated in an environment of temperature 77.0°C.
At what rate does the sphere emit thermal radiation?
At what rate does the sphere absorb thermal radiation?
What is the sphere's net rate of energy exchange?
To answer these questions, we can use the Stefan-Boltzmann law, which relates the rate of thermal radiation emission or absorption to the temperature, area, and emissivity of an object. The formula for the rate of thermal radiation emission or absorption is:
Rate = emissivity * σ * A * ΔT^4
Where:
- Rate is the rate of emission or absorption of thermal radiation in watts (W)
- emissivity is the emissivity of the object (a dimensionless value between 0 and 1)
- σ (sigma) is the Stefan-Boltzmann constant (approximately 5.67 × 10^-8 W/(m^2·K^4))
- A is the surface area of the object in square meters (m^2)
- ΔT is the temperature difference between the object and its surroundings in Kelvin (K)
Let's calculate the answers to each question step by step:
1. Rate of thermal radiation emission:
Given:
- Radius of the sphere = 0.500 m
- Temperature of the sphere = 25.7 °C = 25.7 + 273.15 K (convert to Kelvin)
- Emissivity of the sphere = 0.915
To find the surface area (A) of the sphere, we can use the formula: A = 4πr^2, where r is the radius.
A = 4π(0.500 m)^2
Now, we can calculate the rate of thermal radiation emission using the formula mentioned above.
Rate of thermal radiation emission = emissivity * σ * A * ΔT^4
2. Rate of thermal radiation absorption:
To find the rate at which the sphere absorbs thermal radiation, we assume it absorbs all the radiation incident on it. Therefore, the rate of absorption is the same as the rate of emission.
Rate of thermal radiation absorption = Rate of thermal radiation emission
3. Net rate of energy exchange:
The net rate of energy exchange is simply the difference between the rate of thermal radiation emission and the rate of thermal radiation absorption.
Net rate of energy exchange = Rate of thermal radiation emission - Rate of thermal radiation absorption
Now, you have all the information needed to calculate each value.