Ideal Gases

Step 1: Stefan-Boltzmann law for blackbody radiation.
The energy density of radiation emitted by a blackbody is proportional to the fourth power of its temperature, where is the Stefan-Boltzmann constant and is the temperature. For a material with emissivity , the energy density is given by Step 2: Relation between temperatures.
For the material at temperature , the energy density becomes We are given that this value is , so Substitute into the equation: Simplify and solve for : Final Answer: The emissivity is .

The mean free path is inversely proportional to the number of particles and the volume. The mean free path for the initial state is , and after the change, it becomes . Thus, we have: After halving the number of particles, Using the given condition that , we get: Final Answer: The ratio is .