Scaling temperature up by a factor of 4 scales up all the velocities by a factor of 2 [...] slowing down the playback of a video has the effect of increasing the time between collisions [....]
Oh, good point! But hm, scaling up temperature by 4x should increase velocities by 2x and energy transfer per collision by 4x. And it should increase the rate of collisions per time by 2x. So the rate of energy transfer per time should increase 8x. But that violates Newton’s law as well. What am I missing here?
Material properties such as thermal conductivity can depend on temperature. The actual calculation of thermal conductivity of various materials is very much outside of my area, but Schroeder’s “An Introduction to Thermal Physics” has a somewhat similar derivation showing the thermal conductivity of an ideal gas being proportional to √T based off the rms velocity and mean free path (which can be related to average time between collisions).
Oh, good point! But hm, scaling up temperature by 4x should increase velocities by 2x and energy transfer per collision by 4x. And it should increase the rate of collisions per time by 2x. So the rate of energy transfer per time should increase 8x. But that violates Newton’s law as well. What am I missing here?
Material properties such as thermal conductivity can depend on temperature. The actual calculation of thermal conductivity of various materials is very much outside of my area, but Schroeder’s “An Introduction to Thermal Physics” has a somewhat similar derivation showing the thermal conductivity of an ideal gas being proportional to √T based off the rms velocity and mean free path (which can be related to average time between collisions).