Final answer:
By what factor does the entropy change when the temperature doubles from 200 K to 400 K? The entropy of the conduction electrons in a metal changes by a factor of 2, meaning it doubles, because entropy is directly proportional to temperature for a fixed number of particles.
Step-by-step explanation:
The question asks by what factor does the entropy of the conduction electrons in a metal change when the temperature is changed from 200 K to 400 K. To answer this question, we can use the concept of specific heat and the relationship between entropy and temperature. For electrons in a metal, the entropy S can be approximated as being proportional to the temperature T. This relationship can be expressed as S ∝ T for a fixed number of particles in the metal.
Since the entropy is directly proportional to the temperature, if we double the temperature from 200 K to 400 K, the entropy also doubles. Therefore, by what factor does the entropy change? It changes by a factor of 2, meaning the entropy at 400 K is twice the entropy at 200 K.
This example is analogous to the calculations of entropy change due to heat transfer in the provided reference problems, where the change in entropy is based on the amount of heat transferred and the temperatures of the objects involved. Specific heat is the amount of heat per unit mass required to raise the temperature of a substance by one degree Celsius. For substances with higher specific heats, less change in entropy would be expected for the same amount of heat transfer compared to substances with lower specific heats.
When considering a real process, such as heat conduction through a metal rod, there are additional complexities like the rate of heat transfer and temperature gradients. However, the core principle remains that the entropy change is related to temperature changes according to the specific heat capacity of the material involved.