A cylindrical mug of height 10 cm and inner radius 4.0 cm is filled with hot coffee…
PHYSICS · Multi-part · CBSE · Physics · Thermal Properties of Matter · Thermal Properties of Matter
Problem
A cylindrical mug of height 10 cm and inner radius 4.0 cm is filled with hot coffee at 95 °C. The mug is made of a ceramic whose thermal conductivity is low enough that the exterior surface remains nearly uniform at 35 °C. The mug is initially at rest on a table in a laboratory where the air temperature is 22 °C.
The experiment proceeds in two stages:
Stage I: The mug is left undisturbed in the laboratory for 10 min.
Stage II: The same mug, refilled with coffee at 95 °C, is placed in a vacuum chamber that is evacuated until the residual gas pressure is essentially zero; the chamber walls are maintained at 22 °C.
Neglect energy loss through the mug’s bottom surface in both stages.
Respond to the prompts below regarding how energy leaves the coffee in each stage.
(a) For Stage I, list all significant pathways by which energy is transferred from the coffee to its surroundings. Rank these pathways from greatest to least expected rate of energy transfer and justify your ranking qualitatively.
(b) For Stage II, identify which, if any, of the pathways from part (a) become negligible. Explain why each pathway is or is not affected by evacuation of the chamber.
(c) Derive a symbolic expression for the net rate at which energy leaves the coffee in Stage II as a function of the surface temperature Tc of the coffee, clearly indicating the physical constants and geometric parameters on which your expression depends.
(d) Using your expression from part (c), discuss how the time required for the coffee to cool from 95 °C to 60 °C in Stage II would compare qualitatively with the cooling time over the same temperature range in Stage I. Provide a clear chain of reasoning that references the dominant energy-transfer mechanisms in each stage.