Two identical metal spheres, Sphere X and Sphere Y, are placed in thermal contact in an insulated chamber. Initially Sphere X is at 120∘C and Sphere Y is at 30∘C. A temperature–versus–time graph is recorded for each sphere. The graph (described verbally below) shows two smooth curves that approach one another and eventually merge.
Verbal description of the graph:
• The horizontal axis is time in minutes (0–15 min). The vertical axis is temperature in ∘C (0–120).
• At t=0 the curve for Sphere X starts at 120∘C and decreases exponentially, while the curve for Sphere Y starts at 30∘C and increases exponentially.
• The two curves meet and become indistinguishable after approximately 6 minutes, remaining flat and coincident thereafter.
(1) Estimate the time at which the two spheres reach thermal equilibrium, as indicated by the graph. Explain your reasoning.
(2) Compare the initial slopes (rate of temperature change) of the two curves. What does the relative magnitude of these slopes reveal about the equality or inequality of the spheres’ masses and heat capacities? Provide a detailed explanation.
(3) Suppose the chamber were not perfectly insulated and allowed slow energy exchange with the surroundings, which remain at 25∘C. Describe qualitatively how the long-term behavior of the temperature curves would differ from that shown in the graph, and explain why.