Physics

Physics

Problem

A 0.40 kg0.40\,\text{kg} block of brass at 180∘C180^{\circ}\text{C} is quickly transferred into an insulated cup that contains 0.20 kg0.20\,\text{kg} of water at 20∘C20^{\circ}\text{C}. The specific heat of brass is 380 J kg−1 K−1380\,\text{J}\,\text{kg}^{-1}\,\text{K}^{-1} and the specific heat of water is 4.18×103 J kg−1 K−14.18\times10^{3}\,\text{J}\,\text{kg}^{-1}\,\text{K}^{-1}. After sufficient time, the system reaches a uniform final temperature. Assume the cup and its lid are perfect thermal insulators and have negligible heat capacity. a. Qualitatively describe the direction of energy transfer between the brass block and the water until thermal equilibrium is achieved. In your explanation, include the sign of the temperature changes for both substances. b. Without performing a numerical calculation, explain whether the final equilibrium temperature will be closer to 20∘C20^{\circ}\text{C} or 180∘C180^{\circ}\text{C}. Support your answer by referring to the relative masses and specific heats of the two substances.
Intuitive explanation

This is a thermal equilibrium problem where two objects at different temperatures exchange heat until they reach the same final temperature. The key insight is that in an insulated system, the total heat lost by the hot object equals the total heat gained by the cold object—so you can set up an energy balance equation: mbrasscbrass(Tf−Tbrass,initial)+mwatercwater(Tf−Twater,initial)=0m_{\text{brass}}c_{\text{brass}}(T_f - T_{\text{brass,initial}}) + m_{\text{water}}c_{\text{water}}(T_f - T_{\text{water,initial}}) = 0. Solve for TfT_f algebraically.

Intuitive explanation

This is a thermal equilibrium problem where two objects at different temperatures exchange heat until they reach the same final temperature. The key insight is that in an insulated system, the total heat lost by the hot object equals the total heat gained by the cold object—so you can set up an energy balance equation: mbrasscbrass(Tf−Tbrass,initial)+mwatercwater(Tf−Twater,initial)=0m_{\text{brass}}c_{\text{brass}}(T_f - T_{\text{brass,initial}}) + m_{\text{water}}c_{\text{water}}(T_f - T_{\text{water,initial}}) = 0. Solve for TfT_f algebraically.

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