A solid body at initial temperature T0 is immersed in a bath of water at initial temperature Tw0. Heat is transferred from the solid to the water at a rate 
, where K is a constant and Tw and T are instantaneous values of the temperatures of the water and solid. Develop an expression for T as a function of time ?. Check your result for the limiting cases, ? = 0 and ? = ?. Ignore effects of expansion or contraction, and assume constant specific heats for both water and solid.
What will be an ideal response?
Let symbols without subscripts refer to the solid and symbols with subscript w refer to the water. Heat transfer from the solid to the water results in changes in internal energy of both. Because energy is conserved and there are no energy flows other than the transfer of heat from the solid to the water, ?Ut = -?Utw. If total heat capacity of the solid is Ct (= mC) and total heat capacity of the water is
Ctw (=mwCw), then:
or
This equation relates instantaneous values of Tw and T.
The heat-transfer rate is given as Q = K(Tw ? T ). Thus, writing an energy balance (the first law) with the solid as the system and W = 0, yields
This change in internal energy is related to the change in temperature by the heat capacity:
Combining these equations, we have
Substituting into this our expression for Tw in terms of T gives a single equation for T that is separable:
Integrating from t = 0 (where T = T0) to t gives:
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