Discussion
Two different liquids of having same mass at temperatures \(T_{1}\) and \(T_{2}\) and of specific heat capacities, S and 1.2 S, respectively, are mixed in an insulated system. If \(T_{1} \gt T_{2}\) , then the temperature of the mixture is
Answer: A
The temperature of the mixture is \(\dfrac{5T_{1} + 6T_{2}}{11}\).
In an insulated system the heat lost by the hotter liquid equals the heat gained by the cooler one. With equal masses \(m\) and final temperature \(T\):
\[ \begin{aligned} mS(T_{1} - T) &= m(1.2S)(T - T_{2}) \\ T_{1} + 1.2T_{2} &= 2.2T \\ T &= \frac{5T_{1} + 6T_{2}}{11} \end{aligned} \]
The liquid with the larger specific heat (the cooler one) has more weight in the average, so \(T_{2}\) gets coefficient 6. \(\frac{6T_{1} + 5T_{2}}{11}\) swaps the weights, and \(\frac{T_{1} + T_{2}}{2}\) would be right only for equal specific heats.
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