# Calorimetry

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# The specific heat of a metal at low temperatures varies according to $\mathrm{S}=\left(\frac{4}{5}\right){\mathrm{T}}^{3}$ where T is the absolute temperature. Find the heat energy (in SI unit) needed to raise unit mass of the metal from T = 1 K to T = 2 K.

Moderate
Solution

## $\begin{array}{c}\mathrm{Q}=\int \mathrm{mSdT}={\left[\frac{{\mathrm{mT}}^{4}}{5}\right]}_{1}^{2}\\ ⇒\frac{\mathrm{Q}}{\mathrm{m}}=\frac{15}{5}=3\end{array}$

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A water cooler of storage capacity 120 liters can cool water at a constant rate of P watts. In a closed circulation system (as shown schematically in the figure), the water from the cooler is used to cool an external device that generates constantly 3 kW of heat (thermal load). The temperature of water fed into the device cannot exceed  $30°C$ and the entire stored 120 liters of water is initially cooled to $10°C$ . The entire system is thermally insulated. The minimum value of P (in watts) for which the device can be operated for 3 hours is
(Specific heat of water is 4.2 kJ $k{g}^{-1}{K}^{-1}$  and the density of water is $1000\text{\hspace{0.17em}}kg\text{\hspace{0.17em}}{m}^{-3}$ )